Wick and Anti-Wick Characterizations of Linear Operators on Spaces of Power Series Expansions

Abstract

We study links between Wick, anti-Wick and analytic kernel operators on the Bargmann transform side. We consider classes of kernels, whose corresponding operators agree with the sets of linear and continuous operators on spaces of power series expansions, which are Bargmann images of Pilipović spaces. We show that in several situations, the sets of Wick and kernel operators with symbols and kernels in such classes agree. We also find suitable subclasses to these kernel classes, whose corresponding sets of Wick and anti-Wick operators agree. We also show ring, module and composition properties for such classes.

1 Introduction

A convenient way to represent linear operators acting on functions defined on the configuration space ${\mathbf {R}}^{d}$, is to apply the Bargmann transform. In this process, these operators are transformed into (analytic) integral operators $T_K$ with kernels K, or as analytic pseudo-differential operators ${\text {Op}}_{{\mathfrak {V}}}(a)$ with symbols a, given by

$$\begin{aligned} T_KF(z)&= \pi ^{-d} \int _{{\mathbf {C}}^{d}} K(z,w)F(w)e^{-|w|^2}\, d\lambda (w) \end{aligned}$$

(1.1)

and

$$\begin{aligned} {\text {Op}}_{{\mathfrak {V}}}(a)F (z)&= \pi ^{-d} \int _{{\mathbf {C}}^{d}} a(z,w)F(w)e^{(z-w,w)}\, d\lambda (w), \end{aligned}$$

(1.2)

respectively, when F is a suitable analytic function on ${\mathbf {C}}^{d}$, where $d\lambda$ is the Lebesgue measure and $(\, \cdot \, ,\, \cdot \, )$ is the scalar product on ${\mathbf {C}}^{d}$. Here K(z, w) and a(z, w) are analytic functions or, more generally, power series expansions with respect to $(z,{\overline{w}})\in {\mathbf {C}}^{d}\times {\mathbf {C}}^{d}\simeq {\mathbf {C}}^{2d}$. (See [9] or Sect. 2 for notations.) The operator ${\text {Op}}_{{\mathfrak {V}}}(a)$ is often called the Wick or Berezin operator with (Wick) symbol a.

An important subclass of Wick operators are the anti-Wick operators, ${\text {Op}}_{{\mathfrak {V}}}^{{\text {aw}}}(a)$ with suitable symbols a(z, w) being analytic with respect to $(z,{\overline{w}})\in {\mathbf {C}}^{2d}$, and given by

$$\begin{aligned} {\text {Op}}_{{\mathfrak {V}}}^{{\text {aw}}}(a)F (z)&= \pi ^{-d} \int _{{\mathbf {C}}^{d}} a(w,w)F(w)e^{(z-w,w)}\, d\lambda (w), \end{aligned}$$

(1.3)

where F is a suitable analytic function on ${\mathbf {C}}^{d}$. An important feature for anti-Wick operators is that ${\text {Op}}_{\mathfrak V}^{{\text {aw}}}(a)$ is positive semi-definite when $a(z,z)\ge 0$ for every $z\in {\mathbf {C}}^{d}$, which is essential when performing certain types of energy estimates in quantum physics. Any similar transitions on positivity from symbols to Wick operators are in general not true. On the other hand for suitable a we have the expansion formula

$$\begin{aligned} {\text {Op}}_{{\mathfrak {V}}}(a) = \sum _{|\alpha |<N} \frac{(-1)^{|\alpha |}}{\alpha !} {\text {Op}}_{{\mathfrak {V}}}^{{\text {aw}}}(b_\alpha ) + {\text {Op}}_{\mathfrak V}(c_N) \quad \text {where} \quad b_\alpha = \partial _z^\alpha {\overline{\partial }} _{w}^\alpha a, \end{aligned}$$

(1.4)

of Wick operators into a superposition of anti-Wick operators with error term ${\text {Op}}_{{\mathfrak {V}}}(c_N)$, deduced in [18]. Here the first term ${\text {Op}}_{{\mathfrak {V}}}^{{\text {aw}}}(b_0)={\text {Op}}_{\mathfrak V}^{{\text {aw}}}(a)$ on the right-hand side, possess the convenient positivity transition from symbols to anti-Wick operators, while $c_N$ is a linear combination of expressions of the form

$$\begin{aligned} \int _0^1 (1-t)^{|\alpha |-1}\partial _z^\alpha {\overline{\partial }} _w^\alpha a(w+t(z-w),w)\, dt, \qquad |\alpha |=N. \end{aligned}$$

In several situations, the term ${\text {Op}}_{{\mathfrak {V}}}^{{\text {aw}}}(b_0)$ is dominating, while the reminder term ${\text {Op}}_{{\mathfrak {V}}}(c_N)$ is small compared to the other operators in the expansion. (See e. g. [15, 18].)

Wick and anti-Wick operators act between suitable spaces of analytic functions, which are the Bargmann transforms of spaces of functions or distributions defined on the configuration space ${\mathbf {R}}^{d}$. Some ideas on this approach were performed by G. C. Wick in [25]. Later on, F. Berezin improved and extended the theory in [4, 5], where more general classes of operators were considered, compared to [25]. (See also [6, 15] for some other earlier contributors on Wick and anti-Wick operators.) An important part of the investigations in [4, 5, 25] concerns the links between Wick and anti-Wick operators. Especially in [4, 5], convenient formulae were established which explain such links in the case when the Wick and anti-Wick symbols are polynomials. These formulae are in some sense related to (1.4).

Some recent continuity properties for (analytic) integral, Wick and anti-Wick operators are obtained in [17, 18]. In [17], such continuity properties are obtained in the framework of the spaces of power series expansions,

$$\begin{aligned} {\mathcal {A}}_{0,s}({\mathbf {C}}^{d}),\quad {\mathcal {A}}_s({\mathbf {C}}^{d}), \quad \text {with duals}\quad {\mathcal {A}}_s^{\star }({\mathbf {C}}^{d}), \quad {\mathcal {A}}_{0,s}^{\star }({\mathbf {C}}^{d}), \end{aligned}$$

(1.5)

respectively. These spaces are the Bargmann images of the so-called Pilipović spaces and their distribution (dual) spaces

$$\begin{aligned} {\mathcal {H}}_{0,s}({\mathbf {R}}^{d}),\quad {\mathcal {H}}_s({\mathbf {R}}^{d}),\quad {\mathcal {H}}^{\star }_s({\mathbf {R}}^{d}) \quad \text {and}\quad {\mathcal {H}}^{\star } _{0,s}({\mathbf {R}}^{d}), \end{aligned}$$

(1.6)

a family of function and (ultra-)distribution spaces, which contains the Fourier invariant Gelfand–Shilov spaces and their ultra-distribution spaces, and which are thoroughly investigated in [21]. Some early ideas to some of the spaces ${\mathcal {H}}_s({\mathbf {R}}^{d})$, ${\mathcal {H}}_{0,s}({\mathbf {R}}^{d})$ and their duals is given by S. Pilipović in [12]. For example in [12] it is proved that ${\mathcal {H}}_{s}({\mathbf {R}}^{d})$ and ${\mathcal {H}}_{0,t}({\mathbf {R}}^{d})$ agree with the Fourier invariant Gelfand–Shilov spaces ${\mathcal {S}}_{s}({\mathbf {R}}^{d})$ and $\Sigma _t({\mathbf {R}}^{d})$, respectively, when $s\ge \frac{1}{2}$ and $t>\frac{1}{2}$. (See also Proposition 2.13 in Sect. 2 for a more complete view.)

In [17] it is remarked that the usual distribution kernel results for linear operators from ${\mathcal {H}}_s({\mathbf {R}}^{d})$ or ${\mathcal {H}}_{0,s}({\mathbf {R}}^{d})$ to their duals, give one-to-one correspondence of linear and continuous operators between ${\mathcal {A}}_s({\mathbf {C}}^{d})$ and ${\mathcal {A}}^{\star }_s({\mathbf {C}}^{d})$ and analytic integral operators with kernels in   or in   . Here , if and only if $K(z,{\overline{w}})\in {\mathcal {A}}_s({\mathbf {C}}^{2d})$, and similarly for other spaces. It is then proved that for suitable s, the set of Wick operators with symbols in   agrees with the set of integral operators with kernels in   . Consequently, for such s, there is a one-to-one correspondence of linear and continuous operators between ${\mathcal {A}}_s({\mathbf {C}}^{d})$ and ${\mathcal {A}}^{\,\star }_s({\mathbf {C}}^{d})$ and Wick operators with symbols in   . Similar facts hold true with ${\mathcal {A}}_{0,s}$ in place of ${\mathcal {A}}_s$ at each occurrence. (See Theorems 2.7 and 2.8 in [17].)

In the paper we find characterizations of the kernels to (analytic) integral operators and to symbols of Wick and anti-Wick operators in order for these operators should be continuous on the spaces in (1.5). This leads to consider the classes of power series expansions

(1.7)

encountered above and given in [17], and introduction of certain subclasses

(1.8)

and slightly smaller subclasses

(1.9)

which are defined in similar and convenient ways as   ,   and their duals (see Definition 2.5). In Sect. 3 we prove that $K\mapsto T_K$ is bijective from the spaces in (1.8) into the sets of continuous map**s on the respective spaces in (1.5). In particular,

(see Propositions 3.4 and 3.5). For $s<\frac{1}{2}$ we extend the last equality in Sect. 5 into

(1.10)

and similarly for the other spaces in (1.5), (1.8) and (1.9) (see Theorems 5.8 and 5.9).

For certain s, the spaces in (1.8) and (1.9) are subspaces of   and have simple characterizations in terms of convenient estimates on involved symbols. For example, if $s=\flat _\sigma$ with $\sigma >1$, then

(Cf. [1, 21].)

In several situations it is straight-forward to carry over the characterizations above to characterize linear and continuous operators acting on the spaces in (1.6), by applying the Bargmann transform and its inverse in suitable ways. Here the anti-Wick operator ${\text {Op}}_{{\mathfrak {V}}}^{{\text {aw}}}(a)$ corresponds to the Toeplitz operator ${\text {Tp}}_{{\mathfrak {V}}}(a)$, in the sense that

$$\begin{aligned}{\text {Op}}_{{\mathfrak {V}}}^{{\text {aw}}}(a) = {\mathfrak {V}} _d \circ {\text {Tp}}_{{\mathfrak {V}}}(a) \circ {\mathfrak {V}} _d^{-1}, \end{aligned}$$

when ${\mathfrak {V}}_d$ is the Bargmann transform. Here ${\text {Tp}}_{{\mathfrak {V}}}(a)$ is defined by the formula

$$\begin{aligned}&({\text {Tp}}_{{\mathfrak {V}}}(a)f,g)_{L^2({\mathbf {R}}^{d})} = ({\mathfrak {a}}\cdot V_\phi f,V_\phi g)_{L^2({\mathbf {R}}^{2d})}, \\&\quad \quad {\mathfrak {a}}(\sqrt{2}\, x,-\sqrt{2}\, \xi )=a(z,z),\ z=x+i\xi , \end{aligned}$$

where $V_\phi f$ is the short-time Fourier transform of f with respect to the window function

$$\begin{aligned} \phi (x)=\pi ^{-\frac{d}{4}}e^{-\frac{1}{2}|x|^2} \end{aligned}$$

(see Sect. 2.5). Especially it follows from (1.10) that

(1.11)

(see Theorem 6.1). Here we remark that in (1.11), the former class of operators is, in some sense close to the latter class. In particular several linear and continuous operators on ${\mathcal {H}}_s({\mathbf {R}}^{d})$ can be described as Toeplitz operators.

If instead $s\ge \frac{1}{2}$, then the closed links (1.10) and (1.11) between Wick, anti-Wick and kernel operators are violated. For example, it is shown in Sect. 5.3 that estimates, which seems rather natural, can not be performed when finding an anti-Wick operator to a Wick operator (see Remark 5.25).

The analysis behind (1.10) and the other related results are based on some continuity properties for linear and bilinear binomial type operators of independent interests, which acts on coefficients of power series expansions, given in [17] and in Sect. 4. The linear binomial operators are given by

and their formal adjoints

when c is a suitable sequence on ${{\mathbf {N}}}^{2d}$ and $t\in {\mathbf {C}}$ is fixed.

Let $T_{{\mathcal {A}}}$ be the linear and bijective map which takes the sequence $\{c(\alpha ,\beta )\} _{\alpha ,\beta \in {{\mathbf {N}}}^{d}}$ into the power series expansion

$$\begin{aligned} \sum _{\alpha ,\beta \in {{\mathbf {N}}}^{d}}c(\alpha ,\beta )e_\alpha (z)e_\beta ({\overline{w}}), \qquad e_\alpha (z)= \frac{z^\alpha }{\sqrt{\alpha !}}\, . \end{aligned}$$

Also let

$$\begin{aligned} {\ell ^{\star }_{{\mathcal {A}},s}}({{\mathbf {N}}}^{2d}) \quad \text {and}\quad {\ell ^{\star }_{{\mathcal {A}},0,s}}({{\mathbf {N}}}^{2d}) \end{aligned}$$

be the counter images of   and   , respectively, of $T_{\mathcal {A}}$, and let

$$\begin{aligned}&\ell _{{\mathcal {B}},0,s}({{\mathbf {N}}}^{2d}),&\quad&\ell _{{\mathcal {B}},s}({{\mathbf {N}}}^{2d}),&\quad&\ell _{{\mathcal {B}},s}^\star ({{\mathbf {N}}}^{2d})&\quad&\text {and}&\quad&\ell _{{\mathcal {B}},0,s}^\star ({{\mathbf {N}}}^{2d}) \end{aligned}$$

(1.12)

and

$$\begin{aligned}&\ell _{{\mathcal {C}},0,s}({{\mathbf {N}}}^{2d}),&\quad&\ell _{{\mathcal {C}},s}({{\mathbf {N}}}^{2d}),&\quad&\ell _{{\mathcal {C}},s}^\star ({{\mathbf {N}}}^{2d})&\quad&\text {and}&\quad&\ell _{{\mathcal {C}},0,s}^\star ({{\mathbf {N}}}^{2d}) \end{aligned}$$

(1.13)

be the counter images of the respective spaces in (1.8) and (1.9) under $T_{\mathcal {A}}$. Then it is proved in [17] that ${\mathcal {T}}_{0,t}$ is homeomorphic on $\ell _{{\mathcal {A}},s}^{\star }({{\mathbf {N}}}^{2d})$ when $s<\frac{1}{2}$ and on $\ell _{{\mathcal {A}},0,s}^{\star }({{\mathbf {N}}}^{2d})$ when $s\le \frac{1}{2}$.

In Sects. 4 and 5 we complete these map** properties and prove that for similar s one has that ${\mathcal {T}}_{0,t}$ is homeomorphic on each of the spaces in (1.12), and that ${\mathcal {T}}_{0,t}^{*}$ is homeomorphic on each of the spaces in (1.13). By letting

$$\begin{aligned} {\mathcal {T}}_{t} = T_{{\mathcal {A}}} \circ {\mathcal {T}}_{0,t}\circ T_{{\mathcal {A}}}^{-1} \quad \text {and}\quad {\mathcal {T}}_{t}^{*} = T_{{\mathcal {A}}} \circ {\mathcal {T}}_{0,t}^{*}\circ T_{{\mathcal {A}}}^{-1}, \end{aligned}$$

(1.14)

the relations of the form (1.10) then follows from

$$\begin{aligned} T_K={\text {Op}}_{{\mathfrak {V}}}(a), \quad {\text {Op}}_{{\mathfrak {V}}}({\mathcal {T}}_1^{*}a) = {\text {Op}}_{{\mathfrak {V}}}^{{\text {aw}}}(a) \quad \text {and}\quad {\text {Op}}_{{\mathfrak {V}}}^{{\text {aw}}}({\mathcal {T}}_{-1}^{*}a) = {\text {Op}}_{{\mathfrak {V}}}(a), \end{aligned}$$

when $K={\mathcal {T}}_{1}a$, or equivalently, when $a={\mathcal {T}}_{-1}K$. Here we observe that the inverses of the operators in (1.14) are given by

$$\begin{aligned} {\mathcal {T}}_{t}\,^{-1} = {\mathcal {T}}_{-t} \quad \text{ and }\quad ({\mathcal {T}}_{t}^{*})^{-1} = {\mathcal {T}}_{-t}^{*}. \end{aligned}$$

(See also (4.5) and (4.10) in Sect. 4 for exact formulae for ${\mathcal {T}}_{t}$ and ${\mathcal {T}}_{t}^{*}$.)

We also consider the map**s T, $T_\diamond$ and $T_\#$, given by

In Sect. 4 and 5 we deduce continuity properties for such map**s when acting on elements in the spaces in (1.7)–(1.9).

The paper is organized as follows. In Sect. 2 we set the stage by providing necessary background notions and fixing the notation. It contains useful properties for weight functions, Pilipović spaces, the Bargmann transform, Toeplitz operators, Wick and anti-Wick operators. Here we also introduce the spaces (1.8) and (1.9), and explain some of their basic properties.

In Sect. 3 we deduce kernel results. Especially we show that the spaces of linear and continuous operators on the spaces in (1.5) agree with sets of operators with kernels belonging to the spaces in (1.8).

In Sect. 4 we discuss continuity and bijectivity properties of the binomial operators ${\mathcal {T}}_{0,t}$ and ${\mathcal {T}}_{0,t}^{*}$, especially on the spaces in (1.12) and (1.13). We link these binomial operators to transitions between kernel, Wick and anti-Wick operators. Here we also introduce certain bilinear binomial operators which are linked to multiplications and compositions of Wick symbols, and prove continuity properties for such operators.

In Sect. 5 we apply the results and analysis from Sect. 4 to find continuity and identifications between kernel, Wick and anti-Wick operators, as well as continuity properties for compositions of Wick and anti-Wick symbols. For example, here we show relationships like (1.10).

In Sect. 6 we present some consequences of our investigations concerning linear operators which acts on functions and distributions on ${\mathbf {R}}^{d}$. For example we show that for suitable s, the sets of Toeplitz operators with symbols in (1.9) is in some sense close to the set of linear and continuous operators on ${\mathcal {A}}_s({\mathbf {C}}^{d})$, ${\mathcal {A}}_{0,s}({\mathbf {C}}^{d})$ and their duals.

In Sect. 7 we deduce map** properties for ${\mathcal {A}}_s({\mathbf {C}}^{d})$, ${\mathcal {A}}_{0,s}({\mathbf {C}}^{d})$ and their duals, as well as the spaces in (1.8) and (1.9) under linear pullbacks and trace map**s. These results can be used to get alternative proofs of the multiplication properties in Sect. 5.

Finally some background analyses are presented in Appendices A, B and C. In Appendix A we present some identifications of the spaces in (1.5), (1.8) and some other spaces in terms of spaces of analytic functions. In Appendix B we deduce some basic formulae in the transitions between Wick and anti-Wick symbols.

2 Preliminaries

In this section we recall some facts on involved function and distribution spaces as well as on pseudo-differential operators. In Sects. 2.1 and 2.2 we give definitions and review some basic properties for Gelfand–Shilov spaces, Pilipović spaces and the spaces in (1.8). Thereafter we discuss in Sect. 2.2 the Bargmann transform and recall some topological spaces of entire functions or power series expansions on ${\mathbf {C}}^{d}$. The section is concluded with a review of some facts on pseudo-differential operators, Toeplitz operators, Wick and anti-Wick operators.

2.1 Gelfand–Shilov Spaces

Let $0<s \in {\mathbf {R}}$ be fixed. Then the (Fourier invariant) Gelfand–Shilov space ${\mathcal {S}}_s({\mathbf {R}}^{d})$ ($\Sigma _s({\mathbf {R}}^{d})$) of Roumieu type (Beurling type) consists of all $f\in C^\infty ({\mathbf {R}}^{d})$ such that

$$\begin{aligned} \Vert f\Vert _{{\mathcal {S}}_{s,h}}\equiv \sup \left( \frac{|x^\alpha \partial ^\beta f(x)|}{h^{|\alpha + \beta |}(\alpha !\, \beta !)^s} \right) \end{aligned}$$

(2.1)

is finite for some $h>0$ (for every $h>0$). Here the supremum should be taken over all $\alpha ,\beta \in {\mathbf {N}}^d$ and $x\in {\mathbf {R}}^{d}$. (See [7].) The semi-norms $\Vert \, \cdot \, \Vert _{{\mathcal {S}}_{s,h}}$ induce an inductive limit topology for the space ${\mathcal {S}}_s({\mathbf {R}}^{d})$ and projective limit topology for $\Sigma _s({\mathbf {R}}^{d})$, and the latter space becomes a Fréchet space under this topology.

The space ${\mathcal {S}}_s({\mathbf {R}}^{d})\ne \{ 0\}$ ($\Sigma _s({\mathbf {R}}^{d})\ne \{0\}$), if and only if $s\ge \frac{1}{2}$ ($s> \frac{1}{2}$).

The Gelfand–Shilov distribution spaces $\mathcal {S}^{\,\prime }_s({\mathbf {R}}^{d})$ and $\Sigma _s'({\mathbf {R}}^{d})$ are the (strong) duals of ${\mathcal {S}}_s({\mathbf {R}}^{d})$ and $\Sigma _s({\mathbf {R}}^{d})$, respectively.

Let $(\, \cdot \, ,\, \cdot \, )_{L^2}$ be the scalar product in $L^2({\mathbf {R}}^{d})$. Then the duality between ${\mathcal {S}}_s({\mathbf {R}}^{d})$ and $\mathcal {S}^{\,\prime }_s({\mathbf {R}}^{d})$ can be obtained by straight-forward extensions of the restriction of $(\, \cdot \, ,\, \cdot \, )_{L^2}$ to ${\mathcal {S}}_{1/2}({\mathbf {R}}^{d})$. In this setting we have

$$\begin{aligned} \begin{aligned} {\mathcal {S}}_{1/2} ({\mathbf {R}}^{d})&\hookrightarrow \Sigma _s ({\mathbf {R}}^{d}) \hookrightarrow {\mathcal {S}}_s ({\mathbf {R}}^{d}) \hookrightarrow \Sigma _t({\mathbf {R}}^{d})\\&\hookrightarrow {\mathscr {S}}({\mathbf {R}}^{d}) \hookrightarrow {\mathscr {S}}'({\mathbf {R}}^{d}) \hookrightarrow \Sigma _t' ({\mathbf {R}}^{d})\\&\hookrightarrow \mathcal {S}^{\,\prime }_s({\mathbf {R}}^{d}) \hookrightarrow \Sigma _s'({\mathbf {R}}^{d}) \hookrightarrow \mathcal {S}^{\,\prime }_{1/2}({\mathbf {R}}^{d}), \quad \frac{1}{2}<s<t. \end{aligned} \end{aligned}$$

(2.2)

Here and in what follows we use the notation $A\hookrightarrow B$ when the topological spaces A and B satisfy $A\subseteq B$ with continuous embeddings.

A convenient family of functions concerns the Hermite functions

$$\begin{aligned} h_\alpha (x) = \pi ^{-\frac{d}{4}}(-1)^{|\alpha |} (2^{|\alpha |}\alpha !)^{-\frac{1}{2}}e^{\frac{1}{2}{|x|^2}} (\partial ^\alpha e^{-|x|^2}),\quad \alpha \in {{\mathbf {N}}}^{d}. \end{aligned}$$

The set of Hermite functions on ${\mathbf {R}}^{d}$ is an orthonormal basis for $L^2({\mathbf {R}}^{d})$. It is also a basis for the Schwartz space and its distribution space, and for any $\Sigma _s ({\mathbf {R}}^{d})$ when $s>\frac{1}{2}$, $\mathcal {S} _s ({\mathbf {R}}^{d})$ when $s\ge \frac{1}{2}$ and their distribution spaces. They are also eigenfunctions to the Harmonic oscillator $H=H_d\equiv |x|^2-\Delta$ and to the Fourier transform ${\mathscr {F}}$, given by

$$\begin{aligned} (\mathscr {F}f)(\xi )= {\widehat{f}}(\xi ) \equiv (2\pi )^{-\frac{d}{2}}\int _{{\mathbf {R}}^{d}} f(x)e^{-i\langle x,\xi \rangle }\, dx, \quad \xi \in {\mathbf {R}}^{d}, \end{aligned}$$

when $f\in L^1({\mathbf {R}}^{d})$. Here $\langle \, \cdot \, ,\, \cdot \, \rangle$ denotes the usual scalar product on ${\mathbf {R}}^{d}$. In fact, we have

$$\begin{aligned} H_dh_\alpha = (2|\alpha |+d)h_\alpha . \end{aligned}$$

The Fourier transform ${\mathscr {F}}$ extends uniquely to homeomorphisms on ${\mathscr {S}'}({\mathbf {R}}^{d})$, ${\mathcal {S}}^{\,\prime }_s({\mathbf {R}}^{d})$ and on ${\mathscr {S}}'({\mathbf {R}}^{d})$. Furthermore, ${\mathscr {F}}$ restricts to homeomorphisms on ${\mathscr {S}}({\mathbf {R}}^{d})$, ${\mathcal {S}}_s({\mathbf {R}}^{d})$ and on $\Sigma _s ({\mathbf {R}}^{d})$, and to a unitary operator on $L^2({\mathbf {R}}^{d})$. Similar facts hold true when the Fourier transform is replaced by a partial Fourier transform.

Gelfand–Shilov spaces and their distribution spaces can also be characterized by estimates of short-time Fourier transforms, (see e. g. [8, 16, 21]). They also obey convenient map** properties under short-time Fourier transforms. More precisely, let $\phi \in {\mathscr {S}}({\mathbf {R}}^{d})$ be fixed. Then the short-time Fourier transform $V_\phi f$ of $f\in {\mathscr {S}}' ({\mathbf {R}}^{d})$ with respect to the window function $\phi$ is the tempered distribution on ${\mathbf {R}}^{2d}$, defined by

$$\begin{aligned} V_\phi f(x,\xi ) = {\mathscr {F}}(f \, \overline{\phi (\, \cdot \, -x)})(\xi ), \quad x,\xi \in {\mathbf {R}}^{d}. \end{aligned}$$

(2.3)

If $f ,\phi \in {\mathscr {S}}({\mathbf {R}}^{d})$, then it follows that

$$\begin{aligned} V_\phi f(x,\xi ) = (2\pi )^{-\frac{d}{2}}\int _{{\mathbf {R}}^{d}} f(y)\overline{\phi (y-x)}e^{-i\langle y,\xi \rangle }\, dy, \quad x,\xi \in {\mathbf {R}}^{d}. \end{aligned}$$

By [19, Theorem 2.3] it follows that the definition of the map $(f,\phi )\mapsto V_{\phi } f$ from ${\mathscr {S}}({\mathbf {R}}^{d}) \times {\mathscr {S}}({\mathbf {R}}^{d})$ to ${\mathscr {S}}({\mathbf {R}}^{2d})$ is uniquely extendable to a continuous map from $\mathcal {S}^{\,\prime }_s({\mathbf {R}}^{d})\times \mathcal {S}^{\,\prime }_s({\mathbf {R}}^{d})$ to $\mathcal {S}^{\,\prime }_s({\mathbf {R}}^{2d})$, and restricts to a continuous map from ${\mathcal {S}}_s ({\mathbf {R}}^{d})\times {\mathcal {S}}_s ({\mathbf {R}}^{d})$ to ${\mathcal {S}}_s({\mathbf {R}}^{2d})$. The same conclusions hold with $\Sigma _s$ in place of ${\mathcal {S}}_s$, at each place.

2.2 Spaces of Sequences, Hermite Series and Power Series Expansions

Next we recall the definitions of topological vector spaces of Hermite series expansions, given in [21]. As in [21], it is convenient to use suitable extensions of ${\mathbf {R}}_+$ when indexing our spaces.

Definition 2.1

The sets ${\mathbf {R}}_\flat$, $\overline{{\mathbf {R}}_\flat }$, ${\mathbf {R}}_{\flat ,\infty }$ and $\overline{{\mathbf {R}}_{\flat ,\infty }}$ are given by

$$\begin{aligned} {\textstyle {{\mathbf {R}}_\flat = {\mathbf {R}}_+ \underset{\sigma >0}{\textstyle {\bigcup }} \{ \flat _\sigma \} }}, \quad {\textstyle {\overline{{\mathbf {R}}_\flat } = {\mathbf {R}}_\flat \bigcup \{ 0 \} }}, \quad {\textstyle {{\mathbf {R}}_{\flat ,\infty } = {\mathbf {R}}_\flat \bigcup \{ \infty \} }} \quad \text {and}\quad {\textstyle {\overline{{\mathbf {R}}_{\flat ,\infty }} = \overline{{\mathbf {R}}_\flat } \bigcup \{ \infty \} }}. \end{aligned}$$

Beside the usual ordering in $\overline{{\mathbf {R}}_+}\bigcup \{\infty \}$, the elements $\flat _\sigma$ in these subsets of $\overline{{\mathbf {R}}_{\flat ,\infty }}$ are ordered by the relations $x_1<\flat _{\sigma _1}<\flat _{\sigma _2}<x_2$, when $\sigma _1,\sigma _2,x_1,x_2\in {\mathbf {R}}_+$ satisfy $x_1<\frac{1}{2}$, $x_2\ge \frac{1}{2}$ and $\sigma _1<\sigma _2$.

In order for defining the sequence spaces we shall make use of the weight

$$\begin{aligned} \vartheta _{r,s}(\alpha )&\equiv {\left\{ \begin{array}{ll} 1 &{} \text {when}\quad s=0,\ |\alpha |\le r, \\ \infty &{} \text {when}\quad s=0,\ |\alpha |> r, \\ e^{r|\alpha |^{\frac{1}{2s}}} &{} \text {when}\quad s\in {\mathbf {R}}_+, \\ r^{|\alpha |}(\alpha !)^{\frac{1}{2\sigma }} &{} \text {when}\quad s = \flat _\sigma , \\ \langle \alpha \rangle ^r &{} \text {when}\quad s=\infty , \qquad \alpha \in {{\mathbf {N}}}^{d}, \end{array}\right. } \end{aligned}$$

(2.4)

and we observe that such weights obey estimates given in the following lemma.

Lemma 2.2

Let $\alpha ,\beta \in {{\mathbf {N}}}^{d}$ and $s,\sigma \in {\mathbf {R}}_+$. Then

$$\begin{aligned} \begin{aligned} \vartheta _{c_1r,s}(\alpha )\vartheta _{c_1r,s}(\beta )&\le \vartheta _{r,s}(\alpha +\beta ) \le \vartheta _{c_2r,s}(\alpha )\vartheta _{c_2r,s}(\beta ),\\ c_1&= \min (2^{\frac{1}{2s}-1},1),\ c_2=\max (2^{\frac{1}{2s}-1},1), \end{aligned} \end{aligned}$$

(2.5)

and

$$\begin{aligned} \begin{aligned} \vartheta _{r,\flat _\sigma }(\alpha )\vartheta _{r,s}(\beta )&\le \vartheta _{r,\flat _\sigma }(\alpha +\beta ) \le C\vartheta _{c\, r,\flat _\sigma }(\alpha )\vartheta _{c\, r,s}(\beta ), \\ c&=2^{\frac{1}{2\sigma }}, \quad C= {\left\{ \begin{array}{ll} 1, &{} |\alpha +\beta |=0, \\ 2^{-\frac{1}{2\sigma }}, &{} |\alpha +\beta | > 0. \end{array}\right. } \end{aligned} \end{aligned}$$

(2.6)

If $R\ge 0$ and in addition $s<\frac{1}{2}$, then it also holds

$$\begin{aligned} R^{|\alpha |} \lesssim \vartheta _{r,s}(\alpha ) \quad \text {and}\quad R^{|\alpha |} \lesssim \vartheta _{r,\flat _\sigma }(\alpha ), \quad \alpha \in {{\mathbf {N}}}^{d}. \end{aligned}$$

(2.7)

Proof

Let $\theta =\frac{1}{2s}$. Then the inequalities in (2.5) follows from

$$\begin{aligned} c_1(s^\theta +t^\theta )\le (s+t)^\theta \le c_2(s^\theta +t^\theta ), \quad s,t\ge 0. \end{aligned}$$

The inequalities in (2.6) follows from

and that

The estimates in (2.7) follows from

$$\begin{aligned} R^{|\alpha |}\lesssim \alpha !^{\frac{1}{2\sigma }} \lesssim R^{|\alpha |^{\frac{1}{2s}}} \end{aligned}$$

when $R>1$ and $s<\frac{1}{2}$, and the result follows. $\square$

Definition 2.3

Let $p\in (0,\infty ]$, $s\in \overline{{\mathbf {R}}_{\flat ,\infty }}$, $r\in {\mathbf {R}}$, $\vartheta$ be a map from ${{\mathbf {N}}}^{d}$ to ${\mathbf {R}}_+\bigcup \infty$, and let $\vartheta _{r,s}$ be as in (2.4).

• (1) The set $\ell ^p_{[\vartheta ]}({{\mathbf {N}}}^{d})$ consists of all sequences $\{ c (\alpha ) \} _{\alpha \in {{\mathbf {N}}}^{d}} \subseteq {\mathbf {C}}$ such that

  $$\begin{aligned} \Vert \{ c (\alpha ) \} _{\alpha \in {{\mathbf {N}}}^{d}} \Vert _{\ell ^p_{[\vartheta ]}}\equiv \Vert \{ c (\alpha ) \vartheta (\alpha )\} _{\alpha \in {{\mathbf {N}}}^{d}} \Vert _{\ell ^p} < \infty ; \end{aligned}$$

• (2) if $s<\infty$, then

  $$\begin{aligned} \ell _s({{\mathbf {N}}}^{d})\equiv \underset{r>0}{{\text{ ind } \text{ lim } }}\left( \ell ^p_{[\vartheta _{r,s}]}({{\mathbf {N}}}^{d})\right) \; \text{ and }\; \ell _s^\star ({{\mathbf {N}}}^{d})\equiv \underset{r>0}{{\text{ proj } \text{ lim } }}\left( \ell ^p_{[1/\vartheta _{r,s}]}({{\mathbf {N}}}^{d})\right) \text {; } \end{aligned}$$

  (2.8)

• (3) if $s>0$, then

  $$\begin{aligned} \ell _{0,s}({{\mathbf {N}}}^{d})\equiv \underset{r>0}{{\text{ proj } \text{ lim } }}\left( \ell ^p_{[\vartheta _{r,s}]}({{\mathbf {N}}}^{d}) \right) \; \text{ and } \;\ell _{0,s}^\star ({{\mathbf {N}}}^{d})\equiv \underset{r>0}{{\text{ ind } \text{ lim } }}\left( \ell ^p_{[\vartheta _{r,s}]}({{\mathbf {N}}}^{d})\right) .\quad \quad \end{aligned}$$

  (2.9)

We observe that $\ell ^p_{[\vartheta _{r,s}]}({{\mathbf {N}}}^{d})$ in Definition 2.3 is a quasi-Banach space under the quasi-norm $\Vert \, \cdot \, \Vert _{\ell ^p_{[\vartheta _{r,s}]}}$ when $s>0$. If in addition $p\ge 1$, then $\ell ^p_{[\vartheta _{r,s}]}({{\mathbf {N}}}^{d})$ is a Banach space with norm $\Vert \, \cdot \, \Vert _{\ell ^p_{[\vartheta _{r,s}]}}$.

In several situations we deal with two-parameter version of the spaces in Definition 2.3, which are denoted by

$$\begin{aligned}&\ell _{{\mathcal {A}},0,(s_2,s_1)}(\Lambda ),&\quad&\ell _{{\mathcal {A}},(s_2,s_1)} (\Lambda ),&\quad&\ell _{{\mathcal {A}},0,(s_2,s_1)}^\star (\Lambda ),&\quad \ell _{{\mathcal {A}},(s_2,s_1)}^\star (\Lambda ), \end{aligned}$$

(2.10)

$$\begin{aligned}&\ell _{{\mathcal {B}},0,(s_2,s_1)}(\Lambda ),&\quad&\ell _{{\mathcal {B}},(s_2,s_1)}(\Lambda ),&\quad&\ell _{{\mathcal {B}},0,(s_2,s_1)}^\star (\Lambda ),&\quad \ell _{{\mathcal {B}},(s_2,s_1)}^\star (\Lambda ), \end{aligned}$$

(2.11)

$$\begin{aligned}&\ell _{{\mathcal {C}},0,(s_2,s_1)}(\Lambda ),&\quad&\ell _{{\mathcal {C}},(s_2,s_1)}(\Lambda ),&\quad&\ell _{{\mathcal {C}},0,(s_2,s_1)}^\star (\Lambda ),&\quad \ell _{{\mathcal {C}},(s_2,s_1)}^\star (\Lambda ). \end{aligned}$$

(2.12)

Definition 2.4

Let $\vartheta _{r,s}$ be as in (2.4), $s_1,s_2\in \overline{{\mathbf {R}}_{\flat ,\infty }}$, $\Lambda ={{\mathbf {N}}}^{d_2}\times {{\mathbf {N}}}^{d_1}$, $p\in (0,\infty ]$ and let

$$\begin{aligned} \vartheta _{r,(s_2,s_1)}(\alpha ) = \vartheta _{r,s_1}(\alpha _1) \vartheta _{r,s_2}(\alpha _2),\quad \alpha =(\alpha _2,\alpha _1), \qquad \alpha _1\in {{\mathbf {N}}}^{d_1},\ \alpha _2\in {{\mathbf {N}}}^{d_2}. \end{aligned}$$

Then the spaces in (2.10) are given by

$$\begin{aligned} \ell _{{\mathcal {A}},0,(s_2,s_1)}(\Lambda )&\equiv \underset{r>0}{{\text {proj lim }}}\left( \ell ^p_{[\vartheta _{r,(s_2,s_1)}]}(\Lambda )\right) ,&\qquad s_1,s_2&>0, \end{aligned}$$

(2.13)

$$\begin{aligned} \ell _{{\mathcal {A}},(s_2,s_1)}(\Lambda )&\equiv \underset{r>0}{{\text {ind lim }}}\left( \ell ^p_{[\vartheta _{r,(s_2,s_1)}]}(\Lambda )\right) ,&\qquad s_1,s_2&<\infty , \end{aligned}$$

(2.14)

$$\begin{aligned} \ell _{{\mathcal {A}},0,(s_2,s_1)}^\star (\Lambda )&\equiv \underset{r>0}{{\text {ind lim }}}\left( \ell ^p_{[1/\vartheta _{r,(s_2,s_1)}]}(\Lambda )\right) ,&\qquad s_1,s_2&>0, \end{aligned}$$

(2.15)

and

$$\begin{aligned} \ell _{{\mathcal {A}},(s_2,s_1)}^\star (\Lambda )&\equiv \underset{r>0}{{\text {proj lim }}}\left( \ell ^p_{[1/\vartheta _{r,(s_2,s_1)}]}(\Lambda )\right) ,&\qquad s_1,s_2&<\infty . \end{aligned}$$

(2.16)

Definition 2.5

Let $\vartheta _{r,s}$ be as in Definition 2.4, $s_1,s_2\in \overline{{\mathbf {R}}_{\flat ,\infty }}$, $\Lambda ={{\mathbf {N}}}^{d_2}\times {{\mathbf {N}}}^{d_1}$, $p\in (0,\infty ]$ and let

$$\begin{aligned} \omega _{s_2,s_1;r_2,r_1}(\alpha ) \equiv \frac{\vartheta _{r_2,s_2}(\alpha _2)}{\vartheta _{r_1,s_1}(\alpha _1)}, \quad \alpha = (\alpha _2,\alpha _1), \qquad \alpha _1\in {{\mathbf {N}}}^{d_1},\ \alpha _2\in {{\mathbf {N}}}^{d_2}. \end{aligned}$$

Then the spaces in (2.11) are given by

$$\begin{aligned} \ell _{{\mathcal {B}},0,(s_2,s_1)}(\Lambda )&= \underset{r_2>0}{{\text {proj lim }}}\left( \underset{r_1>0}{{\text {ind lim }}}\left( \ell _{[\omega _{s_2,s_1;r_2,r_1}]}^p (\Lambda ) \right) \right) ,&\quad s_1,s_2&>0 , \end{aligned}$$

(2.17)

$$\begin{aligned} \ell _{{\mathcal {B}},(s_2,s_1)}(\Lambda )&= \underset{r_1>0}{{\text {proj lim }}}\left( \underset{r_2>0}{{\text {ind lim }}}\left( \ell _{[\omega _{s_2,s_1;r_2,r_1}]}^p (\Lambda ) \right) \right) ,&\quad s_1,s_2&<\infty , \end{aligned}$$

(2.18)

$$\begin{aligned} \ell _{{\mathcal {B}},0,(s_2,s_1)}^\star (\Lambda )&= \underset{r_1>0}{{\text {proj lim }}}\left( \underset{r_2>0}{{\text {ind lim }}}\left( \ell _{[1/\omega _{s_2,s_1;r_2,r_1}]}^p (\Lambda ) \right) \right) ,&\quad s_1,s_2&>0. \end{aligned}$$

(2.19)

and

$$\begin{aligned} \ell _{{\mathcal {B}},(s_2,s_1)}^\star (\Lambda )&= \underset{r_2>0}{{\text {proj lim }}}\left( \underset{r_1>0}{{\text {ind lim }}}\left( \ell _{[1/\omega _{s_2,s_1;r_2,r_1}]}^p (\Lambda ) \right) \right) ,&\quad s_1,s_2&<\infty . \end{aligned}$$

(2.20)

The spaces in (2.12) for admissible $s_1,s_2$ are given by the right-hand sides of (2.17)–(2.20), after the orders of inductive and projective limits are swapped.

Remark 2.6

By playing with r,$r_1$ and $r_2$ it follows that the topological vector spaces in (2.8), (2.9) and (2.11)–(2.16) are independent of $p\in (0,\infty ]$.

For conveniency we also complete the spaces in Definitions 2.3, 2.4 and 2.5 with the following.

Definition 2.7

Let $s_1,s_2\in \overline{{\mathbf {R}}_{\flat ,\infty }}$ and $\Lambda = {{\mathbf {N}}}^{d_2}\times {{\mathbf {N}}}^{d_1}$.

• (1) $\ell _\infty ({{\mathbf {N}}}^{d})$ and $\ell _\infty ^\star ({{\mathbf {N}}}^{d})$ are given by (2.8) with $s=\infty$ and $p=2$;

• (2) if $s_1=0$ or $s_2=0$, then

  $$\begin{aligned} \ell _{0,0}(\Lambda )&= \ell _{0,0}^\star (\Lambda ) = \ell _{{\mathcal {A}},0,(s_2,s_1)}(\Lambda ) = \ell _{{\mathcal {B}},0,(s_2,s_1)}(\Lambda ) = \ell _{{\mathcal {C}},0,(s_2,s_1)}(\Lambda ) \\&= \ell _{{\mathcal {A}},0,(s_2,s_1)}^\star (\Lambda ) = \ell _{{\mathcal {B}},0,(s_2,s_1)}^\star (\Lambda ) = \ell _{{\mathcal {C}},0,(s_2,s_1)}^\star (\Lambda ) \equiv \{ 0 \} = \big \{ \{ 0 \}_{\alpha \in \Lambda } \big \} \text {;} \end{aligned}$$

• (2) if $s_1=\infty$ or $s_2=\infty$, then

  $$\begin{aligned} \ell _{{\mathcal {A}},(s_2,s_1)}(\Lambda ), \quad \ell _{{\mathcal {A}},(s_2,s_1)}^\star (\Lambda ), \quad \ell _{{\mathcal {B}},(s_2,s_1)}(\Lambda ) \quad \text {and}\quad \ell _{{\mathcal {B}},(s_2,s_1)}^\star (\Lambda ) \end{aligned}$$

  are defined by (2.14), (2.16), (2.18) and (2.19) with $p=2$, and $\ell _{{\mathcal {C}},s_1,s_2}(\Lambda )$ and $\ell _{{\mathcal {C}},s_1,s_2}^\star (\Lambda )$ are defined by (2.18) and (2.19) with $p=2$ after the orders of inductive and projective limits are swapped.

Remark 2.8

By the definitions it follows that (2.8), (2.9), (2.13)–(2.16) and (2.17)–(2.19) hold true with $\bigcup$ and $\bigcap$ in place of ${\text {ind lim }}$ and ${\text {proj lim }}$, respectively, at each occurrence.

We observe that the following holds true:

• (1) the space $\ell _0^\star ({{\mathbf {N}}}^{d})$ is the set of all sequences $\{c (\alpha ) \} _{\alpha \in {{\mathbf {N}}}^{d}} \subseteq {\mathbf {C}}$ on ${{\mathbf {N}}}^{d}$, and that $\ell _0({{\mathbf {N}}}^{d})$ is the set of all such sequences such that $c (\alpha ) \ne 0$ for at most finite numbers of $\alpha$. In similar ways, the condition $s_1=0$ or $s_2=0$ impose support restrictions of the elements in the spaces (2.10)–(2.12);

• (2) the spaces in (2.10) are complete Hausdorff topological vector spaces, and $\ell _{{\mathcal {A}},0,(s_2,s_1)}(\Lambda )$ and $\ell _{{\mathcal {A}},(s_2,s_1)}^\star (\Lambda )$ are Fréchet spaces. It holds that $\ell _{{\mathcal {A}},(s,s)} = \ell _s$, $\ell _{{\mathcal {A}},0,(s,s)} = \ell _{0,s}$;

• (2) $\ell _0(\Lambda )$ is dense in all of the spaces in (2.10)–(2.12).

Remark 2.9

Let $s_1,s_2\in \overline{{\mathbf {R}}_\flat }$ and $\Lambda ={{\mathbf {N}}}^{d_2}\times {{\mathbf {N}}}^{d_1}$. Then it follows by straight-forward computations that the $\ell ^2(\Lambda )$ form $(\, \cdot \, ,\, \cdot \, )_{\ell ^2(\Lambda )}$ on $\ell _0(\Lambda )$ is uniquely extendable to hypocontinuous map**s from

$$\begin{aligned} \ell _{{\mathcal {A}},(s_2,s_1)}^\star (\Lambda )\times \ell _{{\mathcal {A}},(s_2,s_1)}(\Lambda ) \end{aligned}$$

(2.21)

to ${\mathbf {C}}$, and to separately continuous map**s from

$$\begin{aligned} \ell _{{\mathcal {B}},(s_2,s_1)}^\star (\Lambda )\times \ell _{{\mathcal {C}},(s_2,s_1)}(\Lambda ) \quad \text {or} \qquad \ell _{{\mathcal {C}},(s_2,s_1)}^\star (\Lambda )\times \ell _{{\mathcal {B}},(s_2,s_1)}(\Lambda ) \end{aligned}$$

(2.22)

to ${\mathbf {C}}$. These map**s constitutes dualities in the sense that they are non-degenerate, i. e. if $c_1\in \ell _{{\mathcal {A}},(s_2,s_1)}^\star (\Lambda )\setminus 0$ and $c_2\in \ell _{{\mathcal {A}},(s_2,s_1)} (\Lambda )\setminus 0$, then

$$\begin{aligned} \{ \, (c_1,c)_{\ell ^2}\, ;\, c\in \ell _{{\mathcal {A}},(s_2,s_1)} (\Lambda )\, \} \ne 0 \quad \text {and}\quad \{ \, (c,c_2)_{\ell ^2}\, ;\, c\in \ell _{{\mathcal {A}},(s_2,s_1)}^\star (\Lambda )\, \} \ne 0, \end{aligned}$$

and similarly for the pairs of spaces in (2.22). We also observe that $\ell _{{\mathcal {A}},(s_2,s_1)}^\star (\Lambda )$ and $\ell _{{\mathcal {A}},(s_2,s_1)}(\Lambda )$ are (strong) duals to each others, through the form $(\, \cdot \, ,\, \cdot \, )_{\ell ^2(\Lambda )}$.

If instead $s_1,s_2\in {\mathbf {R}}_{\flat ,\infty }$, then the same holds true with $\ell _{{\mathcal {A}},0,(s_2,s_1)}$, $\ell _{{\mathcal {B}},0,(s_2,s_1)}$ and $\ell _{{\mathcal {C}},0,(s_2,s_1)}$ in place of $\ell _{{\mathcal {A}},(s_2,s_1)}$, $\ell _{{\mathcal {B}},(s_2,s_1)}$ and $\ell _{{\mathcal {C}},(s_2,s_1)}$, respectively, at each occurrence.

Remark 2.10

The spaces in (2.10) are either Fréchet spaces, or so-called LB-spaces, i. e. inductive limits of increasing Banach spaces. For these reasons they possess similar topological properties as for Schwartz spaces or spaces of tempered distributions. Especially we observe that $\ell _{{\mathcal {A}},(s_2,s_1)} (\Lambda )$ are LB-spaces which become strict LB-spaces when $s_1 = s_2 = 0$.

The topological structures of the spaces in (2.11) and (2.12) are more cumbersome since these are mixtures of inductive and projective limits of Banach spaces. We observe that the spaces in (2.11) are projective limits of inductive limits of Banach spaces, i. e. they are projective limits of (ordered) LB-spaces.

For the spaces in (2.12), we observe that they are inductive limits of Fréchet spaces, i. e. they are so-called LF-spaces. We also observe that these Fréchet spaces are not Banach spaces. (See e. g. [10, 11, 14].)

In Sect. 5.5 we refine topological issues on certain bilinear map**s of elements in spaces like (2.10), (2.11) and (2.12). An essential part of these refinements consists of classifying LB-spaces and LF-spaces in terms of barreled spaces and DF-spaces. (See e. g. [10, 11, 14].)

Next we consider, in similar ways as in [21], spaces of formal Hermite series expansions

$$\begin{aligned} f=\sum _{\alpha \in {{\mathbf {N}}}^{d}}c (f;\alpha ) h_\alpha ,\quad \{ c (F;\alpha ) \} _{\alpha \in {{\mathbf {N}}}^{d}} \in \ell _0^\star ({{\mathbf {N}}}^{d}). \end{aligned}$$

(2.23)

and spaces of formal power series expansions

$$\begin{aligned} F=\sum _{\alpha \in {{\mathbf {N}}}^{d}}c (F;\alpha ) e_\alpha , \quad \{c (F; \alpha ) \} _{\alpha \in {{\mathbf {N}}}^{d}} \in \ell _0^\star ({{\mathbf {N}}}^{d}). \end{aligned}$$

(2.24)

which correspond to

$$\begin{aligned} \ell _{0,s}({{\mathbf {N}}}^{d}),\quad \ell _s({{\mathbf {N}}}^{d}),\quad \ell _{0,s}^\star ({{\mathbf {N}}}^{d}) \quad \text {and}\quad \quad \ell _s^\star ({{\mathbf {N}}}^{d}). \end{aligned}$$

(2.25)

Here

$$\begin{aligned} e_\alpha (z) \equiv \frac{z^\alpha }{\sqrt{\alpha !}}, \qquad z\in {\mathbf {C}}^{d},\ \alpha \in {{\mathbf {N}}}^{d}. \end{aligned}$$

(2.26)

We consider the map**s

$$\begin{aligned} T_{{\mathcal {H}}} : \, \{ c (\alpha ) \} _{\alpha \in {{\mathbf {N}}}^{d}} \mapsto \sum _{\alpha \in {{\mathbf {N}}}^{d}} c (\alpha ) h_\alpha \quad \text {and}\quad T_{{\mathcal {A}}} : \, \{ c (\alpha ) \} _{\alpha \in {{\mathbf {N}}}^{d}} \mapsto \sum _{\alpha \in {{\mathbf {N}}}^{d}} c (\alpha ) e_\alpha \qquad \end{aligned}$$

(2.27)

between sequences, and formal Hermite series and power series expansions.

Definition 2.11

If $s\in \overline{{\mathbf {R}}_{\flat ,\infty }}$, then

$$\begin{aligned} {\mathcal {H}}_{0,s}({\mathbf {R}}^{d}),\quad {\mathcal {H}}_s({\mathbf {R}}^{d}), \quad {\mathcal {H}}_{0,s}^\star ({\mathbf {R}}^{d}) \quad \text {and}\quad {\mathcal {H}}_s^\star ({\mathbf {R}}^{d}), \end{aligned}$$

(2.28)

and

$$\begin{aligned} {\mathcal {A}}_{0,s}({\mathbf {C}}^{d}),\quad {\mathcal {A}}_s({\mathbf {C}}^{d}), \quad {\mathcal {A}}_{0,s}^\star ({\mathbf {C}}^{d}) \quad \text {and}\quad {\mathcal {A}}_s^\star ({\mathbf {C}}^{d}), \end{aligned}$$

(2.29)

are the images of $T_{{\mathcal {H}}}$ and $T_{{\mathcal {A}}}$ respectively in (2.27) of corresponding spaces in (2.25). The topologies of the spaces in (2.28) and (2.29) are inherited from the corresponding spaces in (2.25).

By Remark 2.9 it follows that the latter spaces in (2.28) are the (strong) duals of the former spaces with respect to unique extensions of the form $(\, \cdot \, ,\, \cdot \, )_{L^2}$ on ${\mathcal {H}}_0({\mathbf {R}}^{d})$. That is,

$$\begin{aligned} {\mathcal {H}}_s'({\mathbf {R}}^{d}) = {\mathcal {H}}_s^\star ({\mathbf {R}}^{d}) \quad \text {and}\quad {\mathcal {H}}_{0,s} '({\mathbf {R}}^{d}) = {\mathcal {H}}_{0,s} ^\star ({\mathbf {R}}^{d}). \end{aligned}$$

(2.30)

In the same way as in (2.29) we let

be the images of the spaces (2.10) under the map $T_{{\mathcal {A}}}$ when $W={\mathbf {C}}^{d_2}\times {\mathbf {C}}^{d_1}$, $\Lambda ={{\mathbf {N}}}^{d_2}\times {{\mathbf {N}}}^{d_1}$ and $s_1,s_2\in \overline{{\mathbf {R}}_{\flat ,\infty }}$.

Since locally absolutely convergent power series expansions can be identified with entire functions, several of the spaces in (2.29) are identified with topological vector spaces contained in $A({\mathbf {C}}^{d})$ or A(W). (See [21], Theorem A.1 in Appendix A and the introduction.) Here $A(\Omega )$ is the set of all (complex valued) functions which are analytic in the open set $\Omega \subseteq {\mathbf {C}}^{d}$. If $\Omega _0\subseteq {\mathbf {C}}^{d}$ is arbitrary, then $A(\Omega _0)=\cup A(\Omega )$, where the union is taken over all open sets $\Omega \subseteq {\mathbf {C}}^{d}$ such that $\Omega _0\subseteq \Omega$.

We recall that $f\in {\mathscr {S}}({\mathbf {R}}^{d})$ if and only if (2.23) holds with

$$\begin{aligned} |c (f; \alpha ) |\lesssim \langle \alpha \rangle ^{-N}, \end{aligned}$$

for every $N\ge 0$. That is, ${\mathscr {S}}({\mathbf {R}}^{d})={\mathcal {H}}_{0,\infty }({\mathbf {R}}^{d})$. In the same way, ${\mathscr {S}}'({\mathbf {R}}^{d})={\mathcal {H}}_{0,\infty }^\star ({\mathbf {R}}^{d})$, and $f\in L^2({\mathbf {R}}^{d})$, if and only if $\{ c(f;\alpha )\} _{\alpha \in {{\mathbf {N}}}^{d}}\in \ell ^2({{\mathbf {N}}}^{d})$ (cf. e. g. [13]). In particular it follows from the definitions that the inclusions

$$\begin{aligned} {\mathcal {H}}_0({\mathbf {R}}^{d})&\hookrightarrow {\mathcal {H}}_{0,s}({\mathbf {R}}^{d})\hookrightarrow {\mathcal {H}}_{s}({\mathbf {R}}^{d}) \hookrightarrow {\mathcal {H}}_{0,t}({\mathbf {R}}^{d}) \nonumber \\&\hookrightarrow {\mathcal {H}}_{0,t}^\star ({\mathbf {R}}^{d}) \hookrightarrow {\mathcal {H}}_{s}^\star ({\mathbf {R}}^{d}) \hookrightarrow {\mathcal {H}}_{0,s}^\star ({\mathbf {R}}^{d})\hookrightarrow {\mathcal {H}}_0^\star ({\mathbf {R}}^{d}), \nonumber \\ {}&\qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad s,t\in {\mathbf {R}}_{\flat ,\infty } , s<t, \end{aligned}$$

(2.31)

are dense.

Remark 2.12

By the definition it follows that $T_{{\mathcal {H}}}$ in (2.27) is a homeomorphism between any of the spaces in (2.25) and corresponding space in (2.28), and that $T_{{\mathcal {A}}}$ in (2.27) is a homeomorphism between any of the spaces in (2.25) and corresponding space in (2.29).

The next results give some characterizations of ${\mathcal {H}}_s({\mathbf {R}}^{d})$ and ${\mathcal {H}}_{0,s}({\mathbf {R}}^{d})$ when s is a non-negative real number. See also [1] for similar characterizations of ${\mathcal {H}}_{\flat _\sigma }({\mathbf {R}}^{d})$ and ${\mathcal {H}}_{0,\flat _\sigma }({\mathbf {R}}^{d})$.

Proposition 2.13

Let $0\le s,s_1,s_2\in {\mathbf {R}}$ and let $f\in {\mathcal {H}}_0^\star ({\mathbf {R}}^{d})$. Then $f\in {\mathcal {H}}_s({\mathbf {R}}^{d})$ ($f\in {\mathcal {H}}_{0,s}({\mathbf {R}}^{d})$), if and only if $f\in C^\infty ({\mathbf {R}}^{d})$ and satisfies

$$\begin{aligned} \Vert H_d^Nf\Vert _{L^\infty }\lesssim h^NN!^{2s}, \end{aligned}$$

(2.32)

for some (every) $h>0$. If $s_1<\frac{1}{2}$ and $s_2\ge \frac{1}{2}$ ($0<s_1\le \frac{1}{2}$ and $s_2> \frac{1}{2}$), then

$$\begin{aligned} {\mathcal {H}}_{s_1}({\mathbf {R}}^{d})&\ne {\mathcal {S}}_{s_1}({\mathbf {R}}^{d}) = \{ 0\},&\quad \big ( {\mathcal {H}}_{0,s_1}({\mathbf {R}}^{d})&\ne \Sigma _{s_1}({\mathbf {R}}^{d}) \ne \{ 0\} \big ),\\ {\mathcal {H}}_{s_2}({\mathbf {R}}^{d})&= {\mathcal {S}}_{s_2}({\mathbf {R}}^{d}) \ne \{ 0\}&\quad \text {and}\quad \big ( {\mathcal {H}}_{0,s_2}({\mathbf {R}}^{d})&= \Sigma _{s_2}({\mathbf {R}}^{d}) \ne \{ 0\} \big ) . \end{aligned}$$

We refer to [21] for the proof of Proposition 2.13.

Due to the pioneering investigations related to Proposition 2.13 by Pilipović in [12], we call the spaces ${\mathcal {H}}_s({\mathbf {R}}^{d})$ and ${\mathcal {H}}_{0,s}({\mathbf {R}}^{d})$ as Pilipović spaces of Roumieu and Beurling types, respectively. In fact, in the restricted case $s,s_1,s_2\ge \frac{1}{2}$, Proposition 2.13 was proved already in [12].

In what follows we let $F(z_2,{\overline{z}}_1)$ be the formal power series

$$\begin{aligned} \sum c(\alpha _2,\alpha _1)e_{\alpha _2}(z_2)e_{\alpha _1}({\overline{z}}_1), \quad z_j \in {\mathbf {C}}^{d_j}, j=1,2, \end{aligned}$$

(2.33)

when $F(z_2,z_1)$ is the formal power series

$$\begin{aligned} \sum c(\alpha _2,\alpha _1)e_{\alpha _2}(z_2)e_{\alpha _1}(z_1), \quad z_j \in {\mathbf {C}}^{d_j}, \quad j = 1, 2. \end{aligned}$$

(2.34)

Here the sums should be taken over all $(\alpha _2,\alpha _1)\in {{\mathbf {N}}}^{d_2}\times {{\mathbf {N}}}^{d_1}$.

Definition 2.14

Let $s,s_1,s_2\in \overline{{\mathbf {R}}_{\flat ,\infty }}$, $W={\mathbf {C}}^{d_2}\times {\mathbf {C}}^{d_1}$ and $\Theta _{C}$ be the operator

$$\begin{aligned} (\Theta _{C}F)(z_2,z_1)=F(z_2,{\overline{z}}_1) \end{aligned}$$

(2.35)

between formal power series in (2.33) and (2.34), $z_j \in {\mathbf {C}}^{d_j}$, $j=1,2$. Then

(2.36)

and

(2.37)

are the images of (2.29) respectively (2.29’) under $\Theta _{C}$, and   is the image of A(W) under $\Theta _{C}$. The topologies of the spaces in (2.36), (2.37) and   are inherited from the topologies of the spaces (2.29), (2.29’) and A(W) respectively, through the map $\Theta _{C}$.

Remark 2.15

Let $W={\mathbf {C}}^{d_2}\times {\mathbf {C}}^{d_1}$. By letting $d_2=d$ and $d_1=0$, it follows that $A({\mathbf {C}}^{d})$ and the spaces in (2.29) can be considered as special cases of   and the spaces in (2.36).

Since ${\mathcal {A}}_{\flat _1}^\star ({\mathbf {C}}^{d}) = A({\mathbf {C}}^{d})$ and ${\mathcal {A}}_{0,\flat _1}^\star ({\mathbf {C}}^{d}) = A_d(\{ 0\} )$, it follows that

(2.38)

The subspaces of   in the following definition are important when considering analytic kernel operators which are continuous on the spaces in (2.29).

Definition 2.16

Let $s_1,s_2\in \overline{{\mathbf {R}}_{\flat ,\infty }}$, $W={\mathbf {C}}^{d_2}\times {\mathbf {C}}^{d_1}$, $T_{{\mathcal {A}}}$ and $\Theta _C$ be given by (2.27) and (2.35). Then

(2.39)

and

(2.40)

are the images of the spaces in (2.11) and (2.12) under the map $\Theta _C\circ T_{{\mathcal {A}}}$. The topologies of the spaces in (2.39) and (2.40) are inherited from the topologies of the corresponding spaces in (2.11) and (2.12).

For conveniency we set

In particular, if $W={\mathbf {C}}^{d}\times {\mathbf {C}}^{d}\simeq {\mathbf {C}}^{2d}$, then the spaces in (2.39) and (2.40) becomes

and

In Proposition 3.1 in Sect. 3 we present some duality properties of the spaces in Definitions 2.14 and 2.16. In Appendix A we identify the spaces in (2.39) and (2.40) with convenient subspaces of   , for suitable $s_1,s_2\in \overline{{\mathbf {R}}_\flat }$.

2.3 The Bargmann Transform and Spaces of Analytic Functions

Let $p\in [1,\infty ]$. Then the Bargmann transform ${\mathfrak {V}}_df$ of $f\in L^p({\mathbf {R}}^{d})$ is the entire function given by

$$\begin{aligned} ({\mathfrak {V}}_df)(z) =\pi ^{-\frac{d}{4}}\int _{{\mathbf {R}}^{d}}\exp \Big ( -\frac{1}{2}(\langle z,z\rangle +|y|^2)+2^{1/2}\langle z,y\rangle \Big )f(y)\, dy,\quad z \in {\mathbf {C}}^{d}. \end{aligned}$$

We have

$$\begin{aligned} ({\mathfrak {V}}_df)(z) =\int _{{\mathbf {R}}^{d}} {\mathfrak {A}}_d(z,y)f(y)\, dy, \quad z \in {\mathbf {C}}^{d}, \end{aligned}$$

or

$$\begin{aligned} ({\mathfrak {V}}_df)(z) =\langle f,{\mathfrak {A}}_d(z,\, \cdot \, )\rangle , \quad z \in {\mathbf {C}}^{d}, \end{aligned}$$

(2.41)

where the Bargmann kernel ${\mathfrak {A}}_d$ is given by

$$\begin{aligned} {\mathfrak {A}}_d(z,y)=\pi ^{-\frac{d}{4}} \exp \Big ( -\frac{1}{2}(\langle z,z\rangle +|y|^2)+2^{1/2}\langle z,y\rangle \Big ), \quad z \in {\mathbf {C}}^{d}, y \in {\mathbf {R}}^{d}. \end{aligned}$$

(Cf. [2, 3].) Here

$$\begin{aligned} \langle z,w\rangle = \sum _{j=1}^dz_jw_j\quad \text {and} \quad (z,w)= \langle z,{\overline{w}}\rangle \end{aligned}$$

when

$$\begin{aligned} z=(z_1,\dots ,z_d) \in {\mathbf {C}}^{d}\quad \text {and} \quad w=(w_1,\dots ,w_d)\in {\mathbf {C}}^{d}, \end{aligned}$$

and otherwise $\langle \, \cdot \, ,\, \cdot \, \rangle$ denotes the duality between test function spaces and their corresponding duals which is clear from the context. We note that the right-hand side of (2.41) makes sense when $f\in {\mathcal {S}}_{1/2}^{\,\prime }({\mathbf {R}}^{d})$ and defines an element in $A({\mathbf {C}}^{d})$, since $y\mapsto {\mathfrak {A}}_d(z,y)$ can be interpreted as an element in ${\mathcal {S}}_{1/2} ({\mathbf {R}}^{d})$ with values in $A({\mathbf {C}}^{d})$.

It was proved by Bargmann in [2] that $f\mapsto {\mathfrak {V}}_df$ is a bijective and isometric map from $L^2({\mathbf {R}}^{d})$ to the Hilbert space $A^2({\mathbf {C}}^{d})$, the set of entire functions F on ${\mathbf {C}}^{d}$ which fullfils

$$\begin{aligned} \Vert F\Vert _{A^2}\equiv \Big ( \int _{{\mathbf {C}}^{d}}|F(z)|^2d\mu (z) \Big )^{1/2}<\infty . \end{aligned}$$

(2.42)

Here $d\mu (z)=\pi ^{-d} e^{-|z|^2}\, d\lambda (z)$, where $d\lambda (z)$ is the Lebesgue measure on ${\mathbf {C}}^{d}$. The scalar product on $A^2({\mathbf {C}}^{d})$ is given by

$$\begin{aligned} (F,G)_{A^2}\equiv \int _{{\mathbf {C}}^{d}} F(z)\overline{G(z)}\, d\mu (z),\quad F,G\in A^2({\mathbf {C}}^{d}). \end{aligned}$$

(2.43)

For future references we note that the latter scalar product induces the bilinear form

$$\begin{aligned} (F,G)\mapsto \langle F,G\rangle _{A^2}=\langle F,G\rangle _{A^2({\mathbf {C}}^{d})}\equiv \int _{{\mathbf {C}}^{d}} F(z)G(z)\, d\mu (z) \end{aligned}$$

(2.44)

on $A^2({\mathbf {C}}^{d})\times \overline{A^2({\mathbf {C}}^{d})}$.

In [2] it was proved that the Bargmann transform maps the Hermite functions to monomials as

$$\begin{aligned} {\mathfrak {V}}_dh_\alpha = e_\alpha , \quad z\in {\mathbf {C}}^{d},\quad \alpha \in {{\mathbf {N}}}^{d} \end{aligned}$$

(2.45)

(cf. (2.26)). The orthonormal basis $\{ h_\alpha \}_{\alpha \in {{\mathbf {N}}}^{d}} \subseteq L^2({\mathbf {R}}^{d})$ is thus mapped to the orthonormal basis $\{ e_\alpha \} _{\alpha \in {{\mathbf {N}}}^{d}}\subseteq A^2({\mathbf {C}}^{d})$.

In particular it follows that the definition of the Bargmann transform from ${\mathcal {H}}_0({\mathbf {R}}^{d})$ to $A^2({\mathbf {C}}^{d})$ is uniquely extendable to a homeomorphism from ${\mathcal {H}}_0^\star ({\mathbf {R}}^{d})={\mathcal {H}}_0'({\mathbf {R}}^{d})$ to ${\mathcal {A}}_0^\star ({\mathbf {C}}^{d})$, by letting

$$\begin{aligned} {\mathfrak {V}}_d=T_{{\mathcal {A}}}\circ T_{{\mathcal {H}}}^{-1}, \end{aligned}$$

(2.46)

where $T_{{\mathcal {H}}}$ and $T_{{\mathcal {A}}}$ are given by (2.27). From the definitions of ${\mathcal {H}}_s({\mathbf {R}}^{d})$ to ${\mathcal {A}}_s({\mathbf {C}}^{d})$ and their duals it follows that the Bargmann transform restricts to homeomorphisms from ${\mathcal {H}}_s({\mathbf {R}}^{d})$ to ${\mathcal {A}}_s({\mathbf {C}}^{d})$ and from ${\mathcal {H}}_s^\star ({\mathbf {R}}^{d})$ to ${\mathcal {A}}_s^\star ({\mathbf {C}}^{d})$. Similar facts hold true with ${\mathcal {H}}_{0,s}$ and ${\mathcal {A}}_{0,s}$ in place of ${\mathcal {H}}_{s}$ and ${\mathcal {A}}_{s}$, respectively, at each occurrence. (Cf. [21].) In particular, Remark 2.9, (2.45) and (2.46) show that the latter spaces in (2.29) are the (strong) duals of the former spaces with respect to unique extensions of the form $(\, \cdot \, ,\, \cdot \, )_{A^2}$ on ${\mathcal {A}}_0({\mathbf {C}}^{d})$. That is,

$$\begin{aligned} {\mathcal {A}}_s'({\mathbf {C}}^{d}) = {\mathcal {A}}_s^\star ({\mathbf {C}}^{d}) \quad \text {and}\quad {\mathcal {A}}_{0,s} '({\mathbf {C}}^{d}) = {\mathcal {A}}_{0,s} ^\star ({\mathbf {C}}^{d}). \end{aligned}$$

(2.47)

Bargmann also proved that there is a reproducing formula for $A^2({\mathbf {C}}^{d})$. In fact, let $\Pi _A$ be the operator from $L^2(d\mu )$ to $A({\mathbf {C}}^{d})$, given by

$$\begin{aligned} (\Pi _AF)(z)= \int _{{\mathbf {C}}^{d}} F(w)e^{(z,w)}\, d\mu (w),\quad z \in {\mathbf {C}}^{d}. \end{aligned}$$

(2.48)

Then it is proved in [2] that $\Pi _A$ is an orthonormal projection from $L^2(d\mu )$ to $A^2({\mathbf {C}}^{d})$.

From now on we assume that $\phi$ in the short-time Fourier transform (2.3) is given by

$$\begin{aligned} \phi (x)=\pi ^{-\frac{d}{4}}e^{-|x|^2/2}, \quad x\in {\mathbf {R}}^{d}, \end{aligned}$$

(2.49)

if nothing else is stated. For such $\phi$, it follows by straight-forward computations that the relationship between the Bargmann transform and the short-time Fourier transform is given by

$$\begin{aligned} {\mathfrak {V}}_d = U_{{\mathfrak {V}}}\circ V_\phi ,\quad \text {and}\quad U_{{\mathfrak {V}}}^{-1} \circ {\mathfrak {V}}_d = V_\phi , \end{aligned}$$

(2.50)

where $U_{{\mathfrak {V}}}$ is the linear, continuous and bijective operator from ${\mathscr {D}}'({\mathbf {R}}^{2d})$ to ${\mathscr {D}}'({\mathbf {C}}^{d})$, given by

$$\begin{aligned} (U_{{\mathfrak {V}}}F)(x+i\xi ) = (2\pi )^{\frac{d}{2}} e^{\frac{1}{2}(|x|^2+|\xi |^2)}e^{-i\langle x,\xi \rangle } F(\sqrt{2}\, x,-\sqrt{2}\, \xi ), \quad x,\xi \in {\mathbf {R}}^{d}. \end{aligned}$$

(2.51)

We notice that the inverse of $U_{{\mathfrak {V}}}$ is given by

$$\begin{aligned} (U_{{\mathfrak {V}}}^{-1}F)(x,\xi ) = (2\pi )^{-\frac{d}{2}} e^{-\frac{1}{4}(|x|^2+|\xi |^2)} e^{-\frac{i}{2}\langle x,\xi \rangle } F\left( \frac{x-i\xi }{\sqrt{2}} \right) , \quad x,\xi \in {\mathbf {R}}^{d} \end{aligned}$$

(2.52)

(cf. [19]).

If $W={\mathbf {C}}^{d_2}\times {\mathbf {C}}^{d_1}$, then   $(W)$ consists of all $K\in C(W)$ such that $(z_2,z_1)\mapsto K(z_2,{\overline{z}}_1)$ belongs to $A^2(W)$. Then   $(W)$ is a Hilbert space under the scalar product

(2.53)

when

(2.54)

2.4 Wick and Anti-Wick Operators

Next we recall the definition of analytic pseudo-differential operators, so-called Wick operators. Suppose that and $F\in A({\mathbf {C}}^{d})$ satisfy

$$\begin{aligned} w\mapsto a(z,w)F(w)e^{r|w|-|w|^2} \in L^1({\mathbf {C}}^{d}) \end{aligned}$$

(2.55)

for every $r>0$ and $z\in {\mathbf {C}}^{d}$. Then the analytic pseudo-differential operator, or Wick operator ${\text {Op}}_{{\mathfrak {V}}}(a)$ with symbol a, is the linear and continuous operator from ${\mathcal {A}}_0({\mathbf {C}}^{d})$ to $A({\mathbf {C}}^{d})$, defined by the formula

$$\begin{aligned} {\text {Op}}_{{\mathfrak {V}}}(a)F(z)&= \int _{{\mathbf {C}}^{d}} a(z,w)F(w)e^{(z,w)}\, d\mu (w), \quad z \in {\mathbf {C}}^{d}, \end{aligned}$$

(2.56)

when $F\in {\mathcal {A}}_0({\mathbf {C}}^{d})$. (Cf. e. g. [4, 6, 17, 19, 21].) Here we remark that ${\text {Op}}_{{\mathfrak {V}}}(a)F$ is extendable in several ways, allowing a and F to belong to different spaces where (2.55) is violated. For example, in [17] it is proved that the definition of ${\text {Op}}_{{\mathfrak {V}}}(a)$ is uniquely extendable to any , and then ${\text {Op}}_{{\mathfrak {V}}}(a)$ is continuous from ${\mathcal {A}}_0({\mathbf {C}}^{d})$ to ${\mathcal {A}}_0^{\star }({\mathbf {C}}^{d})$.

The definition of the Wick operator in (2.56) resembles on the definition of the classical Kohn-Nirenberg pseudo-differential operators on ${\mathbf {R}}^{d}$. Let A be a real $d\times d$ matrix. Then the pseudo-differential operator ${\text {Op}}_A({\mathfrak {a}})$ with symbol ${\mathfrak {a}}\in {\mathcal {S}}_{1/2} ({\mathbf {R}}^{2d})$ is the linear and continuous operator on ${\mathcal {S}}_{1/2} ({\mathbf {R}}^{d})$, given by

$$\begin{aligned} ({\text {Op}}_A({\mathfrak {a}})f)(x) = (2\pi ) ^{-d}\iint {\mathfrak {a}}(x-A(x-y),\xi ) f(y)e^{i\langle x-y,\xi \rangle }\, dyd\xi , \quad x\in {\mathbf {R}}^{d}. \end{aligned}$$

The definition of ${\text {Op}}_A({\mathfrak {a}})$ extends to any ${\mathfrak {a}}\in \mathcal {S}^{\,\prime }_{1/2}({\mathbf {R}}^{2d})$, and then ${\text {Op}}_A({\mathfrak {a}})$ is continuous from ${\mathcal {S}}_{1/2}({\mathbf {R}}^{d})$ to $\mathcal {S}^{\,\prime }_{1/2}({\mathbf {R}}^{d})$. (See e. g. [9, 20] and the references therein.)

The normal (Kohn-Nirenberg) representation and the Weyl quantization are obtained by choosing $A=0$ and $A=\frac{1}{2}I$ respectively, where $I=I_d$ is the $d\times d$ identity matrix.

In the literature it is also common to consider anti-Wick operators. Suppose that satisfies

$$\begin{aligned} w\mapsto a(w,w)e^{r|w|-|w|^2} \in L^1({\mathbf {C}}^{d}) \end{aligned}$$

(2.57)

for every $r>0$. Then the anti-Wick operator ${\text {Op}}_{{\mathfrak {V}}}^{{\text {aw}}}(a)$ with symbol a is the linear and continuous operator from ${\mathcal {A}}_0({\mathbf {C}}^{d})$ to $A({\mathbf {C}}^{d})$, given by

$$\begin{aligned} {\text {Op}}_{{\mathfrak {V}}}^{{\text {aw}}}(a)F(z)&= \int _{{\mathbf {C}}^{d}} a(w,w)F(w)e^{(z,w)}\, d\mu (w), \quad z \in {\mathbf {C}}^{d}, \end{aligned}$$

(2.58)

when $F\in {\mathcal {A}}_0({\mathbf {C}}^{d})$.

Berezin established in [4] convenient links between Wick and anti-Wick operators (see page 587 in [4]). More precisely, if , then

$$\begin{aligned} {\text {Op}}_{{\mathfrak {V}}}^{{\text {aw}}}(a)&= {\text {Op}}_{{\mathfrak {V}}}(a^{{\text {aw}}}) \end{aligned}$$

(2.59)

when $a^{{\text {aw}}}$ is given by

$$\begin{aligned} a^{{\text {aw}}} (z,w)&= \pi ^{-d} \int _{{\mathbf {C}}^{d}}a(w_1,w_1)e^{-(z-w_1,w-w_1)} \, d\lambda (w_1). \end{aligned}$$

(2.60)

If instead , then (2.59) holds true when is given by

$$\begin{aligned} a(z,-w)&= \pi ^{-d} \int _{{\mathbf {C}}^{d}}a^{{\text {aw}}}(w_1,-w_1)e^{-(z-w_1,w-w_1)}\, d\lambda (w_1). \end{aligned}$$

(2.61)

In fact, if $F\in {\mathcal {A}}_0({\mathbf {C}}^{d})$ and $a^{{\text {aw}}}$ is given by (2.60), then

$$\begin{aligned} {\text {Op}}_{{\mathfrak {V}}}(a^{{\text {aw}}})F(z) {=} \pi ^{-d}\int _{{\mathbf {C}}^{d}} a(w_1,w_1)e^{(z,w_1){-}|w_1|^2} \left( \int _{{\mathbf {C}}^{d}}F(w)e^{(w_1,w)}\, d\mu (w) \right) \, d\lambda (w_1), \end{aligned}$$

and by applying the reproducing formula (2.48) on the inner integral, it follows that the right-hand side becomes ${\text {Op}}_{{\mathfrak {V}}}^{{\text {aw}}}(a)F(z)$ in (2.58). This shows that (2.59) holds true when $a^{{\text {aw}}}$ is given by (2.60). In Sects. 4 and 5 we show the equivalence between (2.60) and (2.61), when a or $a^{{\text {aw}}}$ are allowed to belong to larger classes than   .

Since any element a(z, w) in   is uniquely determined on its values at the diagonal $z=w$ or anti-diagonal $z=-w$, it follows that (2.60) and (2.61) are equivalent to

and

These formulae were especially emphasized on page 587 in [4].

Let . For suitable additional conditions on $a_1$ and $a_2$, it is possible to compose the Wick operators ${\text {Op}}_{{\mathfrak {V}}}(a_1)$ and ${\text {Op}}_{{\mathfrak {V}}}(a_2)$, and the resulting operator is again a Wick operator. The (complex) twisted product $a_1 \#_{{\mathfrak {V}}}a_2$ is then defined by

(2.62)

provided the composition on the left-hand side is well-defined as a continuous operator from ${\mathcal {A}}_0({\mathbf {C}}^{d})$ to ${\mathcal {A}}_0^{\star }({\mathbf {C}}^{d})$. By straight-forward computations it follows that the product $\#_{{\mathfrak {V}}}$ is given by

$$\begin{aligned}&a_1 \#_{{\mathfrak {V}}} a_2 (z,w) \nonumber \\&\quad = \pi ^{-d}\int _{{\mathbf {C}}^{d}} a_1(z,u) a_2 (u,w) e^{-(z-u,w-u)}\, d \lambda (u), \quad z,w \in {\mathbf {C}}^{d}, \end{aligned}$$

(2.63)

when the integrand belongs to $L^1({\mathbf {C}}^{d})$ for every $z,w \in {\mathbf {C}}^{d}$ (see e. g. [4, 18]).

2.5 Toeplitz Operators

Let $\phi _1,\phi _2\in {\mathcal {S}}_{1/2}({\mathbf {R}}^{d})\setminus 0$. Then the Toeplitz operator ${\text {Tp}}_{\phi _1,\phi _2}({\mathfrak {a}})$ with symbol ${\mathfrak {a}}\in \mathcal {S}^{\,\prime }_{1/2}({\mathbf {R}}^{2d})$ is the linear and continuous operator from ${\mathcal {S}}_{1/2}({\mathbf {R}}^{d})$ to $\mathcal {S}_{1/2}^{\,\prime }({\mathbf {R}}^{d})$, given by the formula

$$\begin{aligned} ({\text {Tp}}_{\phi _1,\phi _2}({\mathfrak {a}})f,g)_{L^2({\mathbf {R}}^{d})} = ({\mathfrak {a}}\cdot V_{\phi _1} f,V_{\phi _2} g)_{L^2({\mathbf {R}}^{2d})}. \end{aligned}$$

(2.64)

It is proved in [21] that ${\text {Tp}}_{\phi _1,\phi _2}({\mathfrak {a}})$ is continuous on ${\mathcal {S}}_{1/2}({\mathbf {R}}^{d})$ and uniquely extendable to a continuous operator on $\mathcal {S}^{\,\prime }_{1/2}({\mathbf {R}}^{d})$.

In our situation we have $\phi _1=\phi _2$ is equal to $\phi$ in (2.49), and for conveniency we put ${\text {Tp}}({\mathfrak {a}})={\text {Tp}}_{\phi ,\phi }({\mathfrak {a}})$. In fact, for suitable we shall mainly consider modified Toeplitz operators ${\text {Tp}}_{{\mathfrak {V}}}(a)$ given by

By straight-forward applications of (2.51) and (2.52) it follows

$$\begin{aligned} {\text {Op}}_{{\mathfrak {V}}}^{{\text {aw}}}(a) = {\mathfrak {V}}_d\circ {\text {Tp}}_{{\mathfrak {V}}}(a)\circ {\mathfrak {V}}_d^{-1}. \end{aligned}$$

(2.65)

(See also Section 6 in [19].)

We recall that Toeplitz operators can be formulated as pseudo-differential operators, due to the formula

$$\begin{aligned} {\text {Tp}}({\mathfrak {a}}) = \left( \frac{2}{\pi }\right) ^{\frac{d}{2}} {\text {Op}}^w({\mathfrak {a}}*e^{-|\, \cdot \, |^2}), \end{aligned}$$

(2.66)

which is equivalent to (2.60) and (2.61). (See e. g. [6, 15, 19] and the references therein.)

3 Operator Kernels and Multiplications for Spaces of Power Series Expansions

In this section we deduce kernel theorems for linear operators acting between (different) ${\mathcal {A}}_s$ spaces and their duals. In [17] such kernel results were explained for linear operators which map ${\mathcal {A}}_s({\mathbf {C}}^{d_1})$ into ${\mathcal {A}}_s^{\star }({\mathbf {C}}^{d_2})$ or which map ${\mathcal {A}}_{0,s}({\mathbf {C}}^{d_1})$ into ${\mathcal {A}}_{0,s}^{\star }({\mathbf {C}}^{d_2})$. The latter results can also be obtained by classical kernel results for linear operators acting on nuclear spaces (see e. g. [22]).

3.1 Kernels to linear operators acting on spaces of power series expansions

The continuity results for kernel operators rely on duality properties between the spaces through the   form in (2.53) and (2.54). In fact, we have the following proposition which describes important topological properties of the spaces in Definitions 2.14 and 2.16, and which is a straight-forward consequence of Remarks 2.8 and 2.9, and the definitions. The details are left for the reader.

Proposition 3.1

Let $s_0,s_1,s_2\in \overline{{\mathbf {R}}_\flat }$, $s=(s_2,s_1)$, $W ={\mathbf {C}}^{d_2}\times {\mathbf {C}}^{d_1}$ and let when . Then the following is true:

• (1) the spaces in (2.37) are complete Hausdorff topological vector spaces, and   and   are Fréchet spaces. It holds   and    ;

• (2) is dense in the spaces in (2.37), (2.39) and (2.40);

• (3) the map T from   to ${\mathbf {C}}$ is uniquely extendable to separately continuous and non-degenerate map**s from

  

  (3.1)

  to ${\mathbf {C}}$.

• (4) the map T is hypocontinuous from   to and   are (strong) duals to each others, through the form .

If instead $s_1,s_2\in {\mathbf {R}}_{\flat ,\infty }$, then the same holds true with   ,   and   in place of  ,   and   , respectively, at each occurrence.

Evidently, it follows from (3) in Proposition 3.1 that the duals of   and   are given by

(3.2)

through the form .

The following kernel results characterize linear operators which map (different) ${\mathcal {A}}_s$ or ${\mathcal {A}}_{0,s}$ spaces and their duals into each others. Here and in what follows we let ${\mathcal {L}}(V_1;V_2)$ be the set of linear and continuous map**s from the topological vector space $V_1$ into the topological vector space $V_2$. We also set ${\mathcal {L}}(V)={\mathcal {L}}(V;V)$.

Proposition 3.2

Let $s_1,s_2\in \overline{{\mathbf {R}}_{\flat ,\infty }}$, $s=(s_2,s_1)$, $W={\mathbf {C}}^{d_2}\times {\mathbf {C}}^{d_1}$, and let $T\in {\mathcal {L}}({\mathcal {A}}_0({\mathbf {C}}^{d_1});{\mathcal {A}}_0^{\star }({\mathbf {C}}^{d_2}))$. Then the following is true:

• (1) if $T\in {\mathcal {L}}({\mathcal {A}}_{s_1}^{\star }({\mathbf {C}}^{d_1});{\mathcal {A}}_{s_2}({\mathbf {C}}^{d_2}))$, then there is a unique such that

  $$\begin{aligned} (TF)(z_2) = (F,\, \overline{K(z_2,\, \cdot \, )}\, )_{A^2({\mathbf {C}}^{d_1})}, \qquad z_2\in {\mathbf {C}}^{d_2}, \end{aligned}$$

  (3.3)

  holds true for every $F\in {\mathcal {A}}_{s_1}({\mathbf {C}}^{d_1})$;

• (2) if $T\in {\mathcal {L}}({\mathcal {A}}_{s_1}({\mathbf {C}}^{d_1});{\mathcal {A}}_{s_2}^{\star }({\mathbf {C}}^{d_2}))$, then there is a unique such that (3.3) holds true for every $F\in {\mathcal {A}}_{s_1}'({\mathbf {C}}^{d_1})$.

The same holds true with ${\mathcal {A}}_{0,s_j}$ and   in place of ${\mathcal {A}}_{s_j}$ and   , respectively, $j=1,2$, at each occurrence.

Here (3.3) should be interpreted as

when $F\in {\mathcal {A}}_0({\mathbf {C}}^{d_1})$ and $G\in {\mathcal {A}}_0({\mathbf {C}}^{d_2})$.

Proposition 3.3

Let $s_1,s_2\in \overline{{\mathbf {R}}_{\flat ,\infty }}$, $s=(s_2,s_1)$, $W={\mathbf {C}}^{d_2}\times {\mathbf {C}}^{d_1}$, , and let $T=T_K\in {\mathcal {L}}({\mathcal {A}}_0({\mathbf {C}}^{d_1});{\mathcal {A}}^{\star }_0({\mathbf {C}}^{d_2}))$ be given by

$$\begin{aligned} F\mapsto T_KF = \big ( z_2\mapsto (F,\overline{K(z_2,\, \cdot \, )}) _{A^2({\mathbf {C}}^{d_1})} \big ). \end{aligned}$$

(3.4)

Then the following is true:

• (1) if , then $T_K$ extends uniquely to a linear and continuous map from ${\mathcal {A}}_{s_1}'({\mathbf {C}}^{d_1})$ to ${\mathcal {A}}_{s_2}({\mathbf {C}}^{d_2})$;

• (2) if , then $T_K$ extends uniquely to a linear and continuous map from ${\mathcal {A}}_{s_1}({\mathbf {C}}^{d_1})$ to ${\mathcal {A}}_{s_2}^{\star }({\mathbf {C}}^{d_2})$.

The same holds true with ${\mathcal {A}}_{0,s_j}$ and   in place of ${\mathcal {A}}_{s_j}$ and   , respectively, $j=1,2$, at each occurrence.

We observe that if K and F are the same as in (3.4) and $z_1\mapsto K(z_2,z_1)F(z_1)$ is integrable with respect to $d\mu (z_1)$, then (3.4) is the same as

Propositions 3.2 and 3.3 follow essentially from abstract kernel results for linear operators acting between topological vector spaces. (See e. g. [22]. See also [17] for more explicit approaches.) On the other hand, the following result might be more cumbersome to deduce from such abstract kernel results.

Proposition 3.4

Let $s_1,s_2\in \overline{{\mathbf {R}}_{\flat ,\infty }}$, $s=(s_2,s_1)$, $W={\mathbf {C}}^{d_2}\times {\mathbf {C}}^{d_1}$, and let $T\in {\mathcal {L}}({\mathcal {A}}_0({\mathbf {C}}^{d_1});{\mathcal {A}}_0^{\star }({\mathbf {C}}^{d_2}))$. Then the following is true:

• (1) if $T\in {\mathcal {L}}({\mathcal {A}}_{s_1}({\mathbf {C}}^{d_1});{\mathcal {A}}_{s_2}({\mathbf {C}}^{d_2}))$, then there is a unique such that (3.3) holds true for every $F\in {\mathcal {A}}_{s_1}({\mathbf {C}}^{d_1})$;

• (2) if $T\in {\mathcal {L}}({\mathcal {A}}_{s_1}^{\star }({\mathbf {C}}^{d_1});{\mathcal {A}}_{s_2}^{\star }({\mathbf {C}}^{d_2}))$, then there is a unique such that (3.3) holds true for every $F\in {\mathcal {A}}_{s_1}'({\mathbf {C}}^{d_1})$.

The same holds true with ${\mathcal {A}}_{0,s_j}$ and   in place of ${\mathcal {A}}_{s_j}$ and   , respectively, $j=1,2$, at each occurrence.

We also have the following converse of the preceding proposition.

Proposition 3.5

Let $s_1,s_2\in \overline{{\mathbf {R}}_{\flat ,\infty }}$, $s=(s_2,s_1)$, $W={\mathbf {C}}^{d_2}\times {\mathbf {C}}^{d_1}$, , and let $T=T_K\in {\mathcal {L}}({\mathcal {A}}_0({\mathbf {C}}^{d_1});{\mathcal {A}}_0^{\star }({\mathbf {C}}^{d_2}))$ be given by (3.4). Then the following is true:

• (1) if , then $T_K$ extends uniquely to a linear and continuous map from ${\mathcal {A}}_{s_1}({\mathbf {C}}^{d_1})$ to ${\mathcal {A}}_{s_2}({\mathbf {C}}^{d_2})$;

• (2) if , then $T_K$ extends uniquely to a linear and continuous map from ${\mathcal {A}}_{s_1}^{\star }({\mathbf {C}}^{d_1})$ to ${\mathcal {A}}_{s_2}^{\star }({\mathbf {C}}^{d_2})$.

The same holds true with ${\mathcal {A}}_{0,s_j}$ and   in place of ${\mathcal {A}}_{s_j}$ and   , respectively, $j=1,2$, at each occurrence.

By combining Propositions 3.2–3.5, we get the following.

Corollary 3.6

Let $s_1,s_2 \in \overline{{\mathbf {R}}_\flat }$, $s=(s_1,s_2)$, $W={\mathbf {C}}^{d_2}\times {\mathbf {C}}^{d_1}$ and let $T_K$ be the map in (3.4) when   . Then the map**s

and

are isomorphisms. The same holds true with   ,   , ${\mathcal {A}}_{0,s_1}$ and ${\mathcal {A}}_{0,s_2}$ in place of   ,   , ${\mathcal {A}}_{s_1}$ and ${\mathcal {A}}_{s_2}$, respectively, at each occurrence.

Proposition 3.5 follows by straight-forward computations and is left for the reader.

Proof of Proposition 3.4

We only consider the case when T is continuous from ${\mathcal {A}}_{s_1}({\mathbf {C}}^{d_1})$ to ${\mathcal {A}}_{s_2}({\mathbf {C}}^{d_2})$. The other cases follow by similar arguments and are left for the reader. Let $\vartheta _{r,s}$ be as in Definition 2.3. By Propositions 3.2 and 3.3, there is a unique

(3.5)

such that (3.3) holds true for every $F\in {\mathcal {A}}_0({\mathbf {C}}^{d_1})$. We need to show that .

Let $r>0$ and set $F_{\alpha }(z_1) = \vartheta _{r,s_1}^{-1}(\alpha )e_\alpha (z_1)$ for every $\alpha \in {{\mathbf {N}}}^{d_1}$ and $z_1\in {\mathbf {Z}}^{d_1}$. Then $\{ F_\alpha \} _{\alpha \in {{\mathbf {N}}}^{d_1}}$ is a bounded subset of ${\mathcal {A}}_{s_1}({\mathbf {C}}^{d_1})$. By the continuity of T it follows that $\{ TF_\alpha \} _{\alpha \in {{\mathbf {N}}}^{d_1}}$ is a bounded subset of ${\mathcal {A}}_{s_2}({\mathbf {C}}^{d_2})$. Hence, if

$$\begin{aligned} (TF_\alpha )(z_2) = \sum _{\alpha _2\in {{\mathbf {N}}}^{d_2}}c(TF_\alpha ;\alpha _2) e_{\alpha _2}(z_2), \qquad z_2\in {\mathbf {C}}^{d_2}, \end{aligned}$$

then

$$\begin{aligned} c(TF_\alpha ;\alpha _2) = c(K;\alpha _2,\alpha )\vartheta _{r,s_1}^{-1}(\alpha ) \end{aligned}$$

satisfies

$$\begin{aligned} |c(TF_\alpha ;\alpha _2)|\le C\vartheta _{r_0,s_2}^{-1}(\alpha _2), \end{aligned}$$

for some constants $C,r_0>0$ which are independent of $\alpha _2$ and $\alpha$. This gives,

$$\begin{aligned} |c(K;\alpha _2,\alpha _1)| \lesssim \vartheta _{r,s_1}(\alpha _1)\vartheta _{r_0,s_2}^{-1}(\alpha _2), \qquad \alpha _j\in {\mathbf {C}}^{d_j}. \end{aligned}$$

Since $r>0$ is arbitrary, the latter estimate implies that . $\square$

In Sect. 5 we deduce related results compared to Propositions 3.2–3.5 and Corollary 3.6 with Wick and anti-Wick operators in place of kernel operators.

Example 3.7

Let $\omega \in L^\infty _{{\text {loc}}}({\mathbf {C}}^{d})\simeq L^\infty _{{\text {loc}}}({\mathbf {R}}^{2d})$ be such that

$$\begin{aligned} \omega (z)>0 \quad \text {and}\quad \omega (z+w)\lesssim \omega (z)\langle w\rangle ^r,\qquad z,w\in {\mathbf {C}}^{d}, \end{aligned}$$

(3.6)

for some $r>0$. Then we recall that the Shubin class ${\text {Sh}}^{(\omega )}_\rho ({\mathbf {R}}^{2d})$ (with respect to $\rho$ and $\omega$) is the set of all ${\mathfrak {a}}\in C^\infty ({\mathbf {R}}^{2d})$ such that

$$\begin{aligned} |\partial ^\alpha {\mathfrak {a}}(x,\xi )|\lesssim \omega (x+i\xi )\langle (x,\xi )\rangle ^{-\rho |\alpha |}. \end{aligned}$$

(See [15].)

As in [18], we let   be the class of Wick symbols which consists of all such that

$$\begin{aligned} |\partial _z^\alpha {\overline{\partial }} _w^\beta a(z,w)| \lesssim e^{\frac{1}{2}|z-w|^2}\omega (\sqrt{2}{\overline{z}})\langle z+w\rangle ^{-\rho |\alpha +\beta |} \langle z-w\rangle ^{-N} \end{aligned}$$

(3.7)

for every $\alpha ,\beta \in {{\mathbf {N}}}^{d}$ and $N\ge 0$. Let A be a real $d\times d$ matrix. Then recall that for ${\mathfrak {a}}\in {\mathscr {S}}'({\mathbf {R}}^{2d})$, we have ${\mathfrak {a}}\in {\text {Sh}}^{(\omega )}_\rho ({\mathbf {R}}^{2d})$, if and only if ${\text {Op}}_{{\mathfrak {V}}}(a)={\mathfrak {V}}_d \circ {\text {Op}}_A({\mathfrak {a}})\circ {\mathfrak {V}}_d^{-1}$ for some (cf. [18]).

Let and let K be the kernel of ${\text {Op}}_{{\mathfrak {V}}}(a)$. We claim that

(3.8)

In fact, by (3.7) and that $K(z,w)=e^{(z,w)}a(z,w)$ we get

$$\begin{aligned} |K(z,w)| \lesssim e^{\frac{1}{2}(|z|^2+|w|^2)}\omega (\sqrt{2}{\overline{z}}) \langle z-w\rangle ^{-N} \quad \text {for every}\ N\ge 0. \end{aligned}$$

(3.9)

Since $\omega$ satisfies (3.6) we get

$$\begin{aligned} \omega (\sqrt{2}{\overline{z}}) \lesssim \langle z\rangle ^{N_0} \quad \text {and}\quad \omega (\sqrt{2}{\overline{z}}) \lesssim \omega (\sqrt{2}{\overline{w}})\langle z-w\rangle ^{N_0} \lesssim \langle w\rangle ^{N_0}\langle z-w\rangle ^{N_0}, \end{aligned}$$

for some $N_0\ge 0$. We also have

$$\begin{aligned} \langle z-w\rangle ^{-N} \lesssim \langle z\rangle ^{-N}\langle w\rangle ^N \quad \text {and}\quad \langle z-w\rangle ^{-N} \lesssim \langle z\rangle ^{N}\langle w\rangle ^{-N}. \end{aligned}$$

By playing with $N\ge 0$, a combination of these estimates with (3.9) shows that

$$\begin{aligned} |K(z,w)|&\lesssim e^{\frac{1}{2}(|z|^2+|w|^2)} \langle z\rangle ^{N_0+N}\langle w\rangle ^{-N} \end{aligned}$$

and

$$\begin{aligned} |K(z,w)|&\lesssim e^{\frac{1}{2}(|z|^2+|w|^2)} \langle z\rangle ^{-N}\langle w\rangle ^{N_0+N}, \end{aligned}$$

which implies that (3.8) holds.

By Proposition 3.4 it now follows that ${\text {Op}}_{{\mathfrak {V}}}(a)$ is continuous on both ${\mathcal {A}}_{0,\infty }({\mathbf {C}}^{d})$ and on ${\mathcal {A}}_{0,\infty }^{\star }({\mathbf {C}}^{d})$. This shows that any Shubin operator is continuous on ${\mathscr {S}}({\mathbf {R}}^{d})$ and on ${\mathscr {S}}'({\mathbf {R}}^{d})$ (see e. g. [9, Theorem 18.6.2]).

3.2 Compositions of Analytic Kernel Operators

In what follows we identify a large class of linear and continuous operator T from ${\mathcal {A}}_0({\mathbf {C}}^{d_1})$ to ${\mathcal {A}}_0^{\star }({\mathbf {C}}^{d_2})$ by its kernel in (3.3) and (3.3’). By general continuity properties we get the following theorem concerning the composition map

$$\begin{aligned} (K_2,K_1) \mapsto K_2\circ K_1. \end{aligned}$$

(3.10)

Theorem 3.8

Let $W_j={\mathbf {C}}^{d_{j+1}}\times {\mathbf {C}}^{d_j}$, $j=1,2$, $W_3={\mathbf {C}}^{d_3}\times {\mathbf {C}}^{d_1}$, $s_1,s_2,s_3\in \overline{{\mathbf {R}}_{\flat }}$ and let T be the map from   to   , given by (3.10). Then the following is true:

• (1) T is uniquely extendable to separately continuous bilinear map**s from   or from   to   ;

• (2) T is uniquely extendable to separately continuous bilinear map**s from   or from   ;

If instead $s_1,s_2,s_3\in {\mathbf {R}}_{\flat ,\infty }$, then the same hold true with   and   in place of   and   , respectively, at each occurrence.

Theorem 3.9

Let $W_j={\mathbf {C}}^{d_{j+1}}\times {\mathbf {C}}^{d_j}$, $j=1,2$, $W_3={\mathbf {C}}^{d_3}\times {\mathbf {C}}^{d_1}$, $s_1,s_2,s_3\in \overline{{\mathbf {R}}_{\flat }}$ and let T be the map from   to   , given by (3.10). Then the following is true:

• (1) T is uniquely extendable to separately continuous bilinear map**s from   or from   to   ;

• (2) T is uniquely extendable to separately continuous bilinear map**s from   or from   to   ;

• (3) T is uniquely extendable to continuous bilinear map**s from   or from   to   .

If instead $s_1,s_2,s_3\in {\mathbf {R}}_{\flat ,\infty }$, then the same hold true with   ,   and   in place of   ,   and   , respectively, at each occurrence.

Proof of Theorems 3.8 and 3.9

The assertion, except for those parts which involve   and   spaces, follows from the kernel results Propositions 3.2 to 3.5 and the composition properties of the form

$$\begin{aligned} {\mathcal {L}}({\mathcal {A}}_{s_2}({\mathbf {C}}^{d_2}),{\mathcal {A}}_{s_3}({\mathbf {C}}^{d_3})) \circ {\mathcal {L}}({\mathcal {A}}_{s_1}({\mathbf {C}}^{d_1}),{\mathcal {A}}_{s_2}({\mathbf {C}}^{d_2})) \subseteq {\mathcal {L}}({\mathcal {A}}_{s_1}({\mathbf {C}}^{d_1}),{\mathcal {A}}_{s_3}({\mathbf {C}}^{d_3})) \end{aligned}$$

and

$$\begin{aligned} {\mathcal {L}}({\mathcal {A}}_{s_2}({\mathbf {C}}^{d_2}),{\mathcal {A}}_{s_3}^{\star }({\mathbf {C}}^{d_3})) \circ {\mathcal {L}}({\mathcal {A}}_{s_1}({\mathbf {C}}^{d_1}),{\mathcal {A}}_{s_2}({\mathbf {C}}^{d_2})) \subseteq {\mathcal {L}}({\mathcal {A}}_{s_1}({\mathbf {C}}^{d_1}),{\mathcal {A}}_{s_3}^{\star }({\mathbf {C}}^{d_3})). \end{aligned}$$

Next we prove   . Suppose and , and let $K_3=K_2\circ K_1$, with coefficients $c(K_1;\alpha _2,\alpha _1)$, $c(K_2;\alpha _3,\alpha _2)$ and $c(K_3;\alpha _3,\alpha _1)$, respectively, $\alpha _j\in {{\mathbf {N}}}^{d_j}$, in their power series expansions. Also let $\vartheta _{r,s}$ be the same as in Definition 2.3. Then

$$\begin{aligned} c(K_3;\alpha _3,\alpha _1) = \sum _{\beta \in {{\mathbf {N}}}^{d_2}} c(K_2;\alpha _3,\beta )c(K_1;\beta ,\alpha _1). \end{aligned}$$

By the definitions, there is an $r_{0,1}>0$ such that

$$\begin{aligned} |c(K_1;\alpha _2,\alpha _1)| \lesssim \frac{\vartheta _{r_{1},s_1}(\alpha _1)}{\vartheta _{r_{0,1},s_2}(\alpha _2)} \end{aligned}$$

holds for every $r_{1}>0$, and for every $r_{2}>0$ there is an $r_{0,2}>0$ such that

$$\begin{aligned} |c(K_2;\alpha _3,\alpha _2)| \lesssim \frac{\vartheta _{r_{2},s_2}(\alpha _2)}{\vartheta _{r_{0,2},s_3}(\alpha _3)} \end{aligned}$$

(3.11)

holds. Hence, if we let $r_{2}<r_{0,1}$ and choose $r_{0,2}>0$ such that (3.11) holds, then

$$\begin{aligned} |c(K_3;\alpha _3,\alpha _1)| \lesssim \sum _{\beta \in {{\mathbf {N}}}^{d_2}} \left( \frac{\vartheta _{r_{2},s_2}(\beta )}{\vartheta _{r_{0,1},s_2}(\beta )} \right) \frac{\vartheta _{r_{1},s_1}(\alpha _1)}{\vartheta _{r_{0,2},s_3}(\alpha _3)} \asymp \frac{\vartheta _{r_{1},s_1}(\alpha _1)}{\vartheta _{r_{0,2},s_3}(\alpha _3)} \end{aligned}$$

for every $r_{1}>0$. This is the same as , and the assertion follows. In the same way, we obtain the other assertions. The details are left for the reader. $\square$

In Sect. 5 we present an analogy of Theorem 3.9 for Wick operators (see Theorem 5.18).

4 Binomial Operators and Their Continuity Properties

In this section we consider linear and bilinear operators of binomial types which act on subspaces of $\ell _0^{\star }({{\mathbf {N}}}^{2d})$ and which contain binomial expressions. In the interplay between coefficients of Wick symbols and the kernels to Wick operators, the linear binomial operators are the actions on the coefficients in power series expansions, when passing between operator kernels, Wick symbols and anti-Wick symbols of linear operators. The bilinear binomial operators are in similar ways the actions on the coefficients in power series expansions, which correspond to pure multiplications, their adjoint actions, and compositions of Wick operators.

4.1 Binomial Type Operators

Let $t\in {\mathbf {C}}$. Essential parts of our investigations concerns continuity properties of the linear binomial operators

(4.1)

when $c\in \ell _0^{\star }({{\mathbf {N}}}^{2d})$ and $t \in {\text {C}}$, and their formal $\ell ^2$ adjoint, which are given by ${\mathcal {T}}_{0,{\overline{t}}} ^{*}c$, where

(4.2)

when $c\in \ell _0({{\mathbf {N}}}^{2d})$ and $t \in {\text {C}}$. In terms of the operator

$$\begin{aligned} (S_0c)(\alpha ,\beta )=i^{|\alpha +\beta |}c(\alpha ,\beta ), \qquad \alpha ,\beta \in {{\mathbf {N}}}^{d},\ c\in \ell _0^{\star }({{\mathbf {N}}}^{2d}), \end{aligned}$$

(4.3)

we observe that

$$\begin{aligned} {\mathcal {T}}_{0,-t} = S_0^{-1}\circ {\mathcal {T}}_{0,t} \circ S_0 \quad \text {and}\quad {\mathcal {T}}_{0,-t}^{*} = S_0^{-1}\circ {\mathcal {T}}_{0,t}^{*} \circ S_0 \end{aligned}$$

(4.4)

in the domains of ${\mathcal {T}}_{0,-t}$ and ${\mathcal {T}}_{0,-t}^{*}$.

The operators ${\mathcal {T}}_{0,1}$ and ${\mathcal {T}}_{0,1}^{*}$ are linked to transitions between kernel operators and Wick operators, and to transitions between Wick and anti-Wick operators. In fact, for any fixed $t\in {\mathbf {C}}$, let ${\mathcal {T}}_t$ be the map on   , given by

(4.5)

It follows in particular from (2.56), (3.4) and (3.4’) that

$$\begin{aligned} T_K = {\text {Op}}_{{\mathfrak {V}}}(a) \quad \Leftrightarrow \quad K={\mathcal {T}}_1a, \end{aligned}$$

(4.6)

when , because

As a consequence of the following proposition we have

$$\begin{aligned} ({\mathcal {T}}_{0,t}c)(a;\, \cdot \, ) = c({\mathcal {T}}_ta;\, \cdot \, ) \end{aligned}$$

(4.7)

when   , which gives the link between the operator ${\mathcal {T}}_{0,1}$ and transitions between kernel and Wick operators. We omit the proof because the result is essentially a restatement of [17, Theorem 2.6].

Proposition 4.1

Let $s,s_0\in \overline{{\mathbf {R}}_\flat }$ be such that $s<\frac{1}{2}$ and $s_0\le \frac{1}{2}$, $t\in {\mathbf {C}}$ and let $T_{{\mathcal {A}}}$ be the map in (2.27). Then ${\mathcal {T}}_t$ from   to   extends uniquely to continuous map**s on   and on   , and the diagrams

(4.8)

commute.

Here we observe the misprints in the commutative diagram (2.5) in [17], where $T_{{\mathcal {H}}}$ and   should be replaced by $T_{{\mathcal {A}}}$ and   , respectively, at each occurrence.

By (4.5), the commutative diagrams (4.8) and that ${\mathcal {T}}_{0,{\overline{t}}}^{*}$ is the $\ell ^2$ adjoint of ${\mathcal {T}}_{0,t}$ it follows that

$$\begin{aligned} \begin{aligned} {\mathcal {T}}_{0,t_1}\circ {\mathcal {T}}_{0,t_2}&= {\mathcal {T}}_{0,t_1+t_2},&\quad {\mathcal {T}}_{0,t}^{-1}&= {\mathcal {T}}_{0,-t},\\ {\mathcal {T}}_{0,t_1}^{*}\circ {\mathcal {T}}_{0,t_2}^{*}&= {\mathcal {T}}_{0,t_1+t_2}^{*},&\quad \text {and}\quad ({\mathcal {T}}_{0,t}^{*})^{-1}&= {\mathcal {T}}_{0,-t}^{*}, \quad t,t_1,t_2\in {\mathbf {C}}, \end{aligned} \end{aligned}$$

(4.9)

since similar facts hold true for the operator ${\mathcal {T}}_t$.

The operator ${\mathcal {T}}_{0,t}^{*}$ is linked to the operator ${\mathcal {T}}^{*}_t$, given by

$$\begin{aligned}&({\mathcal {T}}^{*}_ta)(t_0z,\overline{t_0}w)\nonumber \\&\quad = \pi ^{-d} \int _{{\mathbf {C}}^{d}}a({t_0}w_1,\overline{t_0}w_1) e^{-(z-w_1,w-w_1)}\, d\lambda (w_1),\quad t_0^2=t, \end{aligned}$$

(4.10)

when . We observe that due to the definitions it follows that $a^{{\text {aw}}} (z,w)$ in (2.59) is equal to ${\mathcal {T}}_1^{*}a(z,w)$. Hence,

$$\begin{aligned} {\text {Op}}_{{\mathfrak {V}}}({\mathcal {T}}_1^{*}a) = {\text {Op}}_{{\mathfrak {V}}}^{{\text {aw}}}(a) \quad \text {and}\quad {\text {Op}}_{{\mathfrak {V}}}^{{\text {aw}}}({\mathcal {T}}_{-1}^{*}a) = {\text {Op}}_{{\mathfrak {V}}}(a) \end{aligned}$$

(4.11)

when .

By a straight-forward consequence of (B.1) in Appendix B it follows that the diagram

(4.12)

commutes. A combination of (4.11) and (4.12) then gives

$$\begin{aligned} c(a^{{\text {aw}}};\, \cdot \, ) = {\mathcal {T}}_{0,1}^{*}c(a;\, \cdot \, ), \end{aligned}$$

(4.13)

when (2.59) holds and , which in turn is equivalent to

$$\begin{aligned} {\text {Op}}_{{\mathfrak {V}}}(b) = {\text {Op}}_{{\mathfrak {V}}}^{{\text {aw}}}(a) \quad \Leftrightarrow \quad c(b;\, \cdot \, ) = {\mathcal {T}}_{0,1}^{*}c(a;\, \cdot \, ), \end{aligned}$$

(4.14)

when . This gives the link between ${\mathcal {T}}_{0,\pm 1}^{*}$ and transitions between Wick and anti-Wick operators.

Next we introduce certain types of bilinear binomial operators.

Definition 4.2

Let $t\in {\mathbf {C}}$.

• (1) The bilinear operator $(c_1,c_2)\mapsto (c_1\bullet _t c_2)$ from $\ell _0({{\mathbf {N}}}^{d} )\times \ell _0({{\mathbf {N}}}^{d} )$ to $\ell _0({{\mathbf {N}}}^{d} )$ is given by

  

  (4.15)

• (2) The bilinear operator $(c_1,c_2)\mapsto (c_1\bullet _{(t,\diamond )} c_2)$ from $\ell _0({{\mathbf {N}}}^{d} )\times \ell _0({{\mathbf {N}}}^{d} )$ to $\ell _0({{\mathbf {N}}}^{d} )$ is given by

  

  (4.16)

• (3) The bilinear operator $(c_1,c_2)\mapsto (c_1\bullet _{(t,\#)} c_2)$ from $\ell _0({{\mathbf {N}}}^{2d} )\times \ell _0({{\mathbf {N}}}^{2d})$ to $\ell _0({{\mathbf {N}}}^{2d})$ is given by

  $$\begin{aligned}&(c_1\bullet _{(t,\#)} c_2)(\alpha ,\beta ) \nonumber \\&\equiv \sum {C}_{0}(\alpha , {\alpha }_{0}, \beta , {\beta }_{0}, \gamma )t^{|\gamma |} c_1(\alpha _0,\beta -\beta _0+\gamma ) c_2(\alpha -\alpha _0+\gamma ,\beta _0),\qquad \qquad \end{aligned}$$

  (4.17)

  where

  $$\begin{aligned} {C}_{0}(\alpha , {\alpha }_{0}, \beta , {\beta }_{0}, \gamma ) = \left( {{\alpha }\atopwithdelims (){\alpha _0}}{{\beta }\atopwithdelims (){\beta _0}} {{\alpha -\alpha _0+\gamma }\atopwithdelims (){\gamma }} {{\beta -\beta _0+\gamma }\atopwithdelims (){\gamma }} \right) ^\frac{1}{2}.\quad \quad \quad \end{aligned}$$

  (4.18)

  Here the sum is taken over all $\alpha _0,\beta _0,\gamma \in {{\mathbf {N}}}^{d}$ such that $\alpha _0\le \alpha$, $\beta _0\le \beta$.

It follows by straight-forward computations that

$$\begin{aligned} (c_1\bullet _t c_2)(\alpha )&= t^{|\alpha |}(c_2\bullet _{1/t} c_1)(\alpha ) \end{aligned}$$

(4.19)

and

$$\begin{aligned} (c_1\bullet _t c_2,c_3)_{\ell ^2({{\mathbf {N}}}^{d})}&= (c_2,\overline{c_1}\bullet _{({\overline{t}},\diamond )} c_3)_{\ell ^2({{\mathbf {N}}}^{d})}, \end{aligned}$$

(4.20)

when $c_j\in \ell _0(\Lambda )$, $j=1,2,3$.

The different products in Definition 4.2 are linked into different bilinear map**s for spaces of power series expansions. Let $W={\mathbf {C}}^{d_2}\times {\mathbf {C}}^{d_1}$ and . Then

$$\begin{aligned}&K_1(z_2,z_1)K_2(z_2,z_1) = \sum _{{\alpha }_{j},\beta _{j}} c(K_1;\alpha _{2},\alpha _{1}) c(K_2;\beta _{2},\beta _{1}) e_{\alpha _2}(z_2)e_{\beta _2}(z_2) e_{\alpha _1}(z_1)e_{\beta _1}(z_1)\\&\quad = \sum _{\alpha _{1}, \alpha _{2}} \left( \sum _{\gamma _j\le \alpha _j} \left( {{\alpha _1}\atopwithdelims (){\gamma _1}} {{\alpha _2}\atopwithdelims (){\gamma _2}} \right) ^{\frac{1}{2}} c(K_1;\alpha _{2}-\gamma _2,\alpha _{1}-\gamma _1) c(K_2;\gamma _{2},\gamma _{1}) \right) e_{\alpha _{2}}(z_2)e_{\alpha _{1}}(z_1), \end{aligned}$$

and it follows that

$$\begin{aligned} c(K_1\cdot K_2;\, \cdot \, ) = c(K_1;\, \cdot \, ) \bullet _1 c(K_2;\, \cdot \, ). \end{aligned}$$

(4.21)

By (4.20) it follows that $\bullet _{(1,\diamond )}$ is the adjoint operation of $\bullet _1$ on elements in $\ell _0(\Lambda )$. In order to find corresponding relationship for elements in   , we observe that the adjoint operation of $z_j$ is $\partial _{z_j}$. In fact, if $F,G\in {\mathcal {A}}_{0}({\mathbf {C}}^{d})$, then it follows by integration by parts that

$$\begin{aligned} (z_jF,G)_{A^2} = (F,\partial _jG)_{A^2} \quad \text {and}\quad (\partial _jF,G)_{A^2} = (F,z_jG)_{A^2}. \end{aligned}$$

(4.22)

Hence, by letting $\diamond$ be the multiplication on ${\mathcal {A}}_0({\mathbf {C}}^{d})$, given by the formula

$$\begin{aligned} (F_1\diamond F_2,F)_{A^2} = (F_2,F_0\cdot F)_{A^2}, \qquad F_0(z) = \overline{F_1({\overline{z}})}, \end{aligned}$$

(4.23)

when $F,F_1,F_2\in {\mathcal {A}}_0({\mathbf {C}}^{d})$, it follows that

$$\begin{aligned} (F_1\diamond F_2)(z) = \sum _{\alpha \in {{\mathbf {N}}}^{d}} c(F_1,\alpha )\frac{\partial _z^\alpha F_2(z)}{\alpha !^{1/2}}. \end{aligned}$$

(4.24)

Since ${\text {Op}}_{{\mathfrak {V}}}^{{\text {aw}}}({\overline{w}}_j)= \partial _j=\partial _{z_j}$, it also follows that

$$\begin{aligned} F_1\diamond F_2 ={\text {Op}}_{{\mathfrak {V}}}^{{\text {aw}}}(F_1)F_2=F_1(\nabla _z)F_2. \end{aligned}$$

In similar ways, if $W={\mathbf {C}}^{d_2}\times {\mathbf {C}}^{d_1}$ and , then $K_1\diamond K_2$ is defined by

(4.25)

It follows that

(4.26)

when . A combination of (4.20), (4.21) and (4.26), now gives

$$\begin{aligned} c(K_1\diamond K_2;\, \cdot \, ) = c(K_1;\, \cdot \, ) \bullet _{(1,\diamond )} c(K_2;\, \cdot \, ). \end{aligned}$$

(4.27)

Example 4.3

We observe that (4.24) and straight-forward computations give

(4.28)

The following lemma shows that $\bullet _{(1,\#)}$ in Definition 4.2 is linked to compositions of Wick operators on the symbol side.

Lemma 4.4

Let $W={\mathbf {C}}^{d}\times {\mathbf {C}}^{d}$, $\bullet _{(1,\#)}$ be the multiplication on $\ell _0({{\mathbf {N}}}^{d}\times {{\mathbf {N}}}^{d})$ given by (4.17), and let

Then the following is true:

• (1) if $\alpha _j,\beta _j\in {{\mathbf {N}}}^{d}$, $j=1,2$, then

  $$\begin{aligned}&u_{\alpha _1,\beta _1} \#_{{\mathfrak {V}}} u_{\alpha _2,\beta _2} \nonumber \\&\quad = \!\! \sum _{\gamma \le \alpha _2,\beta _1}\!\! \left( {{\alpha _2}\atopwithdelims (){\gamma }}{{\beta _1}\atopwithdelims (){\gamma }} {{\alpha _1+\alpha _2-\gamma }\atopwithdelims (){\alpha _1}} {{\beta _1+\beta _2-\gamma }\atopwithdelims (){\beta _2}} \right) ^{\frac{1}{2}} u_{\alpha _1+\alpha _2-\gamma ,\beta _1+\beta _2-\gamma } \text{; }\nonumber \\ \end{aligned}$$

  (4.29)

• (2) if , then

  $$\begin{aligned} c(a_1\#_{{\mathfrak {V}}}a_2;\, \cdot \, )= c(a_1;\, \cdot \, ) \bullet _{(1,\#)} c(a_2;\, \cdot \, ). \end{aligned}$$

  (4.30)

Proof

By using that

$$\begin{aligned} I_{\gamma ,\delta } = \pi ^{-d}\int _{{\mathbf {C}}^{d}}w^\gamma {{\overline{w}}}^\delta e^{-|w|^2}\, d\lambda (w) = {\left\{ \begin{array}{ll} \gamma ! , &{} \gamma =\delta \\ 0 , &{} \gamma \ne \delta , \end{array}\right. } \end{aligned}$$

we get

$$\begin{aligned}\pi ^{-d}\int _{{\mathbf {C}}^{d}} e_\alpha (w_1)e_\beta ({\overline{w}}_1)&e^{-|z-w_1|^2}\, d\lambda (w_1)\nonumber \\&\quad = \frac{1}{\pi ^{d}(\alpha !\beta !)^{\frac{1}{2}}}\sum _{\gamma \le \alpha } \sum _{\delta \le \beta } {{\alpha }\atopwithdelims (){\gamma }}{{\beta }\atopwithdelims (){\delta }} z^{\alpha -\gamma }{{\overline{z}}}^{\beta -\delta }I_{\gamma ,\delta }\\&\quad = \sum _{\gamma \le \alpha ,\beta } \left( {{\alpha }\atopwithdelims (){\gamma }}{{\beta }\atopwithdelims (){\gamma }} \right) ^{\frac{1}{2}} e_{\alpha -\gamma }(z)e_{\beta -\delta }({\overline{z}}). \end{aligned}$$

This gives

$$\begin{aligned} \pi ^{-d}\int _{{\mathbf {C}}^{d}} e_\alpha (w_1)&e_\beta ({\overline{w}}_1) e^{-(z-w_1,w-w_1)} d\lambda (w_1) \nonumber \\&= \sum _{\gamma \le \alpha ,\beta } \left( {{\alpha }\atopwithdelims (){\gamma }}{{\beta }\atopwithdelims (){\gamma }} \right) ^{\frac{1}{2}} e_{\alpha -\gamma }(z)e_{\beta -\delta }({\overline{w}}). \end{aligned}$$

(4.31)

Hence, if $F_{\alpha ,\beta }(z,w)$ is the left-hand side of (4.31), then we obtain

$$\begin{aligned} (u_{\alpha _1,\beta _1}\#_{{\mathfrak {V}}} u_{\alpha _2,\beta _2})(z,w)&= e_{\alpha _1}(z)e_{\beta _2}({\overline{w}})F_{\alpha _2,\beta _1}(z,w)\\&= \sum _{\gamma \le \alpha _2,\beta _1} \left( {{\alpha _2}\atopwithdelims (){\gamma }}{{\beta _1}\atopwithdelims (){\gamma }} \right) ^{\frac{1}{2}} (e_{\alpha _2-\gamma }(z)e_{\alpha _1}(z)) (e_{\beta _1-\delta }({\overline{w}})e_{\beta _2}({\overline{w}})), \end{aligned}$$

which after some straight-forward computations lead to the right-hand side of (4.29). This gives (1).

Let

and

for admissible $\alpha =(\alpha _1,\alpha _2)\in {{\mathbf {N}}}^{2d}$, $\beta =(\beta _1,\beta _2)\in {{\mathbf {N}}}^{2d}$ and $\gamma \in {{\mathbf {N}}}^{d}$. By (1) we have

$$\begin{aligned}&(a_1\#_{{\mathfrak {V}}}a_2)(z,w)\\&\quad = \sum _{\alpha _j,\beta _j} \sum _{\gamma \le \alpha _2,\beta _1} C_1(\alpha ,\beta ,\gamma )c(a_1;\alpha _1,\beta _1)c(a_2;\alpha _2,\beta _2) e_{\alpha _1+\alpha _2-\gamma }(z)e_{\beta _1+\beta _2-\gamma }({\overline{w}}) \\&\quad =\sum C_1(\alpha ,\beta ,\gamma )c(a_1;\alpha _1,\beta _1)c(a_2;\alpha _2,\beta _2) e_{\alpha _1+\alpha _2-\gamma }(z)e_{\beta _1+\beta _2-\gamma }({\overline{w}}), \end{aligned}$$

where the last sum is taken over all ${\alpha _{1}}, {\alpha }_{2}, \beta _{1}, {\beta }_{2}, \gamma \in {\text {N}^{d}}$ such that $\gamma \le {\alpha _{2}}, \beta _{1}$. By taking

$$\begin{aligned} \alpha _1 ,\quad \alpha _1+\alpha _2-\gamma , \quad \beta _1+\beta _2-\gamma , \quad \beta _2 \quad \text {and}\quad \gamma \end{aligned}$$

as new variables of summations, we get

$$\begin{aligned}&(a_1\#_{{\mathfrak {V}}}a_2)(z,w)\\&\quad = \sum C_2(\alpha ,\beta ,\gamma ) c(a_1; \alpha _1,\beta _1-\beta _2+\gamma ) c(a_2; \alpha _2-\alpha _1+\gamma ,\beta _2) e_{\alpha _2}(z)e_{\beta _1}({\overline{w}}), \end{aligned}$$

where the sum is taken over all ${\alpha _{1}}, {\alpha }_{2}, \beta _{1}, {\beta }_{2}, \gamma \in {\text {N}^{d}}$ such that ${\alpha _{1}}\le {\alpha }_{2}$ and ${\beta _{2}}\le {\beta }_{1}$. This is the same as (4.30), and the result follows. $\square$

4.2 Continuity for Binomial operators

The assertion (1) in the following proposition was deduced in [17], and (2) follows from (1) and duality. The details are left for the reader.

Proposition 4.5

Let $t \in {\mathbf {C}}$, $s,s_0 \in \overline{{\mathbf {R}}_\flat }$ be such that $s < \frac{1}{2}$ and $0 < s_0 \le \frac{1}{2}$, ${\mathcal {T}}_{0,t},S_0\in {\mathcal {L}}(\ell _0^{\star } ({{\mathbf {N}}}^{2d}))$ be given by (4.1) and (4.3), and ${\mathcal {T}}_{0,t}^{*}\in {\mathcal {L}}(\ell _0({{\mathbf {N}}}^{2d}))$ be given by (4.2). Then the following is true:

• (1) ${\mathcal {T}}_{0,t}$ is a homeomorphism on $\ell _0^{\star } ({{\mathbf {N}}}^{2d})$ with the inverse ${\mathcal {T}}_{0,-t}$. Furthermore, ${\mathcal {T}}_{0,t}$ restricts to homeomorphisms on $\ell _s ^{\star } ({{\mathbf {N}}}^{2d})$ and on $\ell _{0,s_0} ^{\star } ({{\mathbf {N}}}^{2d})$, and ${\mathcal {T}}_{0,-t}$ can be obtained from (4.4);

• (2) ${\mathcal {T}}_{0,t}^{*}$ is a homeomorphism on $\ell _0({{\mathbf {N}}}^{2d})$ with the inverse ${\mathcal {T}}_{0,-t}^{*}$. Furthermore, ${\mathcal {T}}_{0,t}^{*}$ is uniquely extendable to homeomorphisms on $\ell _s ({{\mathbf {N}}}^{2d})$ and on $\ell _{0,s_0} ({{\mathbf {N}}}^{2d})$, and ${\mathcal {T}}_{0,-t}^{*}$ can be obtained from (4.4).

We have now the following complementary result to Proposition 4.5.

Proposition 4.6

Let $t \in {\mathbf {C}}$, $s,s_0 \in \overline{{\mathbf {R}}_\flat }$ be such that $s < \frac{1}{2}$ and $0 < s_0 \le \frac{1}{2}$, ${\mathcal {T}}_{0,t}, S_0\in {\mathcal {L}}(\ell _0^{\star } ({{\mathbf {N}}}^{2d}))$ be given by (4.1) and (4.3), and ${\mathcal {T}}_{0,t}^{*}\in {\mathcal {L}}(\ell _0({{\mathbf {N}}}^{2d}))$ given by (4.2). Then the following is true:

• (1) the map ${\mathcal {T}}_{0,t}$ restricts to homeomorphisms on

  $$\begin{aligned}&\ell _{{\mathcal {B}},s} ({{\mathbf {N}}}^{2d}),&\quad&\ell _{{\mathcal {B}},s} ^\star ({{\mathbf {N}}}^{2d}),&\quad&\ell _{{\mathcal {B}},0,s_0} ({{\mathbf {N}}}^{2d}),&\quad&\ell _{{\mathcal {B}},0,s_0} ^\star ({{\mathbf {N}}}^{2d}), \end{aligned}$$

  (4.32)

  with inverse ${\mathcal {T}}_{0,-t}$, and (4.4) holds;

• (2) the map ${\mathcal {T}}_{0,t}^{*}$ on $\ell _0({{\mathbf {N}}}^{2d})$ extends uniquely to homeomorphisms on

  $$\begin{aligned}&\ell _{{\mathcal {C}},s} ({{\mathbf {N}}}^{2d}),&\quad&\ell _{{\mathcal {C}},s} ^\star ({{\mathbf {N}}}^{2d}),&\quad&\ell _{{\mathcal {C}},0,s_0} ({{\mathbf {N}}}^{2d}),&\quad&\ell _{{\mathcal {C}},0,s_0} ^\star ({{\mathbf {N}}}^{2d}), \end{aligned}$$

  (4.33)

  with inverse ${\mathcal {T}}_{0,-t}^{*}$, and (4.4) holds.

Proof

We only prove (1) for the spaces $\ell _{{\mathcal {B}},s} ({{\mathbf {N}}}^{2d})$, as well as for $\ell _{{\mathcal {B}},0,s_0} ({{\mathbf {N}}}^{2d})$ when $s_0=\frac{1}{2}$, and we only prove (2) for $\ell _{{\mathcal {C}},s} ({{\mathbf {N}}}^{2d})$. The other assertions follow by similar arguments and are left for the reader.

First suppose that $s<\frac{1}{2}$, $r_{0,j},r_j,\theta >0$, $j=1,2$, and that

$$\begin{aligned} |c(\alpha ,\beta )|\lesssim \frac{\vartheta _{r_2,s}(\beta )}{\vartheta _{r_{0,2},s}(\alpha )}. \end{aligned}$$

(4.34)

Then it follows from standard estimates of binomial coefficients that

$$\begin{aligned}&|({\mathcal {T}}_{0,t}c)(\alpha ,\beta ) {\vartheta _{r_{0,1},s}(\alpha )}/{\vartheta _{r_1,s}(\beta )}|\nonumber \\&\quad \lesssim \sum _{\gamma \le \alpha ,\beta } {\alpha \atopwithdelims ()\gamma }^{\frac{1}{2}}{\beta \atopwithdelims ()\gamma }^{\frac{1}{2}} |t|^{|\gamma |} \frac{\vartheta _{r_2,s}(\beta -\gamma )\vartheta _{r_{0,1},s}(\alpha )}{\vartheta _{r_{0,2},s}(\alpha -\gamma )\vartheta _{r_1,s}(\beta )} \nonumber \\&\quad \le (2+2|t|)^{|\alpha +\beta |} \sum _{\gamma \le \alpha ,\beta } \frac{\vartheta _{r_2,s}(\beta -\gamma )\vartheta _{r_{0,1},s}(\alpha )}{\vartheta _{r_{0,2},s}(\alpha -\gamma )\vartheta _{r_1,s}(\beta )}. \end{aligned}$$

(4.35)

Suppose that $c\in \ell _{{\mathcal {B}},s}({{\mathbf {N}}}^{2d})$ and let $r_1>0$ be arbitrary but fixed. Let $c_0\in (0,1)$ be a small constant which shall be chosen later on. By Lemma 2.2, we may choose $r_2<r_1$ and then $r_{0,2}\in (0,r_2)$ such that

$$\begin{aligned} \vartheta _{r_2,s}(\beta -\gamma )\vartheta _{r_2,s}(\gamma ) \lesssim \vartheta _{c_0r_1,s}(\beta ) \end{aligned}$$

and that (4.34) holds. Then (4.35) gives

$$\begin{aligned} |({\mathcal {T}}_{0,t}c)(\alpha ,\beta )&{\vartheta _{r_{0,1},s}(\alpha )}/{\vartheta _{r_1,s}(\beta )}|\\&\lesssim (2+2|t|)^{|\alpha +\beta |}\frac{\vartheta _{c_0r_1,s}(\beta )}{\vartheta _{r_1,s}(\beta )} \sum _{\gamma \le \alpha ,\beta } \frac{\vartheta _{r_{0,1},s}(\alpha )}{\vartheta _{r_{0,2},s}(\alpha -\gamma )\vartheta _{r_2,s}(\gamma )}\\&\lesssim (2+2|t|)^{|\alpha +\beta |}\frac{\vartheta _{r_{0,1},s}(\alpha )\vartheta _{c_0r_1,s}(\beta )}{\vartheta _{r_{0,2}/C,s}(\alpha )\vartheta _{r_1,s}(\beta )} \left( \sum _{\gamma \le \alpha ,\beta } 1 \right) \\&\lesssim (3+2|t|)^{|\alpha +\beta |}\frac{\vartheta _{r_{0,1},s}(\alpha )\vartheta _{c_0r_1,s}(\beta )}{\vartheta _{r_{0,2}/C,s}(\alpha )\vartheta _{r_1,s}(\beta )} \end{aligned}$$

Since $s<\frac{1}{2}$, it follows that

$$\begin{aligned} (3+2|t|)^{|\alpha |}\frac{\vartheta _{r_{0,1},s}(\alpha )}{\vartheta _{r_{0,2}/C,s}(\alpha )} \le C_0 \quad \text {and}\quad (3+2|t|)^{|\beta |}\frac{\vartheta _{c_0r_1,s}(\beta )}{\vartheta _{r_1,s}(\beta )}\le C_0, \end{aligned}$$

for some constant $C_0>0$ which is independent of $\alpha$ and $\beta$, provided $c_0$ and $r_{0,1}>0$ are chosen small enough. A combination of these estimates shows that for every $r>0$, there is an $r_0>0$ such that

$$\begin{aligned} |({\mathcal {T}}_{0,t}c)(\alpha ,\beta )| \lesssim \frac{\vartheta _{r,s}(\beta )}{\vartheta _{r_0,s}(\alpha )}, \end{aligned}$$

(4.36)

which implies that ${\mathcal {T}}_{0,t}c \in \ell _{{\mathcal {B}},s}({{\mathbf {N}}}^{2d})$. Since it is clear that the choice of $r_0$ depends continuously of the interplay between $r_2$ and $r_{0,2}$ in (4.34), it also follows that ${\mathcal {T}}_{0,t}$ is continuous on $\ell _{{\mathcal {B}},s}({{\mathbf {N}}}^{2d})$, which proves the asserted continuity for ${\mathcal {T}}_{0,t}$ on $\ell _{{\mathcal {B}},s}({{\mathbf {N}}}^{2d})$.

Next suppose that $c\in \ell _{{\mathcal {B}},0,s_0}({{\mathbf {N}}}^{2d})$ with $s_0=\frac{1}{2}$. That is, for every $r>0$, there is an $r_0>0$ such that

$$\begin{aligned} |c(\alpha ,\beta )|\lesssim e^{-r|\alpha |+r_0|\beta |}. \end{aligned}$$

(4.37)

We shall prove that for every $r>0$, there is an $r_0>0$ such that

$$\begin{aligned} |({\mathcal {T}}_{0,t}c)(\alpha ,\beta )|\lesssim e^{-r|\alpha |+r_0|\beta |}, \end{aligned}$$

(4.38)

and then it suffices to prove this when $r_0\ge r$. By Cauchy-Schwartz inequality we get

$$\begin{aligned} |({\mathcal {T}}_{0,t}c)(\alpha ,\beta )|&\lesssim \sum _{\gamma \le \alpha ,\beta } {\alpha \atopwithdelims ()\gamma }^{\frac{1}{2}}{\beta \atopwithdelims ()\gamma }^{\frac{1}{2}} |t|^{|\gamma |} e^{-r|\alpha -\gamma |+r_0|\beta -\gamma |}\\&\le e^{-r|\alpha |+r_0|\beta |} \left( \sum _{\gamma \le \alpha } {\alpha \atopwithdelims ()\gamma } |t|^{|\gamma |} \right) ^{\frac{1}{2}} \left( \sum _{\gamma \le \beta } {\beta \atopwithdelims ()\gamma } |t|^{|\gamma |} \right) ^{\frac{1}{2}}\\&= e^{-(r-r_t)|\alpha |+(r_0+r_t)|\beta |}, \end{aligned}$$

where

$$\begin{aligned} r_t= \frac{1}{2}\ln (1+|t|). \end{aligned}$$

Since r can be chosen arbitrarily large, it follows that for every $r>0$, there is an $r_0>0$ such that (4.38) holds. Consequently, ${\mathcal {T}}_{0,t}$ is continuous on $\ell _{{\mathcal {B}},0,s_0}({{\mathbf {N}}}^{2d})$.

Next we prove (2) when ${\mathcal {T}}_{0,t}^{*}$ acts on $\ell _{{\mathcal {C}},s} ({{\mathbf {N}}}^{2d})$. Suppose that $c\in \ell _{{\mathcal {C}},s} ({{\mathbf {N}}}^{2d})$. Then there is an $r_0=r_{0,2}>0$ such that (4.34) holds for every $r=r_2>0$. We may assume that $r<c_0r_0$ for some constant $c_0\in (0,1)$ which shall be determined later on.

By Lemma 2.2 we obtain

$$\begin{aligned} |({\mathcal {T}}_{0,t}^{*}c)(\alpha ,\beta )|&\le \sum _{\gamma } \left( {{\alpha +\gamma } \atopwithdelims ()\gamma } {{\beta +\gamma } \atopwithdelims ()\gamma } \right) ^{\frac{1}{2}}|t|^{|\gamma |}|c(\alpha +\gamma ,\beta +\gamma )|\\&\lesssim 2^{\frac{1}{2} |\alpha +\beta |} \sum _{\gamma } |2t|^{|\gamma |} \frac{\vartheta _{r,s}(\beta +\gamma )}{\vartheta _{r_0,s}(\alpha +\gamma )}\\&\lesssim \frac{2^{\frac{1}{2} |\alpha +\beta |}\vartheta _{C_1r,s}(\beta )}{\vartheta _{r_0/C_1,s}(\alpha )}\cdot J \\&\lesssim \frac{\vartheta _{C_2r,s}(\beta )}{\vartheta _{r_0/C_2,s}(\alpha )}\cdot J, \end{aligned}$$

for some constants $C_2>C_1\ge 1$ which are independent of $c_0$, r and $r_0$, where

$$\begin{aligned} J= \sum _{\gamma } |2t|^{|\gamma |}| \frac{\vartheta _{C_1c_0r_0,s}(\gamma )}{\vartheta _{r_0/C_1,s}(\gamma )}. \end{aligned}$$

In the last inequality we have used the fact that $s<\frac{1}{2}$.

By choosing $c_0$ small enough it follows that J is convergent. This shows that for some $r_0>0$, (4.36) holds for every $r>0$. That is, ${\mathcal {T}}_{0,t}c \in \ell _{{\mathcal {C}},s}({{\mathbf {N}}}^{2d})$ when $c \in \ell _{{\mathcal {C}},s}({{\mathbf {N}}}^{2d})$. The continuity assertions of ${\mathcal {T}}_{0,t}$ on $\ell _{{\mathcal {C}},s}({{\mathbf {N}}}^{2d})$ now follows from this fact and that $r_0$ in (4.36) depends continuously on $r_{0,2}$ in (4.34). This gives the result. $\square$

In the following three propositions we show that the bilinear map**s in Definition 4.2 can be extended in suitable ways.

Proposition 4.7

Let $t\in {\mathbf {C}}$, $s_1,s_2\in \overline{{\mathbf {R}}}_{\flat }$ be such that $s_1,s_2< \frac{1}{2}$, $s=(s_1,s_2)$, $\Lambda ={{\mathbf {N}}}^{d_2}\times {{\mathbf {N}}}^{d_1}$ and let $U_t$ be the map from $\ell _0(\Lambda )\times \ell _0(\Lambda )$ to $\ell _0(\Lambda )$, given by $U_t(c_1,c_2)= c_1 \bullet _t c_2$, where $c_1\bullet _t c_2$ is as in Definition 4.2. Then $U_t$ is uniquely extendable to separately continuous map**s

$$\begin{aligned} U_t&: \ell _{{\mathcal {A}},s} (\Lambda )\times \ell _{{\mathcal {A}},s} (\Lambda ) \rightarrow \ell _{{\mathcal {A}},s} (\Lambda ),&U_t&: \ell _{{\mathcal {A}},s}^\star (\Lambda )\times \ell _{{\mathcal {A}},s}^\star (\Lambda ) \rightarrow \ell _{{\mathcal {A}},s} ^\star (\Lambda ), \end{aligned}$$

(4.39)

$$\begin{aligned} U_t&: \ell _{{\mathcal {B}},s} (\Lambda )\times \ell _{{\mathcal {B}},s} (\Lambda ) \rightarrow \ell _{{\mathcal {B}},s} (\Lambda ),&U_t&: \ell _{{\mathcal {B}},s}^\star (\Lambda )\times \ell _{{\mathcal {B}},s}^\star (\Lambda ) \rightarrow \ell _{{\mathcal {B}},s} ^\star (\Lambda ), \end{aligned}$$

(4.40)

$$\begin{aligned} U_t&: \ell _{{\mathcal {C}},s} (\Lambda )\times \ell _{{\mathcal {C}},s} (\Lambda ) \rightarrow \ell _{{\mathcal {C}},s} (\Lambda ),&U_t&: \ell _{{\mathcal {C}},s}^\star (\Lambda )\times \ell _{{\mathcal {C}},s}^\star (\Lambda ) \rightarrow \ell _{{\mathcal {C}},s} ^\star (\Lambda ). \end{aligned}$$

(4.41)

If instead $0<s_1,s_2\le \frac{1}{2}$, then the same holds true with $\ell _{{\mathcal {A}},0,s}$, $\ell _{{\mathcal {B}},0,s}$ and $\ell _{{\mathcal {C}},0,s}$ in place of $\ell _{{\mathcal {A}},s}$, $\ell _{{\mathcal {B}},s}$ and $\ell _{{\mathcal {C}},s}$, respectively, at each occurrence.

Proposition 4.8

Let $t\in {\mathbf {C}}$, $s_1,s_2\in \overline{{\mathbf {R}}}_{\flat }$ be such that $s_1,s_2< \frac{1}{2}$, $s=(s_1,s_2)$, $\Lambda ={{\mathbf {N}}}^{d_2}\times {{\mathbf {N}}}^{d_1}$ and let $U_{t,\diamond }$ be the map from $\ell _0(\Lambda )\times \ell _0(\Lambda )$ to $\ell _0(\Lambda )$, given by $U_{t,\diamond }(c_1,c_2)= c_1 \bullet _{(t ,\diamond )} c_2$, where $c_1\bullet _{(t,\diamond )} c_2$ is as in Definition 4.2. Then $U_{t,\diamond }$ is uniquely extendable to separately continuous map**s

$$\begin{aligned} U_{t,\diamond }&: \ell _{{\mathcal {A}},s} (\Lambda )\times \ell _{{\mathcal {A}},s} ^\star (\Lambda ) \rightarrow \ell _{{\mathcal {A}},s} ^\star (\Lambda ),&U_{t,\diamond }&: \ell _{{\mathcal {A}},s} ^\star (\Lambda )\times \ell _{{\mathcal {A}},s} (\Lambda ) \rightarrow \ell _{{\mathcal {A}},s} (\Lambda ), \end{aligned}$$

(4.42)

$$\begin{aligned} U_{t,\diamond }&: \ell _{{\mathcal {B}},s} (\Lambda ) \times \ell _{{\mathcal {C}},s}^\star (\Lambda ) \rightarrow \ell _{{\mathcal {C}},s}^\star (\Lambda ),&U_{t,\diamond }&: \ell _{{\mathcal {B}},s}^\star (\Lambda )\times \ell _{{\mathcal {C}},s} (\Lambda ) \rightarrow \ell _{{\mathcal {C}},s} (\Lambda ), \end{aligned}$$

(4.43)

$$\begin{aligned} U_{t,\diamond }&: \ell _{{\mathcal {C}},s} (\Lambda ) \times \ell _{{\mathcal {B}},s}^\star (\Lambda ) \rightarrow \ell _{{\mathcal {B}},s}^\star (\Lambda ),&U_{t,\diamond }&: \ell _{{\mathcal {C}},s}^\star (\Lambda )\times \ell _{{\mathcal {B}},s} (\Lambda ) \rightarrow \ell _{{\mathcal {B}},s} (\Lambda ). \end{aligned}$$

(4.44)

If instead $0<s_1,s_2\le \frac{1}{2}$, then the same holds true with $\ell _{{\mathcal {A}},0,s}$, $\ell _{{\mathcal {B}},0,s}$ and $\ell _{{\mathcal {C}},0,s}$ in place of $\ell _{{\mathcal {A}},s}$, $\ell _{{\mathcal {B}},s}$ and $\ell _{{\mathcal {C}},s}$, respectively, at each occurrence.

Proposition 4.9

Let $t\in {\mathbf {C}}$, $s\in \overline{{\mathbf {R}}}_{\flat }$ be such that $s< \frac{1}{2}$, $\Lambda ={{\mathbf {N}}}^{2d}$, and let $U_{t,\#}$ be the map from $\ell _0(\Lambda )\times \ell _0(\Lambda )$ to $\ell _0(\Lambda )$, given by $U_{t,\#}(c_1,c_2)= c_1 \bullet _{(t,\#)} c_2$, where $c_1\bullet _{(t,\#)} c_2$ is as in Definition 4.2. Then $U_{t,\#}$ is uniquely extendable to separately continuous map**s

$$\begin{aligned} U_{t,\#}&: \ell _{{\mathcal {A}},s} (\Lambda ) \times \ell _{{\mathcal {A}},s} (\Lambda ) \rightarrow \ell _{{\mathcal {A}},s} (\Lambda ),&\end{aligned}$$

(4.45)

$$\begin{aligned} U_{t,\#}&: \ell _{{\mathcal {A}},s}^\star (\Lambda )\times \ell _{{\mathcal {B}},s} (\Lambda ) \rightarrow \ell _{{\mathcal {A}},s} ^\star (\Lambda ),&U_{t,\#}&: \ell _{{\mathcal {B}},s}^\star (\Lambda )\times \ell _{{\mathcal {A}},s}^\star (\Lambda ) \rightarrow \ell _{{\mathcal {A}},s}^\star (\Lambda ), \end{aligned}$$

(4.46)

$$\begin{aligned} U_{t,\#}&: \ell _{{\mathcal {B}},s} (\Lambda )\times \ell _{{\mathcal {B}},s} (\Lambda ) \rightarrow \ell _{{\mathcal {B}},s} (\Lambda ),&U_{t,\#}&: \ell _{{\mathcal {B}},s}^\star (\Lambda )\times \ell _{{\mathcal {B}},s}^\star (\Lambda ) \rightarrow \ell _{{\mathcal {B}},s} ^\star (\Lambda ), \end{aligned}$$

(4.47)

$$\begin{aligned} U_{t,\#}&: \ell _{{\mathcal {C}},s} (\Lambda )\times \ell _{{\mathcal {C}},s} (\Lambda ) \rightarrow \ell _{{\mathcal {C}},s} (\Lambda ),&U_{t,\#}&: \ell _{{\mathcal {C}},s}^\star (\Lambda )\times \ell _{{\mathcal {C}},s}^\star (\Lambda ) \rightarrow \ell _{{\mathcal {C}},s} ^\star (\Lambda ), \end{aligned}$$

(4.48)

If instead $0<s_0\le \frac{1}{2}$, then the same holds true with $\ell _{{\mathcal {A}},0,s_0}$, $\ell _{{\mathcal {B}},0,s_0}$ and $\ell _{{\mathcal {C}},0,s_0}$ in place of $\ell _{{\mathcal {A}},s_0}$, $\ell _{{\mathcal {B}},s_0}$ and $\ell _{{\mathcal {C}},s_0}$, respectively, at each occurrence.

In each one of the propositions above, the different statements are proved by similar arguments. We only prove selections of these statements and leave the rest of the verification for the reader. We also remark that for some of the map**s in Propositions 4.7, 4.8 and 4.9, refined continuity properties are given in Sect. 5.5 (see Theorem 5.29).

Proof of Proposition 4.7

We only prove that $U_t$ is uniquely extendable to a separately continuous map from $\ell _{{\mathcal {B}},s} (\Lambda )\times \ell _{{\mathcal {B}},s} (\Lambda )$ to $\ell _{{\mathcal {B}},s} (\Lambda )$ and to a separately continuous map from $\ell _{{\mathcal {C}},s} (\Lambda )\times \ell _{{\mathcal {C}},s} (\Lambda )$ to $\ell _{{\mathcal {C}},s} (\Lambda )$. The other cases follow by similar arguments and are left for the reader.

Let $\vartheta _{r,s}$ be as in Definition 2.3, $\alpha = (\alpha _2,\alpha _1)\in \Lambda$, and suppose that $c_j\in \ell _0^{\star }(\Lambda )$ satisfy

$$\begin{aligned} |c_j(\alpha )| = |c_j(\alpha _2,\alpha _1)| \lesssim \frac{\vartheta _{r_j,s_1}(\alpha _1)}{\vartheta _{r_{0,j},s_2}(\alpha _2)}, \end{aligned}$$

(4.49)

for some $r_j,r_{0,j}>0$, $j=1,2$. By choosing $r=\max (r_1,r_2)>0$ and $r_0=\min (r_{0,1},r_{0,2})>0$, it is clear that (4.49) holds with r and $r_0$ in place of $r_j$ and $r_{0,j}$. By Lemma 2.2 we get

$$\begin{aligned}&|(c_1\bullet _tc_2)(\alpha _2,\alpha _1)| \le \sum _{\gamma \le \alpha } {{\alpha }\atopwithdelims (){\gamma }}^{\frac{1}{2}} |t|^{|\gamma |}|c_1(\alpha -\gamma )c_2(\gamma )| \nonumber \\&\lesssim (1+|t|)^{|\alpha |} \sum _{\gamma \le \alpha } 2^{\frac{1}{2}|\alpha |} \frac{\vartheta _{r,s_1} (\alpha _1-\gamma _1)\vartheta _{r,s_1}(\gamma _1)}{\vartheta _{r_0,s_2}(\alpha _2-\gamma _2) \vartheta _{r_0,s_2}(\gamma _2)} \nonumber \\&\lesssim (2+2|t|)^{|\alpha |} \sum _{\gamma \le \alpha } \frac{\vartheta _{C_0r,s_1}(\alpha _1)}{\vartheta _{r_0/C_0,s_2}(\alpha _2)} \nonumber \\&\lesssim (3+2|t|)^{|\alpha |} \frac{\vartheta _{C_0r,s_1}(\alpha _1)}{\vartheta _{r_0/C_0,s_2}(\alpha _2)} \lesssim \frac{\vartheta _{Cr,s_1}(\alpha _1)}{\vartheta _{r_0/C,s_2}(\alpha _2)}, \end{aligned}$$

(4.50)

for some constants $C>C_0> 1$ which are independent of r and $r_0$. In the last step we have used $s_1,s_2<\frac{1}{2}$.

If $c_j\in \ell _{{\mathcal {B}},s}(\Lambda )$, $j=1,2$, then for every $r=r_j>0$, there is an $r_0=r_{0,j}>0$ such that (4.49) holds. By (4.50) it follows that for every $r>0$, there is an $r_0>0$ such that (4.49) holds with $c_1\bullet _tc_2$ in place of $c_j$. This implies that $U_t$ is extendable to a continuous map from $\ell _{{\mathcal {B}},s} (\Lambda )\times \ell _{{\mathcal {B}},s} (\Lambda )$ to $\ell _{{\mathcal {B}},s} (\Lambda )$. Since $\ell _0(\Lambda )$ is dense in $\ell _{{\mathcal {B}},s} (\Lambda )$, it also follows that the continuous extension of $U_t$ is unique.

If $c_j\in \ell _{{\mathcal {C}},s}(\Lambda )$, $j=1,2$, then there is an $r_0=r_{0,j}>0$ such that (4.49) holds for every $r=r_j>0$. By (4.50) it follows that there is an $r_0>0$ such that (4.49) holds for every $r>0$, with $c_1\bullet _tc_2$ in place of $c_j$. Hence $U_t$ extends to a separately continuous map from $\ell _{{\mathcal {C}},s} (\Lambda )\times \ell _{{\mathcal {C}},s} (\Lambda )$ to $\ell _{{\mathcal {C}},s} (\Lambda )$. Since $\ell _0(\Lambda )$ is dense in $\ell _{{\mathcal {C}},s} (\Lambda )$, it also follows that $U_t$ on $\ell _{{\mathcal {C}},s} (\Lambda )$ is uniquely defined, which gives the assertion. $\square$

Proof of Proposition 4.8

We only prove that $U=U_{t,\diamond }$ extends uniquely to a separately continuous map from $\ell _{{\mathcal {C}},s}^\star (\Lambda )\times \ell _{{\mathcal {B}},s}(\Lambda )$ to $\ell _{{\mathcal {B}},s}(\Lambda )$. The other assertions on continuity follow by similar arguments and are left for the reader.

Since $\ell _{{\mathcal {C}},s}^\star (\Lambda )$ is invariant under the map

$$\begin{aligned} c(\alpha _2,\alpha _1)\mapsto t^{|\alpha _1| + |\alpha _2|}c(\alpha _2,\alpha _1), \end{aligned}$$

it suffices to prove the result for $t=1$.

First we prove the continuity extension of U. Let $c_1\in \ell _{{\mathcal {C}},s}^\star (\Lambda )$ and $c_2\in \ell _{{\mathcal {B}},s} (\Lambda )$. Then it follows that for some $r_0>0$ one has

$$\begin{aligned} |c_1(\alpha _2,\alpha _1)|&\lesssim \frac{\vartheta _{r,s_2}(\alpha _2)}{\vartheta _{r_0,s_1}(\alpha _1)},&\quad (\alpha _2,\alpha _1)&\in \Lambda , \end{aligned}$$

(4.51)

for every $r>0$. It also follows that for every $r>0$, there is an $r_0>0$ such that

$$\begin{aligned} |c_2(\alpha _2,\alpha _1)|&\lesssim \frac{\vartheta _{r,s_1}(\alpha _1)}{\vartheta _{r_0,s_2}(\alpha _2)},&\quad (\alpha _2,\alpha _1)&\in \Lambda . \end{aligned}$$

(4.52)

Let $r_{0,1}>0$ be fixed such that (4.51) holds for every $r>0$, with $r_{0,1}$ in place of $r_0$. Also let $C>1$ be a constant which only dependent on $s_1,s_2$, let $r_2\in (0,r_{0,1}/C)$ be arbitrary and choose $r_{0,2}>0$ such that (4.52) holds with $r_2$ and $r_{0,2}$ in place of r and $r_0$. Finally let $r_1\in (0,r_{0,2}/C)$ be arbitrary.

If C is chosen large enough, and using the same notations as in Lemma 4.4, then it follows from Lemma 2.2 that

$$\begin{aligned} {{\alpha +\gamma }\atopwithdelims (){\gamma }}^{\frac{1}{2}} |c_1(\gamma )c_2(\alpha +\gamma )|&\lesssim 2^{|\alpha +\gamma |}\frac{\vartheta _{r_1,s_2}(\gamma _2)\vartheta _{r_2,s_1}(\alpha _1+\gamma _1)}{\vartheta _{r_{0,1},s_1}(\gamma _1) \vartheta _{r_{0,2},s_2}(\alpha _2+\gamma _2)} \\&\lesssim 2^{-|\alpha +\gamma |}\frac{\vartheta _{Cr_2,s_1}(\alpha _1)}{\vartheta _{r_{0,2}/C,s_2}(\alpha _2)}. \end{aligned}$$

Hence for the series on the right-hand side of (4.16) we have

$$\begin{aligned} \sum _{\gamma \in \Lambda } {{\alpha +\gamma }\atopwithdelims (){\gamma }} ^{\frac{1}{2}} |c_1(\gamma _2,\gamma _1) c_2(\alpha _2+\gamma _2,\alpha _1+\gamma _1)|&\lesssim \sum _{\gamma \in \Lambda } 2^{-|\alpha +\gamma |}\frac{\vartheta _{Cr_2,s_1}(\alpha _1)}{\vartheta _{r_{0,2}/C,s_2}(\alpha _2)} \\&\asymp \frac{\vartheta _{Cr_2,s_1}(\alpha _1)}{\vartheta _{r_{0,2}/C,s_2}(\alpha _2)} . \end{aligned}$$

It follows that the right-hand side of (4.16) is convergent, and by defining $c_1\bullet _{(1,\diamond )}c_2$ by (4.16), it follows that for every $r>0$, there is an $r_0>0$ such that

$$\begin{aligned} |(c_1\bullet _{(1,\diamond )}c_2)(\alpha )| \lesssim \frac{\vartheta _{r,s_1}(\alpha _1)}{\vartheta _{r_0,s_2}(\alpha _2)} . \end{aligned}$$

(4.53)

This implies that $c_1\bullet _{(1,\diamond )}c_2\in \ell _{{\mathcal {B}},s} (\Lambda )$. Furthermore, the involved parameters of the right-hand side of (4.53) depend continuously on the involved parameters of the right-hand sides of (4.51) and (4.52), which proves the continuity assertion.

Finally, the uniqueness of the extension follows from the fact that $\ell _0(\Lambda )$ is dense in $\ell _{{\mathcal {B}},s} (\Lambda )$ and $\ell _{{\mathcal {C}},s}^\star (\Lambda )$, and the result follows. $\square$

Proof of Proposition 4.9

Let $s=s_0$. We only prove that $U=U_{t,\#}$ extends uniquely to continuous products on $\ell _{{\mathcal {B}},s}({{\mathbf {N}}}^{2d} )$ and on $\ell _{{\mathcal {C}},s}({{\mathbf {N}}}^{2d} )$. The other assertions follows by similar arguments and are left for the reader.

Let $C_0(\alpha _{1}, {\alpha }_{2}, \beta _{1}, {\beta }_{2}, \gamma )$ be as in (4.18), when $\alpha , {\alpha }_{0}, \beta , {\beta }_{0}, \gamma \in {\text {N}^{d}}$ satisfy $\alpha _{0} \le \alpha$ and $\beta _{0} \le \beta$. We observe

$$\begin{aligned} |C_0(\alpha ,\alpha _0, \beta ,\beta _0,\gamma )| \le 2^{|\alpha +\beta +\gamma |}. \end{aligned}$$

(4.54)

Suppose that $c_j\in \ell _{{\mathcal {B}},s}({{\mathbf {N}}}^{2d})$, $j=1,2$. Then it follows from Lemma 2.2 and (4.49) for $s_j=s$ that for every $r_j>0$, there is an $r_{0,j}>0$ such that

$$\begin{aligned} |c_1(\alpha _0,\beta -\beta _0+\gamma )|&\lesssim \frac{\vartheta _{r_1,s}(\beta -\beta _0)\vartheta _{r_1,s}(\gamma )}{\vartheta _{r_{0,1,s}}(\alpha _0)} \end{aligned}$$

(4.55)

and

$$\begin{aligned} |c_2(\alpha -\alpha _0+\gamma ,\beta _0)|&\lesssim \frac{\vartheta _{r_2,s}(\beta _0)}{\vartheta _{r_{0,2},s}(\alpha -\alpha _0)\vartheta _{r_{0,2},s}(\gamma )}. \end{aligned}$$

(4.56)

Let $r>0$ be arbitrary, $C>1$ be a constant which is independent of r which is specified later and let $r_2>0$ be such that $Cr_2<r$. Choose $r_{0,2}>0$ such that (4.56) holds. Now choose $r_1>0$ such that $r_1<r_2$ and $r_1<r_{0,2}/2$ and then choose $r_{0,1}>0$ such that $r_{0,1}<r_{0,2}$ and (4.55) hols.

If C is chosen large enough, then it follows from Lemma 2.2 again that

$$\begin{aligned}&|c_1(\alpha _0,\beta -\beta _0+\gamma ) c_2(\alpha -\alpha _0+\gamma ,\beta _0)| \lesssim \frac{\vartheta _{r_1,s}(\beta -\beta _0)\vartheta _{r_1,s}(\gamma ) \vartheta _{r_2,s}(\beta _0)}{\vartheta _{r_{0,1},s}(\alpha _0)\vartheta _{r_{0,2},s}(\alpha -\alpha _0) \vartheta _{r_{0,2},s}(\gamma )}\nonumber \\&\quad \lesssim \frac{2^{-|\alpha +\beta |}2^{-|\alpha _0+\beta _0|}\vartheta _{Cr_2,s}(\beta )}{\vartheta _{r_{0,1}/C,s}(\alpha )\vartheta _{r_1,s}(\gamma )} \lesssim \frac{2^{-|\alpha +\beta |}2^{-|\alpha _0+\beta _0|}\vartheta _{r,s}(\beta )}{\vartheta _{r_0,s}(\alpha )\vartheta _{r_1,s}(\gamma )}, \end{aligned}$$

(4.57)

when $r_0=r_{0,1}/C>0$. By combining these estimates we obtain

$$\begin{aligned} |(c_1 \bullet _{(t,\#)} c_2)(\alpha ,\beta )|&\lesssim \left( \sum _{\gamma } \frac{(2+2|t|)^{|\gamma |}}{\vartheta _{r_1,s}(\gamma )} \right) \left( \sum _{\alpha _0,\beta _0} 2^{-|\alpha _0+\beta _0|} \frac{\vartheta _{r,s}(\beta )}{\vartheta _{r_0,s}(\alpha )} \right) \nonumber \\&\quad \asymp \frac{\vartheta _{r,s}(\beta )}{\vartheta _{r_0,s}(\alpha )}, \end{aligned}$$

(4.58)

and it follows that U on $\ell _0({{\mathbf {N}}}^{2d})$ extends to a continuous products on $\ell _{{\mathcal {B}},s}({{\mathbf {N}}}^{2d} )$.

Suppose instead that $c_1\in \ell _{{\mathcal {C}},s}({{\mathbf {N}}}^{2d} )$, $c_2\in \ell _{{\mathcal {C}},s}({{\mathbf {N}}}^{2d} )$. Then for some $r_{0,1},r_{0,2}>0$ such that $r_{0,1}<r_{0,2}$ one has that (4.55) and (4.56) hold for every $r_1,r_2>0$ such that $r_1<r_2$. Hence, if C and $r_0$ are chosen as in the first part of the proof, then (4.57) and (4.58) show that for some $r_0>0$ one has that

$$\begin{aligned} |(c_1 \bullet _{(t,\#)} c_2)(\alpha ,\beta )| \lesssim \frac{\vartheta _{r,s}(\beta )}{\vartheta _{r_0,s}(\alpha )}, \end{aligned}$$

for every $r>0$. This implies that U on $\ell _0({{\mathbf {N}}}^{2d})$ extends to a separately continuous products on $\ell _{{\mathcal {C}},s}({{\mathbf {N}}}^{2d} )$. The uniqueness assertions follow from the fact that $\ell _0({{\mathbf {N}}}^{2d})$ is dense in $\ell _{{\mathcal {B}},s}({{\mathbf {N}}}^{2d} )$ and in $\ell _{{\mathcal {C}},s}({{\mathbf {N}}}^{2d} )$. $\square$

5 Continuity, Composition and Transitions of Wick and Anti-Wick Operators

In this section we apply the results from the previous sections to deduce estimates on kernels, Wick and anti-Wick symbols of operators. Especially we show that in several situations, any linear and continuous operator on ${\mathcal {A}}_{0,s}({\mathbf {C}}^{d})$, ${\mathcal {A}}_s({\mathbf {C}}^{d})$, ${\mathcal {A}}_s^{\star }({\mathbf {C}}^{d})$ or on ${\mathcal {A}}_{0,s}^{\star }({\mathbf {C}}^{d})$ can be expressed as Wick or anti-Wick operators (see Theorems 5.8 and 5.9). Thereafter we apply the results in Sect. 4.2 on continuity for bilinear binomial operators to deduce separate continuity for multiplication, symbol composition and other related bilinear map**s on the spaces in Definitions 2.14 and 2.16. Some further refined continuity properties for such bilinear operators are thereafter deduced in Sect. 5.5.

In Sect. 5.4 we complete some analysis in Section 3 in [18]. Here we deduce some estimates of Wick symbols of anti-Wick operators with Wick or anti-Wick symbols belonging to Hilbert spaces, related to   .

5.1 Continuity of the operators ${\mathcal {T}}_t$ and ${\mathcal {T}}_t^{*}$

We start by discussing the operator ${\mathcal {T}}_t$. A combination of Proposition 4.6 (1) and (4.8) gives the following result, related to Proposition 4.1. The details are left for the reader.

Theorem 5.1

Let $t \in {\mathbf {C}}$, $s,s_0 \in \overline{{\mathbf {R}}_\flat }$ be such that $s < \frac{1}{2}$ and $s_0 \le \frac{1}{2}$, ${\mathcal {T}}_{0,t}$ be the map on $\ell _0^{\star } ({{\mathbf {N}}}^{2d})$ given by (4.1), and let ${\mathcal {T}}_t$ be the map on   given by (4.5). Then the following is true:

• (1) ${\mathcal {T}}_t$ restricts to homeomorphisms on each of the spaces

  

  (5.1)

• (2) the diagram (4.8) commutes, after $\ell _s^{\star }$ and   are replaced by $\ell _{{\mathcal {B}},s}$ and   and   and   , or by $\ell _{{\mathcal {B}},0,s_0}^\star$ and   , respectively, at each occurrence.

Next we deduce extensions of the operator ${\mathcal {T}}_t^{*}$, and begin with the following, which follows from (2) in Proposition 4.5 and the commutative diagram (4.12). The details are left for the reader.

Theorem 5.2

Let $t \in {\mathbf {C}}$, $s,s_0 \in \overline{{\mathbf {R}}_\flat }$ be such that $s < \frac{1}{2}$ and $s_0 \le \frac{1}{2}$, and let ${\mathcal {T}}_{0,t}^{*}$ be the map on $\ell _0({{\mathbf {N}}}^{2d})$ given by (4.2) and let ${\mathcal {T}}_t^{*}$ be the map on   given by (4.10). Then ${\mathcal {T}}_t^{*}$ is uniquely extendable to homeomorphisms on   and on   . The diagram (4.12) commutes after $\ell _0$ and   have been replaced by $\ell _s$ and   , respectively, or by $\ell _{0,s_0}$ and   , respectively, at each occurrence.

In the same way we get the following by combining Proposition 4.6 (2) and (4.12). Again the details are left for the reader.

Theorem 5.3

Let $t \in {\mathbf {C}}$, $s,s_0 \in \overline{{\mathbf {R}}_\flat }$ be such that $s < \frac{1}{2}$ and $s_0 \le \frac{1}{2}$, ${\mathcal {T}}_{0,t}^{*}$ be the map on $\ell _0({{\mathbf {N}}}^{2d})$ given by (4.2), and let ${\mathcal {T}}_t^{*}$ be the map on   given by (4.10). Then the following is true:

• (1) ${\mathcal {T}}_t^{*}$ is uniquely extendable to homeomorphisms on each of the spaces

  

  (5.2)

• (2) the diagram (4.12) commutes, after $\ell _0$ and   are replaced by $\ell _{{\mathcal {C}},s}$ and    , by $\ell _{{\mathcal {C}},0,s_0}$ and    , by $\ell _{{\mathcal {C}},s}^\star$ and    , or by $\ell _{{\mathcal {C}},0,s_0}^\star$ and    , respectively, at each occurrence.

5.2 Continuity and Relationships Between Kernel, Wick and Anti-Wick Operators

By (2.59)–(2.61), Proposition 4.6, Theorem 5.3, and the fact that (4.11) holds for , we may now extend the domain of anti-Wick operators as in the following definition.

Definition 5.4

Let $s_1,s_2\in \overline{{\mathbf {R}}}_\flat$ be such that $s_1<\frac{1}{2}$ and $s_2\le \frac{1}{2}$.

• (1) If (), then the anti-Wick operator ${\text {Op}}_{{\mathfrak {V}}}^{{\text {aw}}}(a)$ is the linear and continuous operator on ${\mathcal {A}}_{s_1}({\mathbf {C}}^{d})$ (on ${\mathcal {A}}_{s_1}^{\star }({\mathbf {C}}^{d})$), given by (4.11).

• (2) If (), then the anti-Wick operator ${\text {Op}}_{{\mathfrak {V}}}^{{\text {aw}}}(a)$ is the linear and continuous operator on ${\mathcal {A}}_{0,s_2}({\mathbf {C}}^{d})$ (on ${\mathcal {A}}_{0,s_2}^{\star }({\mathbf {C}}^{d})$), given by (4.11).

We can now combine the results in previous sections with the kernel results in Sect. 3 to find equalities between classes of Wick, anti-Wick and kernel operators. As a first taste we have the following result which follows by a combination of (4.6), and Propositions 3.2, 3.3 and 4.1. The details are left for the reader. The result is also an immediate consequence of Theorems 2.7 and 2.8 in [17].

Proposition 5.5

Let $j\in \{ 1,2\}$, $s_1,s_2\in \overline{{\mathbf {R}}}_\flat$ be such that $s_1<\frac{1}{2}$ and $s_2\le \frac{1}{2}$. Then the map**s

and

are isomorphisms.

The corresponding result for anti-Wick operators is the following, which follows by combining (4.11) and Theorem 5.2. The details are left for the reader.

Theorem 5.6

Let $j\in \{ 1,2\}$, $s_1,s_2\in \overline{{\mathbf {R}}}_\flat$ be such that $s_1<\frac{1}{2}$ and $s_2\le \frac{1}{2}$. Then the map**s

(5.3)

and

(5.4)

are isomorphisms.

We notice that if $s_1=0$, then   is the set of all polynomials a(z, w) which are analytic in z and conjugated analytic in w. Hence (5.3) in this case means that the sets of Wick operators with polynomial symbols agree with the set of anti-Wick operators with polynomial symbols, which is well-known (see e. g. [4]). On the other hand, the other cases in Theorem 5.6, seems not to be available in the literature.

Remark 5.7

Note that   and   in Theorem 5.6 are images of those Pilipović spaces, which are not Gelfand–Shilov spaces, under the map $\Theta \circ {\mathfrak {V}}_{2d}$. In particular, Theorem 5.6 gives further motivations for detailed studies of Pilipović spaces.

The next two results follows from (4.6), (4.11), Propositions 3.4 and 3.5, and Theorems 5.1 and 5.3. The details are left for the reader.

Theorem 5.8

Let $s_1,s_2 \in \overline{{\mathbf {R}}_\flat }$ be such that $s_1 < \frac{1}{2}$ and $s_2 \le \frac{1}{2}$. Then the map**s

and

are isomorphisms.

Theorem 5.9

Let $s_1,s_2 \in \overline{{\mathbf {R}}_\flat }$ be such that $s_1 < \frac{1}{2}$ and $s_2 \le \frac{1}{2}$. Then the map**s

are isomorphisms.

Remark 5.10

Let $s_1, s_2\in \overline{{\mathbf {R}}}_\flat$ be such that $s_1<\frac{1}{2}$ and $s_2\le \frac{1}{2}$. By Theorem 5.9 it follows that

(5.5)

and that

$$\begin{aligned} {\text {Op}}_{{\mathfrak {V}}}^{{\text {aw}}}(a_1)\ne {\text {Op}}_{{\mathfrak {V}}}^{{\text {aw}}}(a_2) \quad \Leftrightarrow \quad {\text {Op}}_{{\mathfrak {V}}}(a_1)\ne {\text {Op}}_{{\mathfrak {V}}}(a_2) \quad \Leftrightarrow \quad a_1\ne a_2, \end{aligned}$$

(5.6)

when $a_1$ and $a_2$ belong to any of the spaces in Theorem 5.9.

In Appendix C we have listed some identities of spaces of kernels, Wick and anti-Wick operators which are immediate consequences of Corollary 3.6, Proposition 5.5 and Theorems 5.6–5.9.

For other choices of $s_1$ and $s_2$ it seems that the equalities on the left-hand sides in Theorems 5.8 and 5.9 between sets of Wick and anti-Wick operators are violated. On the other hand, the following result shows that we still may identify the operator classes on the right-hand sides in (5.8) with suitable classes of Wick operators, with Wick symbols bounded by

$$\begin{aligned} \vartheta _{1,s,r_1,r_2}(z,w) = {\left\{ \begin{array}{ll} e^{\frac{1}{2}|z-w|^2-r_1|z|^{\frac{1}{s}}+r_2|w|^{\frac{1}{s}}}, &{} s<\infty ,\\ e^{\frac{1}{2}|z-w|^2}\langle z\rangle ^{-r_1}\langle w\rangle ^{r_2}, &{} s=\infty , \end{array}\right. } \end{aligned}$$

(5.7)

and

$$\begin{aligned} \vartheta _{2,s,r_1,r_2}(z,w) = {\left\{ \begin{array}{ll} e^{\frac{1}{2}|z-w|^2+r_2|z|^{\frac{1}{s}}-r_1|w|^{\frac{1}{s}}}, &{} s<\infty , \\ e^{\frac{1}{2}|z-w|^2}\langle z\rangle ^{r_2}\langle w\rangle ^{-r_1}, &{} s=\infty . \end{array}\right. } \end{aligned}$$

(5.8)

Theorem 5.11

Let $s\in [\frac{1}{2} ,\infty ]$ and $\vartheta _{k,s,r_1,r_2}$ be given by (5.7) and (5.8), $k=1,2$. Then the following is true:

• (1) if $s <\infty$, then ${\mathcal {L}}({\mathcal {A}}_s({\mathbf {C}}^{d}))$ (${\mathcal {L}}({\mathcal {A}}_s^{\star }({\mathbf {C}}^{d}))$) consists of all ${\text {Op}}_{{\mathfrak {V}}}(a)$ such that and for every $r_2>0$, there is an $r_1>0$ such that

  $$\begin{aligned} |a(z,w)|\lesssim \vartheta _{1,s,r_1,r_2}(z,w) \qquad \text{( } \, |a(z,w)|\lesssim \vartheta _{2,s,r_1,r_2}(z,w) \, \text{) } \end{aligned}$$

  (5.9)

  holds;

• (2) if $s>\frac{1}{2}$, then ${\mathcal {L}}({\mathcal {A}}_{0,s}({\mathbf {C}}^{d}))$ (${\mathcal {L}}({\mathcal {A}}_{0,s}^{\star }({\mathbf {C}}^{d}))$) consists of all ${\text {Op}}_{{\mathfrak {V}}}(a)$ such that and for every $r_1>0$, there is an $r_2>0$ such that (5.9) holds.

Proof

We only prove (1). The assertion (2) follows by similar arguments and is left for the reader.

Suppose that T is a linear and continuous map from ${\mathcal {A}}_{\flat _1}({\mathbf {C}}^{d})$ to ${\mathcal {A}}_{\flat _1}^{\star }({\mathbf {C}}^{d})=A({\mathbf {C}}^{d})$ with kernel K and Wick symbol a. Then . By Propositions 3.4 and 3.5 it follows that $T\in {\mathcal {L}}({\mathcal {A}}_s({\mathbf {C}}^{d}))$ ($T\in {\mathcal {L}}({\mathcal {A}}^{\star } _s({\mathbf {C}}^{d}))$), if and only if for every $r_2>0$, there is an $r_1>0$ such that

$$\begin{aligned} |K(z,w)| \lesssim e^{\frac{1}{2}(|z|^2+|w|^2)-r_1|z|^{\frac{1}{s}}+r_2|w|^{\frac{1}{s}}} \quad \big ( |K(z,w)| \lesssim e^{\frac{1}{2}(|z|^2+|w|^2)+r_2|z|^{\frac{1}{s}}-r_1|w|^{\frac{1}{s}}} \big ). \end{aligned}$$

The asserted equivalence now follows from the previous equivalence and the fact that $a(z,w)=K(z,w)e^{-(z,w)}$, which implies that

5.3 Multiplications and Differentiations for Power Series Expansions, and Compositions of Wick Operators

First we combine (4.21) and Proposition 4.7 to deduce ring structures of the spaces in (2.36), (2.37) and in Definition 2.16. The details are left for the reader. Here the ring $({\mathcal {R}},+,\cdot )$ is called a flaccid topological ring, if ${\mathcal {R}}$ is a topological vector space (under ${\mathbf {C}}$) and the multiplication map

$$\begin{aligned} {\mathcal {R}}\times {\mathcal {R}}\ni (x_1,x_2) \mapsto x_1\cdot x_2 \in {\mathcal {R}}\end{aligned}$$

(5.10)

is separately continuous. The flaccid topological ring $({\mathcal {R}},+,\cdot )$ is called a topological ring, if the map (5.10) is continuous.

Proposition 5.12

Let $s_1,s_2\in \overline{{\mathbf {R}}}_{\flat }$, $s=(s_1,s_2)$ and $W={\mathbf {C}}^{d_2}\times {\mathbf {C}}^{d_1}$. Then the following is true:

• (1) if $s_1,s_2<\frac{1}{2}$, then the ring   under addition and multiplication extends uniquely to flaccid topological rings

  

  (5.11)

  under addition and multiplication;

• (2) if $0<s_1,s_2\le \frac{1}{2}$, then the ring   under addition and multiplication extends uniquely to flaccid topological rings

  

  (5.12)

  under addition and multiplication.

In Sect. 5.5 later on we deduce that in some situations, separate continuous bilinear map**s are in fact continuous. In particular it follows that certain parts of the previous proposition can be improved into the following. We postpone the proof till Sect. 5.5.

Theorem 5.13

Let $s_1,s_2\in \overline{{\mathbf {R}}}_{\flat }$, $s=(s_1,s_2)$ and $W={\mathbf {C}}^{d_2}\times {\mathbf {C}}^{d_1}$. Then the following is true:

• (1) if $s_1,s_2<\frac{1}{2}$, then the ring   under addition and multiplication extends uniquely to topological rings   ,   ,   and   under addition and multiplication;

• (2) if $0<s_1,s_2\le \frac{1}{2}$, then the ring   under addition and multiplication extends uniquely to topological rings   ,    ,    and   under addition and multiplication.

Remark 5.14

In Sect. 7 we present trace results and results on linear pullbacks for the spaces in (5.11) and (5.12). The ring assertions in Theorem 5.13 also follows by combining such trace and pullback properties.

From now on we let the spaces in (5.11) and in (5.12) be equipped with the ring structure guaranteed by Theorem 5.13.

Next we discuss extensions of the product $\diamond$ on   , given by (4.25). The following result is a straight-forward consequence of (4.27) and Proposition 4.8. The details are left for the reader.

Theorem 5.15

Let $s_1,s_2\in \overline{{\mathbf {R}}}_{\flat }$, $s=(s_1,s_2)$, $W={\mathbf {C}}^{d_2}\times {\mathbf {C}}^{d_1}$ and let $T_\diamond$ be the map from ${\mathcal {A}}_0({\mathbf {C}}^{d})\times {\mathcal {A}}_0({\mathbf {C}}^{d})$ to ${\mathcal {A}}_0({\mathbf {C}}^{d})$, given by $T_\diamond (K_1,K_2) = K_1\diamond K_2$, where $K_1\diamond K_2$ is given by (4.25). If $s_1,s_2<\frac{1}{2}$, then $T_\diamond$ is uniquely extendable to separately continuous map**s

(5.13)

(5.14)

(5.15)

If instead $0<s_1,s_2\le \frac{1}{2}$, then the same holds true with   ,   and    in place of   ,   and   , respectively, at each occurrence.

Remark 5.16

Let $d_2=d$, $d_1=0$ and $s\in \overline{{\mathbf {R}}} _\flat$ satisfies $s<\frac{1}{2}$. Then it follows from Theorem 5.15 that

(5.16)

are separately continuous. If instead $0<s\le \frac{1}{2}$, then the same holds true with ${\mathcal {A}}_{0,s}$ in place of ${\mathcal {A}}_s$ at each occurrence.

Remark 5.17

It is tempting to proclaim that Theorem 5.15 is the dual result to Theorem 5.13, where the former should follow from the latter by using the adjoint relation (4.23), and that the spaces in (2.12) are close to the duals of (2.11) and vise versa. A problem here is that the literature seems not to support completeness and thereby ensure useful duality properties for some of the involved spaces (see e. g. [14, 23, 24]). It therefore seems not so straight-forward to apply duality arguments here.

Next we discuss extensions of the twisted product $\#_{{\mathfrak {V}}}$ in (2.62) as a map from   to   . The following result is a straight-forward consequence of (4.30) and Proposition 4.9. The details are left for the reader.

Theorem 5.18

Let $s\in \overline{{\mathbf {R}}}_{\flat }$ be such that $s< \frac{1}{2}$, $W ={\mathbf {C}}^{d}\times {\mathbf {C}}^{d}$, and let $T_{\#}$ be the map from   to   , given by $T_{\#}(a_1,a_2)= a_1 \#_{{\mathfrak {V}}} a_2$. Then $T_{\#}$ is uniquely extendable to separately continuous map**s

(5.17)

(5.18)

(5.19)

(5.20)

If instead $0<s\le \frac{1}{2}$, then the same holds true with   ,   and   in place of   ,   and   , respectively, at each occurrence.

By Theorems 5.9 and 5.18 the twisted anti-Wick product $\#_{{\mathfrak {V}}}^{{\text {aw}}}$, defined by

$$\begin{aligned} {\text {Op}}_{{\mathfrak {V}}}^{{\text {aw}}}(a_1)\circ {\text {Op}}_{{\mathfrak {V}}}^{{\text {aw}}}(a_2) = {\text {Op}}_{{\mathfrak {V}}}^{{\text {aw}}}(a_1\#_{{\mathfrak {V}}}^{{\text {aw}}}a_2), \end{aligned}$$

(5.21)

makes sense, when $a_1$ and $a_2$ belong to the symbol classes in Theorem 5.9. The following result now follows by combining Theorems 5.9 and 5.18. The details are left for the reader.

Theorem 5.19

Let $s\in \overline{{\mathbf {R}}}_{\flat }$ be such that $s< \frac{1}{2}$, $W ={\mathbf {C}}^{d}\times {\mathbf {C}}^{d}$, and let $T_{\#}^{{\text {aw}}}$ be the map from   to   , given by $T_{\#}^{{\text {aw}}}(a_1,a_2)= a_1 \#_{{\mathfrak {V}}}^{{\text {aw}}} a_2$. Then $T_{\#}^{{\text {aw}}}$ is uniquely extendable to separately continuous map**s

(5.22)

If instead $0<s\le \frac{1}{2}$, then the same holds true with   and   in place of   and   , respectively, at each occurrence.

We remark that for some of the map**s in Theorems 5.15, 5.18 and 5.19, refined continuity properties are given in Sect. 5.5 (see Theorem 5.29).

5.4 Some Further Relationships Between Operator Kernels, Wick and Anti-Wick Symbols

Next we show some transition properties between Wick, anti-Wick and kernel operators with symbols and kernels belonging to refined classes of   and   . For any $r>0$, let   be the Banach space which consists of all such that

is finite. We observe that   (  ) is the projective (inductive) limit of   with respect to $r>0$. First we have the following two propositions.

Proposition 5.20

Let $r_1\in (0,1)$ and $r_2>r_1/(1-r_1)$. Then the following is true:

• (1) if , then there is a unique such that (2.59) holds. Furthermore,

  

  (5.23)

• (2) if , then there is a unique such that (2.59) holds. Furthermore,

  

  (5.24)

Proposition 5.21

Let $r_1>0$ and $r_2>r_1+1$. Then the following is true:

• (1) if , then there is a unique such that (4.6’) holds. Furthermore,

  

  (5.25)

• (2) if , then there is a unique such that (4.6’) holds. Furthermore,

  

  (5.26)

For the proofs we need the following continuity result for the operators ${\mathcal {T}}_{0,t}$ and ${\mathcal {T}}_{0,t}^{*}$. Here let

$$\begin{aligned} \ell _r^p({{\mathbf {N}}}^{d}) = \ell _{[\vartheta _r]}^p({{\mathbf {N}}}^{d}),\qquad \vartheta _r(\alpha )=r^{-\frac{1}{2}\cdot |\alpha |}, \ r>0,\ \alpha \in {{\mathbf {N}}}^{d} \end{aligned}$$

(cf. Definition 2.3).

Lemma 5.22

Let $p\in [1,\infty ]$, $r_1>0$, $t\in {\mathbf {C}}$ be such that $|t|\le 1$, and let ${\mathcal {T}}_{0,t}$ and ${\mathcal {T}}_{0,t}^{*}$ be given by (4.1) and (4.2), $j=1,2$. Then the following is true:

• (1) if $r_2\ge 1+r_1$ with strict inequality when $p<\infty$, then ${\mathcal {T}}_{0,t}$ restricts to a continuous map from $\ell _{r_1}^p({{\mathbf {N}}}^{2d})$ to $\ell _{r_2}^p({{\mathbf {N}}}^{2d})$;

• (2) if in addition $r_1<1$ and $r_2\ge r_1/(1-r_1)$ with strict inequality when $p>1$, then ${\mathcal {T}}_{0,t}^{*}$ restricts to a continuous map from $\ell _{r_1}^p({{\mathbf {N}}}^{2d})$ to $\ell _{r_2}^p({{\mathbf {N}}}^{2d})$.

For the proof we observe that

$$\begin{aligned} |b(\alpha ,\beta )|\le r^{\frac{1}{2}|\alpha +\beta |}\Vert b\Vert _{\ell _r^p}, \qquad b\in \ell _r^p({{\mathbf {N}}}^{2d}). \end{aligned}$$

(5.27)

Proof

By duality it suffices to prove (1).

Let $r=r_1$ and $b\in \ell _r^p({{\mathbf {N}}}^{2d})$. Then it follows from (5.27), Cauchy-Schwartz inequality and straight-forward computations that

$$\begin{aligned}&\Vert {\mathcal {T}}_{0,t}b\Vert _{\ell _{r_2}^p}^p \le \sum _{\alpha ,\beta } \left( \sum _{\gamma \le \alpha ,\beta } {{\alpha }\atopwithdelims (){\gamma }}^{\frac{1}{2}} {{\beta }\atopwithdelims (){\gamma }}^{\frac{1}{2}}|b(\alpha -\gamma ,\beta -\gamma )| \right) ^pr_2^{-\frac{p}{2}|\alpha +\beta |}\\&\quad \le \Vert b\Vert _{\ell _r^p}^p\sum _{\alpha ,\beta } \left( \sum _{\gamma \le \alpha ,\beta } {{\alpha }\atopwithdelims (){\gamma }}^{\frac{1}{2}} {{\beta }\atopwithdelims (){\gamma }}^{\frac{1}{2}}r^{-|\gamma |} \right) ^pr^{\frac{p}{2} |\alpha +\beta |}r_2^{-\frac{p}{2} |\alpha +\beta |}\\&\quad \le \Vert b\Vert _{\ell _r^p}^p\sum _{\alpha ,\beta } \left( \sum _{\gamma \le \alpha } {{\alpha }\atopwithdelims (){\gamma }} r^{-|\gamma |} \right) ^{\frac{p}{2}} \left( \sum _{\gamma \le \beta } {{\beta }\atopwithdelims (){\gamma }}r^{-|\gamma |} \right) ^{\frac{p}{2}} \left( \frac{r}{r_2} \right) ^{\frac{p}{2}|\alpha +\beta |}\\&\quad = \Vert b\Vert _{\ell _r^p}^p\sum _{\alpha ,\beta } \left( \frac{1+r}{r_2} \right) ^{\frac{p}{2} |\alpha +\beta |}. \end{aligned}$$

Since $1+r<r_2$, it follows that the last series converges, which gives the asserted continuity. $\square$

Proof of Propositions 5.20 and 5.21

If $r>0$ and has the expansion

$$\begin{aligned} a(z,w) = \sum c(a;\alpha ,\beta )e_{\alpha }(z)e_{\beta }({\overline{w}}), \quad z,w \in {\mathbf {C}}^{d}, j=1,2, \end{aligned}$$

(5.28)

then [21, Corollary 3.8] shows that

The result now follows from the latter relation, Lemma 5.22, the fact that $\ell _0({{\mathbf {N}}}^{2d})$ is dense in $\ell ^2_r({{\mathbf {N}}}^{2d})$, leading to that  is dense in  , and that (2.59) and (4.6’) hold true when . $\square$

In [18,  Section 3] some results concerning Wick symbols of anti-Wick operators are presented. For example we have the following result. We omit the proof since the result is a special case of Theorems 3.1, 3.3 and Theorem 3.7 in [18]. Here ${\mathscr {P}}_E({\mathbf {C}}^{d})$ is the set of all $\omega \in L^\infty _{loc}({\mathbf {C}}^{d};{\mathbf {R}}_+)$ such that

$$\begin{aligned} \omega (z+w)\lesssim \omega (z)v(w), \end{aligned}$$

for some $v\in L^\infty _{loc}({\mathbf {C}}^{d};{\mathbf {R}}_+)$.

Theorem 5.23

Let $s\ge \frac{1}{2}$ ($s> \frac{1}{2}$), be such that (2.60’) holds, and let $\omega \in {\mathscr {P}}_E({\mathbf {C}}^{d})$. Then the following is true:

• (1) if $|a (w,w)| \lesssim e^{-r_0 |w|^{\frac{1}{s}}}$ holds for some $r_0>0$ (for every $r_0>0$), then

  $$\begin{aligned} |a^{{\text {aw}}} (z,w)| \lesssim e^{\frac{1}{4}|z-w|^2-r |z+w|^{\frac{1}{s}}} \end{aligned}$$

  (5.29)

  for some $r>0$ (for every $r>0$);

• (2) if $|a (w,w)| \lesssim e^{r_0 |w|^{\frac{1}{s}}}$ holds for every $r_0>0$ (for some $r_0>0$), then

  $$\begin{aligned} |a^{{\text {aw}}} (z,w)| \lesssim e^{\frac{1}{4}|z-w|^2+r |z+w|^{\frac{1}{s}}} \end{aligned}$$

  (5.30)

  for every $r>0$ (for some $r>0$);

• (3) if $|a(w,w)| \lesssim e^{r|w|^2}$ for some $r<1$, then

  $$\begin{aligned} |a^{{\text {aw}}}(z,w)|\lesssim e^{r_0 |z+w|^2-{\text {Re}}(z,w)}, \qquad r_0=4^{-1}(1-r)^{-1}\text{; } \end{aligned}$$

  (5.31)

• (4) if $|a(w,w)|\lesssim \omega (2w)$, then

  $$\begin{aligned} |a^{{\text {aw}}}(z,w)|\lesssim e^{\frac{1}{4}|z-w|^2} \omega (z+w), \quad z,w\in {\mathbf {C}}^{d}. \end{aligned}$$

  (5.32)

By using (2.61) instead of (2.60), we get the following result in the other direction compared to previous result. The details are left for the reader.

Theorem 5.24

Let $s\ge \frac{1}{2}$ ($s> \frac{1}{2}$), be such that (2.60’) holds, and let $\omega \in {\mathscr {P}}_E({\mathbf {C}}^{d})$. Then the following is true:

• (1) if $|a ^{{\text {aw}}}(w,-w)| \lesssim e^{-r_0 |w|^{\frac{1}{s}}}$ holds for some $r_0>0$ (for every $r_0>0$), then

  $$\begin{aligned} |a(z,w)| \lesssim e^{\frac{1}{4}|z+w|^2-r |z-w|^{\frac{1}{s}}} \end{aligned}$$

  (5.33)

  for some $r>0$ (for every $r>0$);

• (2) if $|a^{{\text {aw}}} (w,-w)| \lesssim e^{r_0 |w|^{\frac{1}{s}}}$ holds for every $r_0>0$ (for some $r_0>0$), then

  $$\begin{aligned} |a(z,w)| \lesssim e^{\frac{1}{4}|z+w|^2+r |z-w|^{\frac{1}{s}}} \end{aligned}$$

  (5.34)

  for every $r>0$ (for some $r>0$);

• (3) if $|a^{{\text {aw}}}(w,-w)| \lesssim e^{r|w|^2}$ for some $r<1$, then

  $$\begin{aligned} |a(z,w)|\lesssim e^{r_0 |z-w|^2+{\text {Re}}(z,w)}, \qquad r_0=4^{-1}(1-r)^{-1}\text{; } \end{aligned}$$

  (5.35)

• (4) if $|a^{{\text {aw}}}(w,-w)|\lesssim \omega (2w)$, then

  $$\begin{aligned} |a(z,w)|\lesssim e^{\frac{1}{4}|z+w|^2} \omega (z-w), \quad z,w\in {\mathbf {C}}^{d}. \end{aligned}$$

  (5.36)

Remark 5.25

The relationships between the Wick and anti-Wick symbols in Theorems 5.23 and 5.24 are similar, and one might believe that transitions of continuity properties for Wick operators to anti-Wick operators are similar as for transitions in reversed directions. Here we notice that this is not the case.

For example, suppose that $a^{{\text {aw}}}$ satisfies (5.30) for every $r>0$, $0<t_0<\frac{3}{4}$ and that $F\in A({\mathbf {C}}^{d})$ satisfies

$$\begin{aligned} |F(z)|\lesssim e^{t_0|z|^2+r|z|^{\frac{1}{s}}} \end{aligned}$$

(5.37)

for every $r>0$. (The case when $a^{{\text {aw}}}$ satisfies conditions of the form (5.29) is treated in similar ways and leads to even stronger continuity properties.) If

$$\begin{aligned} t_1= \frac{1-t_0}{3-4t_0} \end{aligned}$$

(which is bounded from below by $t_0$), then the size of the integrand in

$$\begin{aligned} |({\text {Op}}_{{\mathfrak {V}}}(a^{{\text {aw}}})F)(z)| \le \pi ^{-d} \int _{{\mathbf {C}}^{d}}|a^{{\text {aw}}}(z,w)F(w)|e^{{\text {Re}}(z,w)-|w|^2}\, d\lambda (w) \end{aligned}$$

can be estimated as

$$\begin{aligned} |a^{{\text {aw}}}(z,w)F(w)&|e^{{\text {Re}}(z,w)-|w|^2} \nonumber \\&\lesssim e^{\frac{1}{4}|z-w|^2+r|z+w|^{\frac{1}{s}}}e^{{\text {Re}}(z,w)-|w|^2}e^{t_0|w|^2+r|w|^{\frac{1}{s}}} \nonumber \\&= e^{t_1|z|^2-\frac{3-4t_0}{4}|w-\frac{z}{3-4t_0}|^2+ r(|z+w|^{\frac{1}{s}}+|w|^{\frac{1}{s}}}\nonumber \\&\lesssim e^{t_1|z|^2+cr|z|^{\frac{1}{s}}} e^{-\frac{3-4t_0}{4}|w-\frac{z}{3-4t_0}|^2+ cr|w-\frac{z}{3-4t_0}|^{\frac{1}{s}}}, \end{aligned}$$

(5.38)

for some $c\ge 1$ which is independent of $z,w\in {\mathbf {C}}^{d}$ and $r>0$. By integrating the last estimate with respect to w, it follows that (5.37) holds true for every $r>0$ with ${\text {Op}}_{{\mathfrak {V}}}(a^{{\text {aw}}})F$ and $t_1$ in place of F and $t_0$.

By choosing $t_0=\frac{1}{2}$, then $t_1=\frac{1}{2}$, and we have proved that ${\text {Op}}_{{\mathfrak {V}}}(a^{{\text {aw}}})$ is continuous on ${\mathcal {A}}^{\star }_s({\mathbf {C}}^{d})$ for such $t_0$.

As side remark we observe that stronger continuity properties for anti-Wick operators with symbols satisfying estimates of the form $a(w,w)e^{\pm r|w|^{\frac{1}{s}}}$ are obtained with more direct computational methods, without rewriting ${\text {Op}}_{{\mathfrak {V}}}^{{\text {aw}}}(a)$ as a Wick operator. (See Proposition 3.6 to Corollary 3.10 in [18].)

The corresponding estimates (5.33) and (5.34) when passing from Wick to anti-Wick operators lead to strongly different conclusions. In fact, let $F\in A({\mathbf {C}}^{d})$. For the integrand in

$$\begin{aligned} |({\text {Op}}_{{\mathfrak {V}}}^{{\text {aw}}}(a)F)(z)| \le \pi ^{-d} \int _{{\mathbf {C}}^{d}}|a(w,w)F(w)|e^{{\text {Re}}(z,w)-|w|^2}\, d\lambda (w) \end{aligned}$$

a similar type of estimate gives

$$\begin{aligned} |a(w,w)F(w)|e^{{\text {Re}}(z,w)-|w|^2}\lesssim & {} e^{\frac{1}{4}|w+w|^2\pm r|w-w|^{\frac{1}{s}}}e^{{\text {Re}}(z,w)-|w|^2}|F(w)| \nonumber \\= & {} e^{{\text {Re}}(z,w)}|F(w)|. \end{aligned}$$

(5.39)

Since F should be entire, the right-hand side is integrable with respect to w, only when F is identically zero. Consequently, (5.33) and (5.34) might be useful only when F is the trivial zero function.

In this context we observe that continuity properties for ${\text {Op}}_{{\mathfrak {V}}}(a^{{\text {aw}}})$ are reachable with more direct computational methods, without rewriting it as an anti-Wick operator. (See [17, 18].)

5.5 Some Refined Topological Properties of Bilinear Map**s

Next we refine some of the continuity assertions in Propositions 4.7, 4.8, 4.9 and 5.12, and Theorems 5.15, 5.18 and 5.19, by putting the involved topological vector spaces in the framework of barreled spaces and DF-spaces. In the following proposition we list certain properties on topological vector spaces which we need. We refer to [10, 11, 14, 23] for definitions.

Proposition 5.26

Let ${\mathcal {V}}_1$, ${\mathcal {V}}_2$ be barreled, ${\mathcal {W}}$ local convex topological vector space and let T be a separately continuous bilinear map from ${\mathcal {V}}_1\times {\mathcal {V}}_2$ to ${\mathcal {W}}$. Then the following is true:

• (1) every LF-space and LB-space are barreled;

• (2) every Banach space is a DF-space;

• (3) a countable inductive limit of barreled spaces is barreled;

• (4) a countable inductive limit of DF-spaces is a DF-space;

• (5) every LB-space is a DF-space;

• (6) T is hypocontinuous;

• (7) if in addition ${\mathcal {V}}_1$ and ${\mathcal {V}}_2$ are DF-spaces, then T is continuous.

As remarked above, the assertions in the previous proposition are well-known. In order to assist the reader we link the assertions to the literature.

Proof

By 7.1 in [14,  Chapter II] it follows that each F-space is barreled, and by 7.2 in [14,  Chapter II] it follows that inductive limits of barreled spaces are barreled. The assertion (1) now follows by combining these properties, and using the fact that every Banach space is a Fréchet space.

The assertion (2) follows essentially from the definitions (see also 3 on page 396 in [10]), and (3) follows from Corollary 1 in 7.1 on page 61 in [14,  Chapter II].

The assertion (4) follows from (4) on page 402 in [10] (see also Exercise 24 e) on pages 196–197 in [14]). The assertion (5) follows by combining (2) and (3).

Finally, (6) and (7) follows from (5) and (11) on pages 159 and 161 in [11]. $\square$

We have now the following.

Proposition 5.27

Let $j=1,2,3$, $k\in {\mathbf {N}}$, $\mathcal {A}_{j,k}$ and $\mathcal {A}_{j,k}'$ be Banach spaces such that $\mathcal {A}_{j,k}\hookrightarrow \mathcal {A}_{j,k+1}$ and $\mathcal {A}_{j,k+1}'\hookrightarrow \mathcal {A}_{j,k}'$ for every $k\in {\mathbf {N}}$. Also let $\mathcal {A}_j$ be the inductive limit of $\mathcal {A}_{j,k}$ with respect to k, $\mathcal {A}_j'$ be the projective limit of $\mathcal {A}_{j,k}'$ with respect to k, and suppose that the bilinear map**s

$$\begin{aligned} T_{\mathcal {A}} : \, \mathcal {A}_1\, \times \mathcal {A}_2\, \rightarrow \, \mathcal {A}_3, \qquad T_{\mathcal {A}}' : \, \mathcal {A}_1'\, \times \, \mathcal {A}_2' \, \rightarrow \, \mathcal {A}_3', \end{aligned}$$

(5.40)

are separately continuous. Then the map**s in (5.40) are continuous.

Proof

The continuity of the first map in (5.40) follows by combining (2), (4) and (7) in Proposition 5.26. The continuity of the second map in (5.40) follows from Banach-Steinhaus theorem on bilinear map**s, and the fact that $\mathcal {A}_j'$ are Fréchet spaces. This gives the result. $\square$

In the next proposition we extend the previous proposition to allow projective limits of LB-spaces $\mathcal {B}_{j,k}$. It is then assumed that the bilinear map**s

$$\begin{aligned} \begin{aligned} T_{\mathcal {A},\mathcal {B}} : \, \mathcal {A}_1\, \times \, \mathcal {B}_{2,\nu (k)}\, \rightarrow \, \mathcal {B}_{3,k}, \qquad T_{\mathcal {A},\mathcal {B}}'&:&\, \mathcal {A}_{1,\nu (k)}' \,&\times \, \mathcal {B}_{2,\nu (k)}\,&\rightarrow \, \mathcal {B}_{3,k} \\ \text {and}\qquad \qquad T_{\mathcal {B}}&:&\, \mathcal {B}_{1,\nu (k)}\,&\times \, \mathcal {B}_{2,\nu (k)}\,&\rightarrow \, \mathcal {B}_{3,k}, \end{aligned}\nonumber \\ \end{aligned}$$

(5.41)

are separately continuous, where $\nu$ should satisfy

$$\begin{aligned} \lim _{k\rightarrow \infty }\nu (k) =\infty , \end{aligned}$$

(5.42)

Proposition 5.28

Let $j=1,2,3$, $k\in {\mathbf {N}}$, $\mathcal {A}_j$, $\mathcal {A}_j'$ and $\mathcal {A}_{j,k}'$ be the same as in Proposition 5.27, and let $\mathcal {B}_{j,k}$ be an LB-space such that $\mathcal {B}_{j,k+1}\hookrightarrow \mathcal {B}_{j,k}$ for every $k\in {\mathbf {N}}$. Also let $\mathcal {B}_j$ be the projective limit of $\mathcal {B}_{j,k}$ with respect to k, $\mathcal {C}_j$ be an LF-space, $\nu$ from ${\mathbf {N}}$ to ${\mathbf {N}}$ satisfies (5.42) and suppose that the bilinear map**s in (5.41) and in

$$\begin{aligned} T_{\mathcal {A},\mathcal {A}'}&:&\, \mathcal {A}_1\,&\times \, \mathcal {A}_2'\,&\rightarrow \, \mathcal {A}_3',&\qquad T_{\mathcal {A}',\mathcal {A}}'&:&\, \mathcal {A}_1' \,&\times \, \mathcal {A}_2\,&\rightarrow \, \mathcal {A}_3, \end{aligned}$$

(5.43)

$$\begin{aligned} T_{\mathcal {A},\mathcal {B}}&:&\, \mathcal {A}_1\,&\times \, \mathcal {B}_2\,&\rightarrow \, \mathcal {B}_3,&\qquad T_{\mathcal {A},\mathcal {B}}'&:&\, \mathcal {A}_1' \,&\times \, \mathcal {B}_2\,&\rightarrow \, \mathcal {B}_3, \end{aligned}$$

(5.44)

$$\begin{aligned} T_{\mathcal {B}}&:&\, \mathcal {B}_1\,&\times \, \mathcal {B}_2\,&\rightarrow \, \mathcal {B}_3,&\qquad \text {and}\qquad T_{\mathcal {C}}&:&\, \mathcal {C}_1\,&\times \, \mathcal {C}_2\,&\rightarrow \, \mathcal {C}_3, \end{aligned}$$

(5.45)

are separately continuous. Then the following is true:

• (1) the map**s in (5.44) and the first map in (5.45) are continuous;

• (2) the map**s in (5.43) and the second map in (5.45) are hypocontinuous.

Proof

By combining (2), (5) and (7) in Proposition 5.26, it follows that the map**s in (5.41) are continuous. The continuity assertions in (1) now follows from general relationships between continuity for sequences of spaces and their projective limits (see e. g. [14, Chapter II]).

The hypocontinuity of the second map in (5.45) follows from (1) and (6) in Proposition 5.26, and the result follows. $\square$

Proof of Theorem 5.13

The result follows by combining Propositions 5.12, 5.27 and 5.28. $\square$

We have now the following result.

Theorem 5.29

Let $s_j\in \overline{{\mathbf {R}}}_\flat$ be such that $s_j<\frac{1}{2}$, $j=0,1,2$. Then the following is true:

• (1) The map**s (4.39), (4.40), (4.45), (4.47), (5.17), (5.19) and the first map in (5.22) are continuous;

• (2) the map**s (4.41), (4.42), (4.48), (5.13), (5.16), (5.20) and the last two map**s in (5.22) are hypocontinuous.

If instead $0<s_j\le \frac{1}{2}$, $j=0,1,2$, then the same holds true with

$$\begin{aligned}&\ell _{{\mathcal {A}},0,s},&\quad&\ell _{{\mathcal {A}},0,s_0},&\quad&\ell _{{\mathcal {B}},0,s},&\quad&\ell _{{\mathcal {B}},0,s_0},&\quad&\ell _{{\mathcal {C}},0,s},&\quad&\ell _{{\mathcal {C}},0,s_0}, \end{aligned}$$

(5.46)

and corresponding spaces in Definitions 2.14 and 2.16 linked by $\Theta _C\circ T_{{\mathcal {A}}}$, in place of

$$\begin{aligned}&\ell _{{\mathcal {A}},s},&\quad&\ell _{{\mathcal {A}},s_0},&\quad&\ell _{{\mathcal {B}},s},&\quad&\ell _{{\mathcal {B}},s_0},&\quad&\ell _{{\mathcal {C}},s},&\quad&\ell _{{\mathcal {C}},s_0}, \end{aligned}$$

(5.47)

and corresponding spaces in Definitions 2.14 and 2.16 linked by $\Theta _C\circ T_{{\mathcal {A}}}$, respectively, at each occurrence.

Proof

We only prove the result for bilinear map**s involving spaces of the form (5.47). The other assertions follow by similar arguments and from Definitions 2.14 and 2.16, and are left for the reader.

By Remark 2.10 and the estimates (4.50) and (4.58), it follows that

$$\begin{aligned} \mathcal {A}_j = \ell _{{\mathcal {A}},s}, \quad \mathcal {A}_j^{\star '} = {\ell ^{\star }_{{\mathcal {A}},s}}, \quad \mathcal {B}_j = \ell _{{\mathcal {B}},s}, \quad \text {and}\quad \mathcal {C}_j = \ell _{{\mathcal {C}},s}, \end{aligned}$$

fulfills the hypotheses in Propositions 5.27 and 5.28. The result now follows by combining Propositions 4.7, 4.8, 4.9, 5.27 and 5.28. $\square$

6 Applications to Linear Operators on Functions and Distributions Defined on ${\mathbf {R}}^{d}$

In this section we use the results in previous sections to extend the definition of Toeplitz operators. In the end we find that if $s\in \overline{{\mathbf {R}}_\flat }$ is suitable, then a large class of linear and continuous operators on ${\mathcal {A}}_s({\mathbf {C}}^{d})$, ${\mathcal {A}}_{0,s}({\mathbf {C}}^{d})$ and their duals, can be expressed as Toeplitz operators.

If , then ${\text {Tp}}_{{\mathfrak {V}}}(a)$ is the linear and continuous operator on ${\mathcal {S}}_{1/2}({\mathbf {R}}^{d})$, given by (2.64)$''$. For such operators we have the following extensions.

Theorem 6.1

Let $s_1,s_2\in \overline{{\mathbf {R}}_\flat }$ be such that $s_1<\frac{1}{2}$ and $0<s_2\le \frac{1}{2}$. Then the following is true:

• (1) the map $(a,f)\mapsto {\text {Tp}}_{{\mathfrak {V}}}(a)f$ from  to ${\mathcal {H}}_0^{\star }({\mathbf {R}}^{d})$ is uniquely extendable to a separately continuous map from   to ${\mathcal {H}}_{s_1}({\mathbf {R}}^{d})$, and from   to ${\mathcal {H}}^\star _{s_1}({\mathbf {R}}^{d})$;

• (2) the map $(a,f)\mapsto {\text {Tp}}_{{\mathfrak {V}}}(a)f$ from  to ${\mathcal {H}}^{\star }_0({\mathbf {R}}^{d})$ is uniquely extendable to a separately continuous map from to ${\mathcal {H}}_{0,s_2}({\mathbf {R}}^{d})$, and from   to ${\mathcal {H}}^{\star }_{0,s_2}({\mathbf {R}}^{d})$.

Proof

The asserted continuity extensions follow from (2.64’), (2.65), Theorems 5.8 and 5.9, and the facts that the Bargmann transform is homeomorphic from the spaces in (2.28) to the spaces in (2.29). We need to prove the uniqueness.

By playing with $r_1$ and $r_2$ in Definition 2.5, it follows that $\ell _0(\Lambda )$ is dense in the spaces in (2.11), which implies that   is dense in the spaces in (2.39). The uniqueness in Theorem 6.1 now follows by combining these density properties with the fact that ${\mathcal {H}}_0({\mathbf {R}}^{d})$ is dense in the spaces in (2.28). $\square$

Theorem 6.2

Let $s_1,s_2\in \overline{{\mathbf {R}}_\flat }$ be such that $s_1<\frac{1}{2}$ and $0<s_2\le \frac{1}{2}$. If and ${\text {Tp}}_{{\mathfrak {V}}}(a_1)f={\text {Tp}}_{{\mathfrak {V}}}(a_2)f$ for every $f\in {\mathcal {H}}_0({\mathbf {R}}^{d})$, then $a_1=a_2$. The same holds true for    and    in place of    .

Proof

Suppose that satisfy ${\text {Tp}}_{{\mathfrak {V}}}(a_1)f={\text {Tp}}_{{\mathfrak {V}}}(a_2)f$ for every $f\in {\mathcal {H}}_0({\mathbf {R}}^{d})$. Then Theorem 5.3 gives

$$\begin{aligned} {\text {Tp}}_{{\mathfrak {V}}}(a_1) =&{\text {Tp}}_{{\mathfrak {V}}}(a_2) \quad \quad \Leftrightarrow {\text {Op}}_{{\mathfrak {V}}}^{{\text {aw}}}(a_1) = {\text {Op}}_{{\mathfrak {V}}}^{{\text {aw}}}(a_2) \Leftrightarrow \\ {\text {Op}}_{{\mathfrak {V}}}({\mathcal {T}}_1^{*}a_1) =&{\text {Op}}_{{\mathfrak {V}}}({\mathcal {T}}_1^{*}a_2)\, \Leftrightarrow \quad \quad \, {\mathcal {T}}_1^{*}a_1 = {\mathcal {T}}_1^{*}a_2 \qquad \Leftrightarrow \quad a_1 = a_2. \end{aligned}$$

$\square$

7 Linear Pullbacks and Trace Map**s for Spaces of Power Series Expansions

In this section we show that for $0\le s< \frac{1}{2}$ (for $0< s\le \frac{1}{2}$), then linear pullbacks and trace map**s are continuous map**s on the spaces in (5.11) (in (5.12)).

For $B_j\in \mathbf {M}(d_j,{\mathbf {C}})$, $j=1,2$, we consider linear pullbacks of the form

$$\begin{aligned} K(z_2,z_1)&\mapsto K(z_2,B_1z_1) \end{aligned}$$

(7.1)

$$\begin{aligned} K(z_2,z_1)&\mapsto K(B_2z_2,z_1) \end{aligned}$$

(7.2)

and

$$\begin{aligned} K(z_2,z_1)&\mapsto K(B_2z_2,B_1z_1). \end{aligned}$$

(7.3)

Here $\mathbf {M}(d,{\mathbf {C}})$ is the set of all $d\times d$ matrices with entries in ${\mathbf {C}}$. For such pullbacks we have the following.

Theorem 7.1

Let $s_1,s_2\in \overline{{\mathbf {R}}_{\flat ,\infty }}$, $s=(s_2,s_1)$, $W={\mathbf {C}}^{d_2}\times {\mathbf {C}}^{d_1}$ and $B_j\in \mathbf {M}(d_j,{\mathbf {C}})$. Then the following is true:

• (1) the map (7.3) on extends uniquely to a continuous map on   ;

• (2) if in addition $s_1<\frac{1}{2}$ ($0<s_1\le \frac{1}{2}$), then (7.1) on   restricts to continuous map**s on the spaces in (5.11) (in (5.12));

• (3) if in addition $s_2<\frac{1}{2}$ ($0<s_2\le \frac{1}{2}$), then (7.2) on  restricts to continuous map**s on the spaces in (5.11) (in (5.12));

• (4) if in addition $s_1,s_2< \frac{1}{2}$ ($0<s_1,s_2\le \frac{1}{2}$), then (7.3) on  restricts to continuous map**s on the spaces in (5.11) (in (5.12)).

Proof

We only prove (1) and (3). The assertion (2) follows from similar arguments and then (4) follows by combining (2) and (3). The details are left for the reader.

Let $B=B_2$, $d=d_2$ and $z=z_2\in {\mathbf {C}}^{d}$. For any integer $N\ge 0$ we let

$$\begin{aligned} \Omega _N = \Omega _{N,d} = \{ \, \alpha \in {{\mathbf {N}}}^{d}\, ;\, |\alpha |=N\, \} . \end{aligned}$$

Then

$$\begin{aligned} e_\alpha (Bz) = \sum _{\beta \in \Omega _N} C_B(\beta ,\alpha )e_\beta (z), \quad \alpha \in \Omega _N,\ z\in {\mathbf {C}}^{d}, \end{aligned}$$

(7.4)

for some constants $C_B(\beta ,\alpha )$ which are uniquely defined and only depend on $\alpha$, $\beta$ and B. If is given by (3.5), then

$$\begin{aligned} K_B(z_2,z_1)\equiv K(Bz_2,z_1)&= \sum _{\alpha _j\in {{\mathbf {N}}}^{d_j}} c(K_B;\alpha _2,\alpha _1)e_{\alpha _2}(z_2)e_{\alpha _1}(\overline{z_1}), \end{aligned}$$

(7.5)

where

$$\begin{aligned} c(K_B;\alpha _2,\alpha _1)&= \sum _{\beta \in \Omega _{|\alpha _2|}}C_B(\alpha _2,\beta ) c(K;\beta ,\alpha _1). \end{aligned}$$

(7.6)

Hence, if, more generally   , then the only possibility to define $K(B_2z_2,z_1)$ is given by (7.5) and (7.6). Since $\Omega _N$ is a finite set for every N, it follows that (7.6) is well-defined and that the map

$$\begin{aligned} \{ c(K;\alpha _2,\alpha _1)\} _{(\alpha _2,\alpha _1)\in {{\mathbf {N}}}^{d_2}\times {{\mathbf {N}}}^{d_1}} \mapsto \{ c(K_B;\alpha _2,\alpha _1)\} _{(\alpha _2,\alpha _1)\in {{\mathbf {N}}}^{d_2}\times {{\mathbf {N}}}^{d_1}} \end{aligned}$$

is continuous on

$$\begin{aligned}&\ell _{{\mathcal {A}}_0}({{\mathbf {N}}}^{d_2}\times {{\mathbf {N}}}^{d_1}),&\quad&\ell _{{\mathcal {A}}_0^{\star }}({{\mathbf {N}}}^{d_2}\times {{\mathbf {N}}}^{d_1}),&\quad&\ell _{{\mathcal {B}}_{(0,0)}}({{\mathbf {N}}}^{d_2}\times {{\mathbf {N}}}^{d_1}), \\&\ell _{{\mathcal {B}}_{(0,0)}}^\star ({{\mathbf {N}}}^{d_2}\times {{\mathbf {N}}}^{d_1}),&\quad&\ell _{{\mathcal {C}}_{(0,0)}} ({{\mathbf {N}}}^{d_2}\times {{\mathbf {N}}}^{d_1})&\quad&\text {and on}\quad \ell _{{\mathcal {C}}_{(0,0)}}^\star ({{\mathbf {N}}}^{d_2}\times {{\mathbf {N}}}^{d_1}). \end{aligned}$$

Hence the map (7.3) on is uniquely extendable to a continuous map on   , which in turn restricts to continuous map**s on the spaces in (5.11) when $s_1=s_2=0$. This gives (1) and (3) for $s_1=s_2=0$.

We only prove (3) in the case when $s_1,s_2>0$. The case when $s_1=s_2=0$ is already treated and the case when one of $s_1$ and $s_2$ is zero follows by similar arguments and is left for the reader.

Let $\{ b(j,k) \} _{k=1}^d$be the entries in the row j of B and let

Then

$$\begin{aligned} e_\alpha (Bz)&= \frac{1}{\alpha !^{\frac{1}{2}}} \prod _{j=1}^d (b(j,1)z_1+\cdots +b(j,d)z_d)^{\alpha _j} \\&= \frac{1}{\alpha !^{\frac{1}{2}}} \prod _{j=1}^d \left( \sum _{\gamma _j\in \Omega _{\alpha _j}} {{\alpha _j}\atopwithdelims (){\gamma _j}} \prod _{m=1}^{d} \left( b(j,m)^{\gamma _{j,m}}z_m^{\gamma _{j,m}} \right) \right) \\&= \frac{1}{\alpha !^{\frac{1}{2}}} \sum _{\gamma _1\in \Omega _{\alpha _1}} \cdots \sum _{\gamma _d\in \Omega _{\alpha _d}} \left( \prod _{j=1}^d {{\alpha _j}\atopwithdelims (){\gamma _j}} \prod _{m=1}^d \left( b(j,m)^{\gamma _{j,m}}z_m^{\gamma _{j,m}} \right) \right) . \end{aligned}$$

Here $\gamma _j=(\gamma _{j,1},\dots ,\gamma _{j,d})$ in the expressions above. If we let

$$\begin{aligned} c(\gamma ) = \prod _{j,m=1}^d b(j,m)^{\gamma _{j,m}}, \end{aligned}$$

(7.7)

and using (2.26), then it follows by straight-forward computations that

(7.8)

where

$$\begin{aligned} \delta _j&= (\gamma _{1,j},\dots ,\gamma _{d,j}) \quad \text {and}\quad \beta _j=\gamma _{1,j}+\cdots +\gamma _{d,j}. \end{aligned}$$

(7.9)

This in turn implies that

where the sum is taken over all

$$\begin{aligned} \alpha \in {{\mathbf {N}}}^{d},\quad \varrho \in {{\mathbf {N}}}^{d_1} \quad \text {and}\quad \gamma _j\in \Omega _{\alpha _j},\ j=1,\dots ,d. \end{aligned}$$

(7.10)

Hence

(7.11)

where the sum is taken over all $\alpha$ and $\gamma _j$ in (7.10) such that (7.9) holds. Here we observe that

$$\begin{aligned} |\alpha | = |\beta | = \sum _{j,k=1}^d \gamma _{j,k}. \end{aligned}$$

(7.12)

This implies that

for some constant $C\ge 1$. Here the last estimate follows from (7.7). Since it is clear that the number of elements in the sum in (7.11) is bounded by

$$\begin{aligned} (1+\alpha _1)^d\cdots (1+\alpha _d)^d\lesssim C^{|\alpha |}, \end{aligned}$$

for some constant $C\ge 1$, it follows by combining these estimates and (7.12) that

$$\begin{aligned} \max _{|\alpha |=N}|c(K_B;\alpha ,\varrho )| \le C^N \max _{|\alpha |=N}|c(K;\alpha ,\varrho )| \end{aligned}$$

for some C which is independent of N and $\varrho$.

If $\vartheta _{r,(s_2,s_1)}$ and $\omega _{s_2,s_1;r_2,r_1}$ are the same as in Definitions 2.4 and 2.5, it follows by the assumptions on $s_2$ that

$$\begin{aligned} \sup _{\alpha \in {{\mathbf {N}}}^{d}} |c(K_B;\alpha ,\varrho )\vartheta _{r,(s_2,s_1)}(\alpha ,\varrho )|&\le C\sup _{\alpha \in {{\mathbf {N}}}^{d}} |c(K;\alpha ,\varrho )\vartheta _{c_1r,(s_2,s_1)}(\alpha ,\varrho )|,\\ \sup _{\alpha \in {{\mathbf {N}}}^{d}} |c(K_B;\alpha ,\varrho )/\vartheta _{r,(s_2,s_1)}(\alpha ,\varrho )|&\le C\sup _{\alpha \in {{\mathbf {N}}}^{d}} |c(K;\alpha ,\varrho )/\vartheta _{c_2r,(s_2,s_1)}(\alpha ,\varrho )|,\\ \sup _{\alpha \in {{\mathbf {N}}}^{d}} |c(K_B;\alpha ,\varrho )\omega _{r_2,r_1;s_2,s_1}(\alpha ,\varrho )|&\le C\sup _{\alpha \in {{\mathbf {N}}}^{d}} |c(K;\alpha ,\varrho )\omega _{c_1r_2,r_1;s_2,s_1}(\alpha ,\varrho )|, \end{aligned}$$

and

$$\begin{aligned} \sup _{\alpha \in {{\mathbf {N}}}^{d}} |c(K_B;\alpha ,\varrho )/\omega _{r_2,r_1;s_2,s_1}(\alpha ,\varrho )|&\le C\sup _{\alpha \in {{\mathbf {N}}}^{d}} |c(K;\alpha ,\varrho )/\omega _{c_2r_2,r_1;s_2,s_1}(\alpha ,\varrho )|, \end{aligned}$$

for some constants $c_1,c_2,C>0$ which only depend on $s_2$ and d. The asserted continuity properties in (3) now follows from these estimates. $\square$

Next we consider generalized and twisted forms of trace map**s on the spaces in (5.11) and (5.12), given in the following definition.

Definition 7.2

Let $d,d_j,n,n_j\in {\mathbf {N}}$, $\{ e_{j,d}\} _{j=1}^d$ be the canonical basis on ${\mathbf {C}}^{d}$ and let $S_d$ be the permutation group on $\{ 1,\dots ,d\}$.

• (1) For any $\tau \in S_{d+n}$, $\iota _{\tau ,d}$ is the linear map from ${\mathbf {C}}^{d}$ to ${\mathbf {C}}^{d+n}$ such that

  $$\begin{aligned} \iota _{\tau ,d}e_{j,d} = e_{\tau (j),d+n} , \quad j=1,\dots ,d \text{. } \end{aligned}$$

• (2) The twisted trace map ${\text {Tr}}_{\tau ,d}$ with respect to $\tau \in S_{d+n}$, d and n is the continuous (pullback) map from  ${\mathcal {A}}^{\star }_0({\mathbf {C}}^{d+n})$ to ${\mathcal {A}}^{\star }_0({\mathbf {C}}^{d})$ given by

  $$\begin{aligned} ({\text {Tr}}_{\tau ,d}F)(z) \equiv F(\iota _{\tau ,d}(z)), \qquad F \in {\mathcal {A}}_0^\star ({\mathbf {C}}^{d+n}),\ z\in {\mathbf {C}}^{d} \text{. } \end{aligned}$$

• (3) The twisted trace map ${\text {Tr}}_{\tau ,d}$ with respect to $\tau =(\tau _2,\tau _1)$ and $d=(d_2,d_1)$, $\tau _j\in S_{d_j+n_j}$, $j=1,2$, is the continuous (pullback) map from    to   given by

  

If $\tau \in S_{d+n}$ and $\iota ^*_{\tau ,d}$ is the map from ${{\mathbf {N}}}^{d}$ to ${{\mathbf {N}}}^{n+d}$, given by

$$\begin{aligned} \iota ^*_{\tau ,d}(\alpha ) = (\beta _1,\dots ,\beta _{d+n}), \quad \text {where}\quad \beta _j = {\left\{ \begin{array}{ll} \alpha _{\tau ^{-1}(j)}, &{} 1\le \tau ^{-1}(j)\le d, \\ 0, &{} d< \tau ^{-1}(j)\le d+n, \end{array}\right. } \end{aligned}$$

(7.13)

then it follows that

$$\begin{aligned} ({\text {Tr}}_{\tau ,d}F)(z) = \sum _{\alpha \in {{\mathbf {N}}}^{d}} c(F;\iota ^*_{\tau ,d}(\alpha ))e_\alpha (z), \qquad z\in {\mathbf {C}}^{d}, \end{aligned}$$

(7.14)

when

$$\begin{aligned} F(z) = \sum _{\alpha \in {{\mathbf {N}}}^{d+n}} c(F;\alpha )e_\alpha (z), \qquad z\in {\mathbf {C}}^{d+n}. \end{aligned}$$

Now let $\vartheta _{r,(s_2,s_1)}$ and $\omega _{s_2,s_1;r_2,r_1}$ be the same as in Definitions 2.4 and 2.5. Then it follows that

$$\begin{aligned}&\sup _{\alpha _j\in {{\mathbf {N}}}^{d_j}} |c(K;\iota ^*_{\tau ,d_2}(\alpha _2),\iota ^*_{\tau ,d_1}(\alpha _1)) \vartheta _{r,(s_2,s_1)} (\iota ^*_{\tau ,d_2}(\alpha _2),\iota ^*_{\tau ,d_1}(\alpha _1))|\\&\qquad \qquad \qquad \qquad \qquad \qquad \qquad \le \sup _{\alpha _j\in {{\mathbf {N}}}^{d_j+n_j}} |c(K;\alpha _2,\alpha _1) \vartheta _{r,(s_2,s_1)} (\alpha _2,\alpha _1)|, \end{aligned}$$

when , and similarly with

$$\begin{aligned} 1/\vartheta _{r,(s_2,s_1)},\quad \omega _{s_2,s_1;r_2,r_1} \quad \text {or}\quad 1/\omega _{s_2,s_1;r_2,r_1} \end{aligned}$$

in place of $\vartheta _{r,(s_2,s_1)}$. The following twisted trace result is now a straight-forward consequence of these observations. The details are left for the reader.

Proposition 7.3

Let $s_1,s_2\in \overline{{\mathbf {R}}_{\flat }}$, $s=(s_1,s_2)$, ${\text {Tr}}_{\tau ,d}$ be as in (3) in Definition 7.2, $W_1={\mathbf {C}}^{d_2}\times {\mathbf {C}}^{d_1}$ and $W_2={\mathbf {C}}^{d_2+n_2}\times {\mathbf {C}}^{d_1+n_1}$. Then ${\text {Tr}}_{\tau ,d}$ restricts to continuous map**s from

to

respectively.

If instead $s_1,s_2\in {\mathbf {R}}_{\flat ,\infty }$, then the same holds true with   ,   and   in place of    ,   and    , respectivley, at each occurrence.

We observe that we may combine Proposition 7.3 and Theorem 7.1 to get an alternative proof of Theorem 5.13.

Appendices

Appendix A. Identifications of spaces of power series expansions in terms of spaces of analytic functions

In this appendix we identify the spaces in (2.39) and (2.40) by convenient subspaces of , for suitable $s\in \overline{{\mathbf {R}}_\flat }$.

We start with the following. We omit the proof since the result is an immediate consequences of Theorems 4.1, 4.2, 5.2 and 5.3 in [21] and Definition 2.14. Here let

$$\begin{aligned} \kappa _{1,r,s}(z)&= {\left\{ \begin{array}{ll} e^{r(\log \langle z\rangle )^{\frac{1}{1-2s}}}, &{} s<\frac{1}{2} \\ e^{r|z|^{\frac{2\sigma }{\sigma +1}}}, &{} s=\flat _\sigma ,\ \sigma >0, \\ e^{\frac{|z|^2}{2}-r|z|^{\frac{1}{s}}}, &{} \frac{1}{2} \le s<\infty , \\ e^{\frac{|z|^2}{2}}\langle z\rangle ^{-r}, &{} s=\infty , \end{array}\right. } \end{aligned}$$

(A.1)

$$\begin{aligned} \kappa _{2,r,s}(z)&= {\left\{ \begin{array}{ll} e^{r|z|^{\frac{2\sigma }{\sigma -1}}}, &{} s=\flat _\sigma ,\ \sigma >1, \\ e^{\frac{|z|^2}{2}+r|z|^{\frac{1}{s}}}, &{} \frac{1}{2} \le s<\infty , \\ e^{\frac{|z|^2}{2}}\langle z\rangle ^{r}, &{} s=\infty , \end{array}\right. } \end{aligned}$$

(A.2)

and

$$\begin{aligned} \kappa _{j,r,s}^0(z)&= {\left\{ \begin{array}{ll} \kappa _{j,r,s}(z), &{} s\ne \frac{1}{2}, \\ e^{r|z|^2}, &{} s=\frac{1}{2}, \end{array}\right. } \qquad j=1,2. \end{aligned}$$

(A.3)

Theorem A.1

Let $s_j,t_j\in \overline{{\mathbf {R}}_{\flat ,\infty }}$ be such that $t_j>\flat _1$, $j=1,2$, $W={\mathbf {C}}^{d_2}\times {\mathbf {C}}^{d_1}$ and let $\kappa _{j,r,s}$ and $\kappa _{j,r,s}^0$ be given by (A.1)–(A.3), $j=1,2$, when $r>0$. Then the following is true:

• (1) if $s_1,s_2<\infty$, then   consists of all such that $|K(z_2,z_1)|\lesssim \kappa _{1,r,s_2}(z_2)\kappa _{1,r,s_1}(z_1)$ for some $r>0$;

• (2) if $s_1,s_2>0$, then   consists of all such that $|K(z_2,z_1)|\lesssim \kappa _{1,r,s_2}^0(z_2)\kappa _{1,r,s_1}^0(z_1)$ for every $r>0$;

• (3) if $t_1,t_2<\infty$, then   consists of all such that $|K(z_2,z_1)|\lesssim \kappa _{2,r,t_2}(z_2)\kappa _{2,r,t_1}(z_1)$ for every $r>0$;

• (4) consists of all such that $|K(z_2,z_1)|\lesssim \kappa _{2,r,t_2}^0(z_2)\kappa _{2,r,t_1}^0(z_1)$ for some $r>0$.

By Remark 2.15 it follows that Theorem A.1 remains true after the spaces in (2.36) are replaced by corresponding spaces in (2.29).

By similar arguments as in the proofs of Theorems 4.1, 4.2, 5.2 and 5.3 in [21] we get the following two theorems. The details are left for the reader.

Theorem A.2

Let $s_1,s_2\in \overline{{\mathbf {R}}_{\flat ,\infty }}$, $W={\mathbf {C}}^{d_2}\times {\mathbf {C}}^{d_1}$, and let $\kappa _{j,r,s}$ and $\kappa _{j,r,s}^0$ be given by (A.1)–(A.3), $j=1,2$, when $r>0$. Then the following is true:

• (1) if $s_1>\flat _1$ and $s_1,s_2<\infty$, then consists of all such that for every $r_1>0$ there is an $r_2>0$ such that $|K(z_2,z_1)|\lesssim \kappa _{1,r_2,s_2}(z_2)\kappa _{2,r_1,s_1}(z_1)$;

• (2) if $s_1>\flat _1$ and $s_2>0$, then consists of all such that for every $r_2>0$ there is an $r_1>0$ such that $|K(z_2,z_1)|\lesssim \kappa _{1,r_2,s_2}^0(z_2)\kappa _{2,r_1,s_1}^0(z_1)$;

• (3) if $s_2>\flat _1$ and $s_1,s_2<\infty$, then consists of all such that for every $r_2>0$ there is an $r_1>0$ such that $|K(z_2,z_1)|\lesssim \kappa _{2,r_2,s_2}^0(z_2)\kappa _{1,r_1,s_1}^0(z_1)$;

• (4) if $s_1>0$ and $s_2>\flat _1$, then consists of all such that for every $r_1>0$ there is an $r_2>0$ such that $|K(z_2,z_1)|\lesssim \kappa _{2,r_2,s_2}^0(z_2)\kappa _{1,r_1,s_1}^0(z_1)$.

Theorem A.3

Let $s_1,s_2\in \overline{{\mathbf {R}}_{\flat ,\infty }}$, $W={\mathbf {C}}^{d_2}\times {\mathbf {C}}^{d_1}$, and let $\kappa _{j,r,s}$ and $\kappa _{j,r,s}^0$ be given by (A.1)–(A.3), $j=1,2$, when $r>0$. Then the following is true:

• (1) if $s_1>\flat _1$ and $s_1,s_2<\infty$, then     consists of all such that for some $r_2>0$ it holds $|K(z_2,z_1)|\lesssim \kappa _{1,r_2,s_2}(z_2)\kappa _{2,r_1,s_1}(z_1)$ for every $r_1>0$;

• (2) if $s_1>\flat _1$ and $s_2>0$, then     consists of all such that for some $r_1>0$ it holds $|K(z_2,z_1)|\lesssim \kappa _{1,r_2,s_2}^0(z_2)\kappa _{2,r_1,s_1}^0(z_1)$ for every $r_2>0$;

• (3) if $s_2>\flat _1$ and $s_1,s_2<\infty$, then     consists of all such that for some $r_1>0$ it holds $|K(z_2,z_1)|\lesssim \kappa _{2,r_2,s_2}^0(z_2)\kappa _{1,r_1,s_1}^0(z_1)$ for every $r_2>0$;

• (4) if $s_1>0$ and $s_2>\flat _1$, then     consists of all such that for some $r_2>0$ it holds $|K(z_2,z_1)|\lesssim \kappa _{2,r_2,s_2}^0(z_2)\kappa _{1,r_1,s_1}^0(z_1)$ for every $r_1>0$.

Appendix B: The Link Between Wick and Anti-Wick Symbols of Rank One

In this appendix we show that if

$$\begin{aligned} b_{z,w}(\alpha ,\beta ) =e_\alpha (z)e_\beta ({\overline{w}}), \end{aligned}$$

$t,t_0\in {\mathbf {C}}\setminus 0$ are such that $t_0^2=t$ and

$$\begin{aligned} b_{t_0z,\overline{t_0}w}^t(\alpha ,\beta )&\equiv \pi ^{-d} \int _{{\mathbf {C}}^{d}}b_{t_0w_1,\overline{t_0}w_1}(\alpha ,\beta ) e^{-(z-w_1,w-w_1)} \, d\lambda (w_1) \\&= \pi ^{-d} \int _{{\mathbf {C}}^{d}}e_\alpha (t_0w_1)e_\beta (t_0\overline{w_1}) e^{-(z-w_1,w-w_1)} \, d\lambda (w_1) \end{aligned}$$

(cf. (2.60)), then

$$\begin{aligned} b_{z,w}^t(\alpha ,\beta )&= ({\mathcal {T}}_{0,t}b_{z,w})(\alpha ,\beta ) , \qquad \alpha ,\beta \in {{\mathbf {N}}}^{d}. \end{aligned}$$

(B.1)

In fact, since all integrations can be done coordinate wise, we may assume that $d=1$.

First we prove (B.1) for $t=1$ and consider first the case when $\alpha =\beta =0$, which is the same as

$$\begin{aligned} \pi ^{-1}\int _{{\mathbf {C}}} e^{(z-w_1)({\overline{w}}-{\overline{w}}_1)}\, d\lambda (w_1)=1. \end{aligned}$$

(B.2)

In fact, if z and w are real, then the latter integral is equal to

$$\begin{aligned} \pi ^{-1} e^{-zw} \iint _{{\mathbf {R}}^{2}}e^{(z+w)x}e^{-i(z-w)y}e^{-x^2-y^2}\, dxdy = e^{-zw} e^{\frac{1}{4}(z+w)^2}e^{-\frac{1}{4}(z-w)^2} = 1. \end{aligned}$$

The searched identity now follows by analytic continuation.

For general $\alpha$ and $\beta$, we need to simplify

$$\begin{aligned} I_{z,w}(\alpha ,\beta )= & {} \int _{{\mathbf {C}}} w_1^\alpha {\overline{w}}_1^\beta e^{\overline{w}{w_1}}\,e^{z\overline{w_1}} d\mu (w_1). \end{aligned}$$

By integrating by parts and using Leibnitz formula we obtain

$$\begin{aligned} I_{z,w}(\alpha ,\beta )= & {} \pi ^{-1} (-1)^\beta \int _{{\mathbf {C}}} w_1^\alpha e^{\overline{w}{w_1}}\,e^{z\overline{w_1}} \big ( \partial ^\beta _{w_1} (e^{-|w_1|^2}) \big )\, d\lambda (w_1)\\= & {} \sum _{\gamma \le \beta } {{\beta } \atopwithdelims (){\gamma }}\pi ^{-1} \int _{{\mathbf {C}}} \big ( \partial ^\gamma _{w_1} w_1^\alpha \big ) \big ( \partial _{{{w}}_1}^{\beta -\gamma } e^{{\overline{w}}w_1} \big )e^{z\overline{w_1}} e^{-|w_1|^2} \, d\lambda (w_1)\\= & {} \sum _{\gamma \le \alpha ,\beta } {{\alpha } \atopwithdelims (){\gamma }} {{\beta } \atopwithdelims (){\gamma }} \gamma ! {\overline{w}}^{\beta -\gamma } \int _{{\mathbf {C}}} w_1^{\alpha -\gamma } e^{{\overline{w}}w_1} e^{z{\overline{w}}_1} \, d\mu (w_1)\\= & {} e^{z{\overline{w}}} \sum _{\gamma \le \alpha ,\beta } {{\alpha } \atopwithdelims (){\gamma }} {{\beta } \atopwithdelims (){\gamma }} \gamma ! z^{\alpha -\gamma }{\overline{w}}^{\beta -\gamma } . \end{aligned}$$

Here the last equality follows from the reproducing kernel. This gives

$$\begin{aligned} (\alpha !\beta !)^{\frac{1}{2}}b^1_{z,w}(\alpha , \beta )= & {} \pi ^{-1} \int _{{\mathbf {C}}}w_1^\alpha {\overline{w}}_1^\beta e^{-(z-w_1)(\overline{w}-\overline{w}_1)}d \lambda (w_1)\\= & {} e^{-z{\overline{w}}}\int _{{\mathbf {C}}}w_1^\alpha {\overline{w}}_1^\beta e^{\overline{w}{w_1}}\,e^{z\overline{w_1}} d\mu (w_1)\\= & {} e^{-z{\overline{w}}} I_{z,w}(\alpha , \beta )\\= & {} \sum _{\gamma \le \alpha ,\beta } {{\alpha } \atopwithdelims (){\gamma }} {{\beta } \atopwithdelims (){\gamma }} \gamma ! z^{\alpha -\gamma }{\overline{w}}^{\beta -\gamma }\\= & {} (\alpha !\beta !)^{\frac{1}{2}} \sum _{\gamma \le \alpha ,\beta } {{\alpha } \atopwithdelims (){\gamma }}^{\frac{1}{2}} {{\beta } \atopwithdelims (){\gamma }}^{\frac{1}{2}} e_{\alpha -\gamma }(z)e_{\beta -\gamma }(\overline{w}), \end{aligned}$$

and (B.1) follows for general $\alpha$ and $\beta$. Here the last equality follows from (B.2) and the identity

and (B.1) follows in the case when $t=1$.

Next suppose that $t\in {\mathbf {C}}\setminus 0$ and recall that $t_0^2=t$. By the definitions it follows that

$$\begin{aligned} ({\mathcal {T}}_{0,t}b_{z,w})(\alpha ,\beta ) = t_0^{|\alpha +\beta |} ({\mathcal {T}}_{0,1} b_{z/t_0,w/\overline{t_0}})(\alpha ,\beta ), \end{aligned}$$

and the case $t=1$ shows that

$$\begin{aligned} ({\mathcal {T}}_{0,t}b_{t_0z,\overline{t_0}w})(\alpha ,\beta )&= \pi ^{-d} t_0^{|\alpha +\beta |} \int _{{\mathbf {C}}^{d}} b_{w_1,w_1}(\alpha ,\beta ) e^{-(z-w_1,w-w_1)}\, d\lambda (w_1)\\&=\pi ^{-d} \int _{{\mathbf {C}}^{d}} b_{t_0w_1,\overline{t_0}w_1}(\alpha ,\beta ) e^{-(z-w_1,w-w_1)}\, d\lambda (w_1), \end{aligned}$$

which gives the assertion.

Appendix C: Identities of Spaces of Kernel Operators, Wick Operators and Anti-Wick Operators

Let $s_1,s_2 \in \overline{{\mathbf {R}}_\flat }$ be such that $s_1 < \frac{1}{2}$ and $s_2 \le \frac{1}{2}$. By Corollary 3.6, Proposition 5.5 and Theorems 5.6–5.9, it follows that

(C.1)

(C.2)

(C.3)

(C.4)

(C.5)

(C.6)

(C.7)

(C.8)