Abstract
Let E and F be complex Banach spaces, U be an open subset of E and \(1\le p\le \infty .\) We introduce and study the notion of a Cohen strongly p-summing holomorphic map** from U to F, a holomorphic version of a strongly p-summing linear operator. For such map**s, we establish both Pietsch Domination/Factorization Theorems and analyse their linearizations from (the canonical predual of
) and their transpositions on
Concerning the space
formed by such map**s and endowed with a natural norm
we show that it is a regular Banach ideal of bounded holomorphic map**s generated by composition with the ideal of strongly p-summing linear operators. Moreover, we identify the space
with the dual of the completion of tensor product space
endowed with the Chevet–Saphar norm \(g_p.\)
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1 Introduction
The linear theory of absolutely summing operators between Banach spaces was initiated by Grothendieck [11] in 1950 with the introduction of the concept of 1-summing operator. In 1967, Pietsch [22] defined the class of absolutely p-summing operators for any \(p>0\) and established many of their fundamental properties.
The nonlinear theory for such operators started with Pietsch [23] in 1983. Since then, the idea of extending the theory of absolutely p-summing operators to other settings has been developed by various authors, namely, the polynomial, multilinear, Lipschitz and holomorphic settings (see, for example, [1, 2, 7, 8, 19, 27, 28]).
Summability for holomorphic map**s was first considered by Matos in a series of papers (see e.g. [13, 14]). Our approach in this paper is different from that of Matos. Moreover, strong p-summability in the sense of Dimant [7] was also addressed for subspaces of holomorphic map**s as polynomials and multilinear map**s under the name of factorable strongly p-summing (see [20, 24, 25]). In these papers, it was proved that the ideal of factorable strongly p-summing polynomials (multilinear map**s) coincides with the ideal formed by composition with p-summing linear operators. Ideals of polynomial map**s were also studied by Floret and García [9, 10].
In 1973, Cohen [5] introduced the concept of a strongly p-summing linear operator to characterize those operators whose adjoints are absolutely \(p^*\)-summing operators, where \(p^*\) denotes the conjugate index of \(p\in (1,\infty ].\) Influenced by this class of operators, we introduce and study a new concept of summability in the category of bounded holomorphic map**s, which yields the called Cohen strongly p-summing holomorphic map**s.
We now describe the contents of the paper. Let E and F be complex Banach spaces, U be an open subset of E and \(1\le p\le \infty .\) We denote by the Banach space of all bounded holomorphic map**s from U to F, equipped with the supremum norm. In particular,
stands for the space
It is known that
is a dual Banach space whose canonical predual, denoted
is the norm-closed linear subspace of
generated by the evaluation functionals at the points of U.
In Sect. 1, we fix the notation and recall some results on the space essentially, a remarkable linearization theorem due to Mujica [16] which is a key tool to establish our results.
In Sect. 2, we show that the space of all Cohen strongly p-summing holomorphic map**s denoted and equipped with a natural norm
is a regular Banach ideal of bounded holomorphic map**s. Furthermore,
with
The elements of the tensor product of two linear spaces can be viewed as linear map**s or bilinear forms (see [26, Section 1.3]). Following this idea, in Sect. 3 we introduce the tensor product \(\Delta (U)\otimes F\) as a space of linear functionals on the space and equip this space with the known Chevet–Saphar norms \(g_p\) and \(d_p.\)
Section 4 addresses the duality theory: the space is canonically isometrically isomorphic to the dual of the completion of the tensor product space
In particular, we deduce that
is a dual space.
Pietsch [22] established a Domination/Factorization Theorem for p-summing linear operators between Banach spaces. Characterizing previously the elements of the dual space of \(\Delta (U)\otimes _{g_p} F,\) we present for Cohen strongly p-summing holomorphic map**s both versions of Pietsch Domination Theorem and Pietsch Factorization Theorem in Sects. 5 and 6, respectively.
Moreover, in Sect. 5, we prove that a map** \(f:U\rightarrow F\) is Cohen strongly p-summing holomorphic if and only if Mujica’s linearization is a strongly p-summing operator. Several interesting applications of this fact are obtained.
In addition, we show that the ideal is generated by composition with the ideal
of strongly p-summing linear operators, that is, every map**
admits a factorization in the form \(f=T\circ g,\) for some complex Banach space G,
and
Moreover,
coincides with \(\inf \{d_p(T)\left\| g\right\| _{\infty }\},\) where the infimum is extended over all such factorizations of f, and, curiously, this infimum is attained at Mujica’s factorization of f. We also show that every
factors through a Hilbert space whenever F is reflexive, and establish some inclusion and coincidence properties of spaces
These results represent advances in the research program initiated by Aron et al. [4] on the factorization of bounded holomorphic map**s in terms of an element of an operator ideal and a bounded holomorphic map**.
Finally, we analyse holomorphic transposition of their elements and prove that every member of has relatively weakly compact range that becomes relatively compact whenever F is reflexive. We thus contribute to the study of holomorphic map**s with relatively (weakly) compact range, begun by Mujica [16] and continued in [12].
2 Notation and preliminaries
Throughout this paper, unless otherwise stated, E and F will denote complex Banach spaces and U an open subset of E.
We first introduce some notation. As usual, \(B_E\) denotes the closed unit ball of E. For two vector spaces E and F, L(E, F) stands for the vector space of all linear operators from E into F. In the case that E and F are normed spaces, represents the normed space of all bounded linear operators from E to F endowed with the canonical norm of operators. In particular, the algebraic dual \(L(E,{\mathbb {K}})\) and the topological dual
are denoted by \(E^{\prime }\) and \(E^*,\) respectively. For each \(e\in E\) and \(e^*\in E^{\prime },\) we frequently will write \(\langle e^*,e\rangle \) instead of \(e^*(e).\) We denote by \(\kappa _E\) the canonical isometric embedding of E into \(E^{**}\) defined by \(\left\langle \kappa _E(e),e^*\right\rangle =\left\langle e^*,e\right\rangle \) for \(e\in E\) and \(e^*\in E^*.\) For a set \(A\subseteq E,\) \({\textrm{co}}(A)\) denotes the convex hull of A.
We now recall some concepts and results of the theory of holomorphic map**s on Banach spaces.
Theorem 1.1
(See [18, 7 Theorem] and [15, Theorem 8.7]) Let E and F be complex Banach spaces and let U be an open set in E. For a map** \(f:U\rightarrow F,\) the following conditions are equivalent :
-
(i)
For each \(a\in U,\) there is an operator
such that
$$\begin{aligned} \lim _{x\rightarrow a}\frac{f(x)-f(a)-T(x-a)}{\left\| x-a\right\| }=0. \end{aligned}$$ -
(ii)
For each \(a\in U,\) there exist an open ball \(B(a,r)\subseteq U\) and a sequence of continuous m-homogeneous polynomials \((P_{m,a})_{m\in {\mathbb {N}}_0}\) from E into F such that
$$\begin{aligned} f(x)=\sum _{m=0}^\infty P_{m,a}(x-a), \end{aligned}$$where the series converges uniformly for \(x\in B(a,r).\)
-
(iii)
f is G-holomorphic (that is, for all \(a\in U\) and \(b\in E,\) the map \(\lambda \mapsto f(a+\lambda b)\) is holomorphic on the open set \(\{\lambda \in {\mathbb {C}}:a+\lambda b\in U\})\) and continuous. \(\square \)
A map** \(f:U\rightarrow F\) is said to be holomorphic if it verifies the equivalent conditions in Theorem 1.1. The map** T in condition (i) is uniquely determined by f and a, and is called the differential of f at a and denoted by Df(a).
A map** \(f:U\rightarrow F\) is locally bounded if f is bounded on a suitable neighborhood of each point of U. Given a Banach space E, a subset \(N\subseteq B_{E^*}\) is said to be norming for E if the function
defines the norm on E.
If \(U\subseteq E\) and \(V\subseteq F\) are open sets, will represent the set of all holomorphic map**s from U to V. We will denote by
the linear space of all holomorphic map**s from U into F and by
the subspace of all
such that f(U) is bounded in F. When \(F={\mathbb {C}},\) then we will write
It is easy to prove that the linear space equipped with the supremum norm:
![](http://media.springernature.com/lw343/springer-static/image/art%3A10.1007%2Fs43037-023-00269-y/MediaObjects/43037_2023_269_Equ73_HTML.png)
is a Banach space. Let denote the norm-closed linear hull in
of the set \(\left\{ \delta (x):x\in U\right\} \) of evaluation functionals defined by
![](http://media.springernature.com/lw238/springer-static/image/art%3A10.1007%2Fs43037-023-00269-y/MediaObjects/43037_2023_269_Equ74_HTML.png)
In [16, 17], Mujica established the following properties of
Theorem 1.2
[16, Theorem 2.1] Let E be a complex Banach space and let U be an open set in E.
-
(i)
is isometrically isomorphic to
via the evaluation map**
given by
-
(ii)
The map**
defined by \(g_U(x)=\delta (x)\) is holomorphic with \(\left\| g_U(x)\right\| =1\) for all \(x\in U.\)
-
(iii)
For each complex Banach space F and each map**
there exists a unique operator
such that \(T_f\circ g_U=f.\) Furthermore, \(\left\| T_f\right\| =\left\| f\right\| _{\infty }.\)
-
(iv)
The map** \(f\mapsto T_f\) is an isometric isomorphism from
onto
-
(v)
[16, Corollary 4.12] (see also [17, Theorem 5.1]).
consists of all functionals
of the form \(\gamma =\sum _{i=1}^{\infty }\lambda _i\delta (x_i)\) with \((\lambda _i)_{i\ge 1}\in \ell _1\) and \((x_i)_{i\ge 1}\in U^\mathbb {N}.\) Moreover, \(\left\| \gamma \right\| =\inf \left\{ \sum _{i=1}^{\infty }\left| \lambda _i\right| \right\} \) where the infimum is taken over all such representations of \(\gamma .\) \(\square \)
3 Cohen strongly p-summing holomorphic map**s
Let E and F be Banach spaces and \(1\le p\le \infty .\) Let us recall [6] that an operator is p-summing if there exists a constant \(C\ge 0\) such that, regardless of the natural number n and regardless of the choice of vectors \(x_1,\ldots ,x_n\) in E, we have the inequalities:
The infimum of such constants C is denoted by \(\pi _p(T)\) and the linear space of all p-summing operators from E into F by \(\Pi _p(E,F).\)
The analogous notion for holomorphic map**s could be introduced as follows.
Definition 2.1
Let E and F be complex Banach spaces, let U be an open subset of E, and let \(1\le p\le \infty .\) A holomorphic map** \(f:U\rightarrow F\) is said to be p-summing if there exists a constant \(C\ge 0\) such that for all \(n\in {\mathbb {N}}\) and \(x_1,\ldots ,x_n\in U,\) we have
![](http://media.springernature.com/lw484/springer-static/image/art%3A10.1007%2Fs43037-023-00269-y/MediaObjects/43037_2023_269_Equ76_HTML.png)
We denote by the infimum of such constants C, and by
the set of all p-summing holomorphic map**s from U into F.
p-Summing holomorphic map**s are of little interest to us as with
for all
and furthermore the subclass of p-summing holomorphic map**s that we will study in this paper includes this case.
Let \(1\le p\le \infty \) and let \(p^*\) denote the conjugate index of p given by
In [5], Cohen introduced the following subclass of p-summing operators between Banach spaces: an operator is strongly p-summing if there exists a constant \(C\ge 0\) such that for all \(n\in {\mathbb {N}},\) \(x_1,\ldots ,x_n\in E\) and \(y^*_1,\ldots ,y^*_n\in F^*,\) we have
The infimum of such constants C is denoted by \(d_p(T),\) and the space of all strongly p-summing operators from E into F by If \(p=1,\) we have
We now introduce a version of this concept in the setting of holomorphic map**s.
Definition 2.2
Let E and F be complex Banach spaces, let U be an open subset of E, and let \(1\le p\le \infty .\) A holomorphic map** \(f:U\rightarrow F\) is said to be Cohen strongly p-summing if there exists a constant \(C\ge 0\) such that for all \(n\in {\mathbb {N}},\) \(\lambda _1,\ldots ,\lambda _n\in {\mathbb {C}},\) \(x_1,\ldots ,x_n\in U\) and \(y^*_1,\ldots ,y^*_n\in F^*,\) we have
We denote by the infimum of such constants C, and by
the set of all Cohen strongly p-summing holomorphic map**s from U into F.
The introduction of the scalars \(\lambda _i\) in the previous definition is justified by the assertion (v) of Theorem 1.2. Proposition 2.5 shows that
The concept of an ideal of bounded holomorphic map**s is inspired by the analogous one for bounded linear operators between Banach spaces [26, Section 8.2].
Definition 2.3
An ideal of bounded holomorphic map**s (or simply, a bounded-holomorphic ideal) is a subclass of the class
of all bounded holomorphic map**s such that for each complex Banach space E, each open subset U of E and each complex Banach space F, the components
![](http://media.springernature.com/lw238/springer-static/image/art%3A10.1007%2Fs43037-023-00269-y/MediaObjects/43037_2023_269_Equ77_HTML.png)
satisfy the following properties:
-
(I1)
is a linear subspace of
-
(I2)
For any
and \(y\in F,\) the map** \(g\cdot y:x\mapsto g(x)y\) from U to F is in
-
(I3)
The ideal property: If H, G are complex Banach spaces, V is an open subset of H,
and
then \(S\circ f\circ h\) is in
A bounded-holomorphic ideal is said to be normed (Banach) if there exists a function
such that for every complex Banach space E, every open subset U of E and every complex Banach space F, the following conditions are satisfied:
-
(N1)
is a normed (Banach) space with
for all
-
(N2)
for every
and \(y\in F,\)
-
(N3)
If H, G are complex Banach spaces, V is an open subset of H,
and
then
A normed bounded-holomorphic ideal is said to be regular if for any
we have that
with
whenever
The following class of bounded holomorphic map**s appears involved in Definition 2.3.
Lemma 2.4
Let and \(y\in F.\) The map** \(g\cdot y:U\rightarrow F,\) given by \((g\cdot y)(x)=g(x)y,\) belongs to
with \(\left\| g\cdot y\right\| _{\infty }=\left\| g\right\| _{\infty }\left\| y\right\| .\) \(\square \)
We are now ready to establish the main result of this section.
Proposition 2.5
is a regular Banach ideal of bounded holomorphic map**s. Furthermore,
with
Proof
We will only prove the case \(1<p<\infty .\) The cases \(p=1\) and \(p=\infty \) follow similarly.
(N1) We first show that with
for all
Indeed, given
we have
![](http://media.springernature.com/lw393/springer-static/image/art%3A10.1007%2Fs43037-023-00269-y/MediaObjects/43037_2023_269_Equ78_HTML.png)
for all \(x\in U\) and \(y^*\in F^*.\) By Hahn–Banach Theorem, we obtain that for all \(x\in U.\) Hence
with
Let Given \(n\in {\mathbb {N}},\) \(\lambda _1,\ldots ,\lambda _n\in {\mathbb {C}},\) \(x_1,\ldots ,x_n\in U\) and \(y^*_1,\ldots ,y^*_n\in F^*,\) we have
![](http://media.springernature.com/lw530/springer-static/image/art%3A10.1007%2Fs43037-023-00269-y/MediaObjects/43037_2023_269_Equ79_HTML.png)
Using the two inequalities above, we obtain
![](http://media.springernature.com/lw507/springer-static/image/art%3A10.1007%2Fs43037-023-00269-y/MediaObjects/43037_2023_269_Equ80_HTML.png)
This tells us that with
Let \(\lambda \in {\mathbb {C}}\) and Given \(n\in {\mathbb {N}},\) \(\lambda _i\in {\mathbb {C}},\) \(x_i\in U\) and \(y^*_i\in F^*\) for \(i=1,\ldots ,n,\) we have
![](http://media.springernature.com/lw564/springer-static/image/art%3A10.1007%2Fs43037-023-00269-y/MediaObjects/43037_2023_269_Equ81_HTML.png)
and thus with
This implies that
if \(\lambda =0.\) For \(\lambda \ne 0,\) we have
hence
and so
Moreover, if and
then \(\left\| f\right\| _{\infty }=0\) by (N1), and so \(f=0.\) Thus,
is a normed space.
To prove that is complete, it suffices to prove that every absolutely convergent series is convergent. So let \((f_n)_{n\in {\mathbb {N}}}\) be a sequence in
such that
is convergent. Since
for all \(n\in {\mathbb {N}}\) and
is a Banach space, then \(\sum _{n\in {\mathbb {N}}}f_n\) converges in
to a function
Given \(m\in {\mathbb {N}},\) \(x_1,\ldots ,x_m\in U,\) \(y^*_1,\ldots ,y^*_m\in F^*\) and \(\lambda _1,\ldots ,\lambda _m\in {\mathbb {C}},\) we have
![](http://media.springernature.com/lw441/springer-static/image/art%3A10.1007%2Fs43037-023-00269-y/MediaObjects/43037_2023_269_Equ82_HTML.png)
for all \(n\in {\mathbb {N}},\) and by taking limits with \(n\rightarrow \infty \) yields
![](http://media.springernature.com/lw581/springer-static/image/art%3A10.1007%2Fs43037-023-00269-y/MediaObjects/43037_2023_269_Equ83_HTML.png)
Hence with
Moreover, we have
![](http://media.springernature.com/lw389/springer-static/image/art%3A10.1007%2Fs43037-023-00269-y/MediaObjects/43037_2023_269_Equ84_HTML.png)
for all \(n\in {\mathbb {N}},\) and thus f is the -limit of the series \(\sum _{n\in {\mathbb {N}}}f_n.\)
(N2) Let and \(y\in F.\) If \(y=0,\) there is nothing to prove. Assume \(y\ne 0.\) By Lemma 2.4,
Given \(n\in {\mathbb {N}},\) \(x_1,\ldots ,x_n\in U,\) \(y^*_1,\ldots ,y^*_n\in F^*\) and \(\lambda _1,\ldots ,\lambda _n\in {\mathbb {C}},\) we have
by applying the Hölder inequality, and therefore with
Conversely, by applying what was proved in (N1), we have
(N3) Let H, G be complex Banach spaces, V be an open subset of H,
and
We can suppose \(S\ne 0.\) Given \(n\in {\mathbb {N}},\) \(x_1,\ldots ,x_n\in U,\) \(y^*_1,\ldots ,y^*_n\in G^*\) and \(\lambda _1,\ldots ,\lambda _n\in {\mathbb {C}},\) we have
![](http://media.springernature.com/lw488/springer-static/image/art%3A10.1007%2Fs43037-023-00269-y/MediaObjects/43037_2023_269_Equ85_HTML.png)
and therefore with
We now prove that the ideal is regular. Let
and assume that
Given \(n\in {\mathbb {N}},\) \(x_1,\ldots ,x_n\in U,\) \(y^*_1,\ldots ,y^*_n\in F^*\) and \(\lambda _1,\ldots ,\lambda _n\in {\mathbb {C}},\) we have
![](http://media.springernature.com/lw492/springer-static/image/art%3A10.1007%2Fs43037-023-00269-y/MediaObjects/43037_2023_269_Equ86_HTML.png)
and thus with
The reverse inequality follows from (N3).
Finally, we have seen in (N1) that with
for all
For the converse, let
If \(f=0,\) there is nothing to prove. Assume \(f\ne 0.\) Given \(n\in {\mathbb {N}},\) \(x_1,\ldots ,x_n\in U,\) \(y^*_1,\ldots ,y^*_n\in F^*\) and \(\lambda _1,\ldots ,\lambda _n\in {\mathbb {C}},\) we have
and therefore with
\(\square \)
4 The tensor product \(\Delta (U)\otimes F\)
We introduce \(\Delta (U)\otimes F\) as a space of linear functionals on
Definition 3.1
Let E and F be complex Banach spaces and let U be an open subset of E. For each \(x\in U,\) let be the linear functional defined by
![](http://media.springernature.com/lw238/springer-static/image/art%3A10.1007%2Fs43037-023-00269-y/MediaObjects/43037_2023_269_Equ87_HTML.png)
Let \(\Delta (U)\) be the linear subspace of spanned by the set \(\left\{ \delta (x):x\in U\right\} .\)
For any \(x\in U\) and \(y\in F,\) let be the linear functional given by
![](http://media.springernature.com/lw347/springer-static/image/art%3A10.1007%2Fs43037-023-00269-y/MediaObjects/43037_2023_269_Equ88_HTML.png)
We define the tensor product \(\Delta (U)\otimes F\) as the linear subspace of spanned by the set
We say that \(\delta (x)\otimes y\) is an elementary tensor of \(\Delta (U)\otimes F.\) Note that each element u in \(\Delta (U)\otimes F\) is of the form \(u=\sum _{i=1}^n\lambda _i(\delta (x_i)\otimes y_i),\) where \(n\in {\mathbb {N}},\) \(\lambda _i\in {\mathbb {C}},\) \(x_i\in U\) and \(y_i\in F\) for \(i=1,\ldots ,n.\) This representation of u is not unique. It is worth noting that each element u of \(\Delta (U)\otimes F\) can be represented as \(u=\sum _{i=1}^n \delta (x_i)\otimes y_i\) since \(\lambda (\delta (x)\otimes y)=\delta (x)\otimes (\lambda y).\)
As a straightforward consequence from Definition 3.1, we describe the action of a tensor u in \(\Delta (U)\otimes F\) on a function f in :
Lemma 3.2
Let \(u=\sum _{i=1}^n \lambda _i\delta (x_i)\otimes y_i\in \Delta (U)\otimes F\) and Then
\(\Box \)
The following characterization of the zero tensor of \(\Delta (U)\otimes F\) follows immediately from [26, Proposition 1.2].
Proposition 3.3
If \(u=\sum _{i=1}^n\delta (x_i)\otimes y_i\in \Delta (U)\otimes F,\) the following are equivalent :
-
(i)
\(u=0.\)
-
(ii)
\(\sum _{i=1}^ng(x_i)\phi (y_i)=0\) for every
and \(\phi \in B_{F^*}.\) \(\Box \)
By Definition 3.1, \(\Delta (U)\otimes F\) is a linear subspace of In fact, we have:
Proposition 3.4
forms a dual pair, where the bilinear form \(\left\langle \cdot ,\cdot \right\rangle \) associated to the dual pair is given by
for \(u=\sum _{i=1}^n\lambda _i \delta (x_i)\otimes y_i\in \Delta (U)\otimes F\) and
Proof
Since \(\langle u,f\rangle =u(f)\) by Lemma 3.2, it is immediate that \(\langle \cdot ,\cdot \rangle \) is a well-defined bilinear map from to \({\mathbb {C}.}\) On the one hand, if \(u\in \Delta (U)\otimes F\) and \(\langle u,f\rangle =0\) for all
then \(u=0\) follows easily from Proposition 3.3, and thus
separates points of \(\Delta (U)\otimes F.\) On the other hand, if
and \(\langle u,f\rangle =0\) for all \(u\in \Delta (U)\otimes F,\) then \(\left\langle f(x),y\right\rangle =\left\langle \delta (x)\otimes y,f\right\rangle =0\) for all \(x\in U\) and \(y\in F,\) this means that \(f=0\) and thus \(\Delta (U)\otimes F\) separates points of
\(\square \)
Since is a dual pair, we can identify
with a linear subspace of \((\Delta (U)\otimes F)^{\prime }\) as follows.
Corollary 3.5
For each the functional \(\Lambda _0(f):\Delta (U)\otimes F\rightarrow {\mathbb {C}},\) given by
for \(u=\sum _{i=1}^n \lambda _i\delta (x_i)\otimes y_i\in \Delta (U)\otimes F,\) is linear. We will say that \(\Lambda _0(f)\) is the linear functional on \(\Delta (U)\otimes F\) associated to f. Furthermore, the map \(f\mapsto \Lambda _0(f)\) is a linear monomorphism from into \((\Delta (U)\otimes F)^{\prime }.\)
Proof
Let Note that \(\Lambda _0(f)(u)=\left\langle u,f\right\rangle \) for all \(u\in \Delta (U)\otimes F.\) It is immediate that \(\Lambda _0(f)\) is a well-defined linear functional on \(\Delta (U)\otimes F\) and that \(f\mapsto \Lambda _0(f)\) from
into \((\Delta (U)\otimes F)^{\prime }\) is a well-defined linear map. Finally, let
and assume that \(\Lambda _0(f)=0.\) Then \(\left\langle u,f\right\rangle =0\) for all \(u\in \Delta (U)\otimes F.\) Since \(\Delta (U)\otimes F\) separates points of
by Proposition 3.4, it follows that \(f=0\) and this proves that \(\Lambda _0\) is one-to-one. \(\square \)
Given two linear spaces E and F, the tensor product space \(E\otimes F\) equipped with a norm \(\alpha \) will be denoted by \(E\otimes _{\alpha } F,\) and the completion of \(E\otimes _{\alpha } F\) by \(E{\widehat{\otimes }}_{\alpha } F.\) If E and F are normed spaces, a cross-norm on \(E\otimes F\) is a norm \(\alpha \) such that \(\alpha (x\otimes y)=\left\| x\right\| \left\| y\right\| \) for all \(x\in E\) and \(y\in F.\)
Given two normed spaces E and F, the projective norm \(\pi \) on \(E\otimes F\) (see [26, Chapter 2]) takes the following form on \(\Delta (U)\otimes F\):
where the infimum is taken over all the representations of u as above.
We next see that, on the space \(\Delta (U)\otimes F,\) the projective norm and the norm induced by the dual norm of the supremum norm of coincide.
Theorem 3.6
The linear space \(\Delta (U)\otimes F\) is contained in Moreover, \(\pi (u)=H(u)\) for all \(u\in \Delta (U)\otimes F,\) where H is the norm on \(\Delta (U)\otimes F\) defined by
![](http://media.springernature.com/lw493/springer-static/image/art%3A10.1007%2Fs43037-023-00269-y/MediaObjects/43037_2023_269_Equ89_HTML.png)
Proof
Let \(\lambda \in {\mathbb {C}},\) \(x\in U\) and \(y\in F.\) Since \(\lambda \delta (x)\otimes y\) is a linear map on and
for all then
with \(\left\| \lambda \delta (x)\otimes y\right\| \le \left| \lambda \right| \left\| y\right\| ,\) and thus
Let \(u\in \Delta (U)\otimes F\) and let \(\sum _{i=1}^n\lambda _i\delta (x_i)\otimes y_i\) be a representation of u. Since u is linear and
for all we deduce that \(H(u)\le \sum _{i=1}^n\left| \lambda _i\right| \left\| y_i\right\| .\) Since this holds for each representation of u, it follows that \(H(u)\le \pi (u).\) Hence, \(H\le \pi .\) To prove that the reverse inequality, suppose by contradiction that \(H(u_0)<1<\pi (u_0)\) for some \(u_0\in \Delta (U)\otimes F.\) Denote \(B=\{u\in \Delta (U)\otimes F:\pi (u)\le 1\}.\) Clearly, B is a closed and convex set in \(\Delta (U)\otimes _\pi F.\) Applying the Hahn–Banach Separation Theorem to B and \(\{u_0\},\) we obtain a functional \(\eta \in (\Delta (U)\otimes _\pi F)^*\) such that
Define \(f:U\rightarrow F^*\) by \(\langle f(x),y\rangle =\eta \left( \delta (x)\otimes y\right) \) for all \(y\in F\) and \(x\in U.\) It is easy to prove that f is well defined and with \(\left\| f\right\| _{\infty }\le 1.\) Moreover, \(u(f)=\eta (u)\) for all \(u\in \Delta (U)\otimes F.\) Therefore \(H(u_0)\ge |u_0(f)|\ge {\textrm{Re}}\,u_0(f)={\textrm{Re}}\,\eta (u_0),\) so \(H(u_0)>1\) and this is a contradiction. \(\square \)
We now will define the Chevet–Saphar norms on the tensor product \(E\otimes F.\) Let E and F be normed spaces and let \(1\le p\le \infty .\) Given \(u=\sum _{i=1}^n x_i\otimes y_i\in E\otimes F,\) denote
and
If \(E=F={\mathbb {C}},\) we write \(\ell ^n_p(E)=\ell ^n_p\) and \(\ell ^{n,w}_{p^*}(F)=\ell ^{n,w}_{p^*}.\) According to [26, Section 6.2], the Chevet–Saphar norms are defined on \(E\otimes F\) by
the infimum being taken over all representations of u as \(u=\sum _{i=1}^nx_i\otimes y_i\in E\otimes F.\)
Since \(\left\| \delta (x)\right\| =1\) for all \(x\in U,\) the norm \(g_p\) on \(\Delta (U)\otimes F\) takes the form:
Notice that \(g_p\) is a cross-norm on \(\Delta (U)\otimes F.\)
We next show that \(g_1\) on \(\Delta (U)\otimes F\) is just the projective tensor norm \(\pi .\)
Proposition 3.7
\(g_1(u)=\pi (u)\) for all \(u\in \Delta (U)\otimes F.\)
Proof
Let \(u\in \Delta (U)\otimes F\) and let \(\sum _{i=1}^n\lambda _i\delta (x_i)\otimes y_i\) be a representation of u. We have
and taking the infimum over all representations of u gives \(\pi (u)\le g_1(u).\) For the reverse inequality, notice that \(g_1(\lambda \delta (x)\otimes y)\le |\lambda |\left\| y\right\| \) for all \(\lambda \in {\mathbb {C}},\) \(x\in U\) and \(y\in F.\) Since \(g_1\) is a norm on \(\Delta (U)\otimes F,\) it follows that
and taking the infimum over all representations of u yields \(g_1(u)\le \pi (u).\) \(\square \)
5 Duality for Cohen strongly p-summing holomorphic map**s
We now show that the duals of the tensor product can be canonically identified as spaces of Cohen strongly p-summing holomorphic map**s.
Theorem 4.1
Let \(1\le p\le \infty .\) Then is isometrically isomorphic to
via the map**
defined by
for and \(u=\sum _{i=1}^n\lambda _i\delta (x_i)\otimes y_i\in \Delta (U)\otimes F.\) Furthermore, its inverse is given by
for \(x\in U\) and \(y\in F.\)
Proof
We prove it for \(1<p\le \infty .\) The case \(p=1\) is similarly proved.
Let and let \(\Lambda _0(f):\Delta (U)\otimes F\rightarrow {\mathbb {C}}\) be its associate linear functional. We claim that \(\Lambda _0(f)\in (\Delta (U)\otimes _{g_p} F)^*\) with
Indeed, given \(u=\sum _{i=1}^n\lambda _i\delta (x_i)\otimes y_i\in \Delta (U)\otimes F,\) we have
![](http://media.springernature.com/lw517/springer-static/image/art%3A10.1007%2Fs43037-023-00269-y/MediaObjects/43037_2023_269_Equ90_HTML.png)
and taking infimum over all the representations of u, we deduce that Since u was arbitrary, then \(\Lambda _0(f)\) is continuous on \(\Delta (U) \otimes _{g_p} F\) with
as claimed.
Since \(\Delta (U)\) is a norm-dense linear subspace of and \(g_p\) is a cross-norm on
then \(\Delta (U)\otimes F\) is a dense linear subspace of
and therefore also of its completion
Hence there is a unique continuous map** \(\Lambda (f)\) from
to \( {\mathbb {C}}\) that extends \(\Lambda _0(f).\) Further, \(\Lambda (f)\) is linear and \(\left\| \Lambda (f)\right\| =\left\| \Lambda _0(f)\right\| .\)
Let be the map** so defined. Since the map**
is a linear monomorphism by Corollary 3.5, it follows easily that \(\Lambda \) is so. To prove that \(\Lambda \) is a surjective isometry, let
and define \(f_{\varphi }:U\rightarrow F^*\) by
Given \(x\in U,\) the linearity of both \(\varphi \) and the product tensor in the second variable yields that the functional \(f_{\varphi }(x):F\rightarrow {\mathbb {C}}\) is linear, and since
for all \(y\in F,\) we deduce that \(f_{\varphi }(x)\in F^*\) with \(\Vert f_{\varphi }(x)\Vert \le \left\| \varphi \right\| .\) Since x was arbitrary, we have that \(f_{\varphi }\) is bounded with \(\left\| f_{\varphi }\right\| _{\infty }\le \left\| \varphi \right\| .\)
We now prove that \(f_{\varphi }:U\rightarrow F^*\) is holomorphic. To this end, we first claim that, for every \(y\in F,\) the function \(f_y:U\rightarrow {\mathbb {C}}\) defined by
is holomorphic. Let \(a\in U.\) Since is holomorphic by Theorem 1.2, there exists
such that
Consider the function \(T(a):E\rightarrow {\mathbb {C}}\) given by
Clearly, \(T(a)\in E^*\) and since
it follows that
Hence, \(f_y\) is holomorphic at a with \(Df_y(a)=T(a),\) and this proves our claim. Now, notice that the set \(\left\{ \kappa _F(y):y\in B_F\right\} \subseteq B_{F^{**}}\) is norming for \(F^*\) since
for every \(y^*\in F^*,\) and that \(\kappa _F(y)\circ f_{\varphi }=f_y\) for every \(y\in F\) since
for all \(x\in U.\)
We are now ready to show that \(f_{\varphi }:U\rightarrow F^*\) is holomorphic. Indeed, let \(a\in U\) and \(b\in E.\) Denote \(V=\left\{ \lambda \in {\mathbb {C}} :a+\lambda b\in U\right\} .\) Clearly, the map** \(h:V\rightarrow U\) given by \(h(\lambda )=a+\lambda b\) is holomorphic. Since \(f_{\varphi }\circ h\) is locally bounded and \(\kappa _F(y)\circ (f_{\varphi }\circ h)=f_y\circ h\) is holomorphic on the open set \(V\subseteq {\mathbb {C}}\) for all \(y\in F,\) Proposition A.3 in [3] assures that \(f_{\varphi }\circ h\) is holomorphic. This means that \(f_{\varphi }\) is G-holomorphic but since it is also locally bounded, we deduce that \(f_{\varphi }\) is continuous by [15, Proposition 8.6]. Now, we conclude that \(f_{\varphi }\) is holomorphic by Theorem 1.1.
We now prove that To see this, take \(n\in {\mathbb {N}},\) \(\lambda _i\in {\mathbb {C}},\) \(x_i\in U\) and \(y^{**}_i\in F^{**}\) for \(i=1,\ldots ,n.\) Let \(\varepsilon >0\) and consider the finite-dimensional subspaces \(V={\textrm{lin}}\{y^{**}_1,\ldots ,y^{**}_n\} \subseteq F^{**}\) and \(W={\textrm{lin}}\{f_{\varphi }(x_1),\ldots ,f_{\varphi }(x_n) \}\subseteq F^{*}.\) The principle of local reflexivity [6, Theorem 8.16] gives us a bounded linear operator \(T_{(\varepsilon , V,W)}:V\rightarrow F\) such that
-
(i)
\(T_{(\varepsilon , V,W)}(y^{**})=y^{**}\) for every \(y^{**}\in V\cap \kappa _F(F),\)
-
(ii)
\((1-\varepsilon )\left\| y^{**}\right\| \le \left\| T_{(\varepsilon ,V,W)}(y^{**})\right\| \le (1+\varepsilon )\left\| y^{**}\right\| \) for every \(y^{**}\in V,\)
-
(iii)
\(\left\langle y^*,T_{(\varepsilon , V,W)}(y^{**})\right\rangle =\left\langle y^{**},y^* \right\rangle \) for every \(y^{**}\in V\) and \(y^*\in W.\)
Using (iii) and taking \(y_i=T_{(\varepsilon , V,W)}(y_i^{**}),\) we first have
Since
it follows that
By the arbitrariness of \(\varepsilon ,\) we deduce that
and this implies that with
For any \(u=\sum _{i=1}^n \lambda _i\delta (x_i)\otimes y_i\in \Delta (U)\otimes F,\) we get
Hence \(\Lambda (f_{\varphi })=\varphi \) on a dense subspace of and, consequently, \(\Lambda (f_{\varphi })=\varphi ,\) which shows the last statement of the theorem. Moreover,
and the theorem holds. \(\square \)
In particular, in view of Theorem 4.1 and taking into account Propositions 2.5, 3.6 and 3.7, we can identify the space with the dual space of
Corollary 4.2
is isometrically isomorphic to
via the map**
given by
for and \(u=\sum _{i=1}^n\lambda _i\delta (x_i)\otimes y_i\in \Delta (U)\otimes F.\) Furthermore, its inverse is given by
for \(x\in U\) and \(y\in F.\) \(\square \)
Remark 4.3
It is known (see [26, p. 24]) that if E and F are Banach spaces, then is isometrically isomorphic to \((E{\widehat{\otimes }}_\pi F)^*,\) via
given by
for and \(\sum _{i=1}^n x_i\otimes y_i\in E\otimes F.\) Notice that the identification \(\Lambda \) in Corollary 4.2 is just \(\Phi \circ \Phi _0,\) where \(\Phi _0:f\mapsto T_f\) is the isometric isomorphism from
onto
given in Theorem 1.2.
6 Pietsch domination for Cohen strongly p-summing holomorphic map**s
In [22], Pietsch established a domination theorem for p-summing linear operators between Banach spaces. To present a version of this theorem for Cohen strongly p-summing holomorphic map**s on Banach spaces, we first characterize the elements of the dual space of \(\Delta (U)\otimes _{g_p} F.\)
Theorem 5.1
Let \(\varphi \in (\Delta (U)\otimes F)^{\prime },\) \(C>0\) and \(1<p\le \infty .\) The following conditions are equivalent :
-
(i)
\(\left| \varphi (u)\right| \le Cg_p(u)\) for all \(u\in \Delta (U)\otimes F.\)
-
(ii)
For any representation \(\sum _{i=1}^n\lambda _i\delta (x_i)\otimes y_i\) of \(u\in \Delta (U)\otimes F,\) we have
$$\begin{aligned} \sum _{i=1}^n\left| \varphi (\lambda _i\delta (x_i)\otimes y_i)\right| \le Cg_p(u). \end{aligned}$$ -
(iii)
There exists a Borel regular probability measure \(\mu \) on \(B_{F^{*}}\) such that
$$\begin{aligned} \left| \varphi (\lambda \delta (x)\otimes y)\right| \le C\left| \lambda \right| \left\| y\right\| _{L_{p^*}(\mu )} \end{aligned}$$for all \(\lambda \in {\mathbb {C}},\) \(x\in U\) and \(y\in F,\) where
$$\begin{aligned} \left\| y\right\| _{L_{p^*}(\mu )}=\left( \int _{B_{F^{*}}}\left| y^*(y)\right| ^{p^*}\ {\textrm{d}}\mu (y^*)\right) ^{\frac{1}{p^*}}. \end{aligned}$$
Proof
\((\text {i})\ \Rightarrow \ (\text {ii})\): Let \(u\in \Delta (U)\otimes F\) and let \(\sum _{i=1}^n\lambda _i\delta (x_i)\otimes y_i\) be a representation of u. It is elementary that the function \(T:{\mathbb {C}}^n\rightarrow {\mathbb {C}}\) defined by
is linear and continuous on \(({\mathbb {C}}^n,\left\| \cdot \right\| _{\ell _{\infty }^n})\) with \(\left\| T\right\| =\sum _{i=1}^n\left| \varphi (\lambda _i\delta (x_i)\otimes y_i)\right| .\)
For any \((t_1,\ldots ,t_n)\in {\mathbb {C}}^n\) with \(\left\| (t_1,\ldots ,t_n)\right\| _{\ell _{\infty }^n}\le 1,\) by (i) we have
and, therefore,
Taking infimum over all the representations of u, we deduce that
\((\text {ii})\ \Rightarrow \ (\text {iii})\): Let be the set of all Borel regular probability measures \(\mu \) on \(B_{F^*}.\) Clearly, it is a convex compact subset of \((C(B_{F^*})^*,w^*).\) Assume first \(1<p<\infty .\) Let M be set of all functions from
to \({\mathbb {R}}\) of the form
where \(n\in {\mathbb {N}},\) \(\lambda _{i}\in {\mathbb {C}},\) \(x_{i}\in U\) and \(y_{i}\in F\) for \(i=1,\ldots ,n.\)
It is easy check that M satisfies the three conditions of Ky Fan’s Lemma (see [6, 9.10]):
-
(a)
Each \(f_{((\lambda _{i})_{i=1}^{n},(x_{i})_{i=1}^{n},(y_{i})_{i=1}^{n})}\in M\) is convex and lower semicontinuous.
-
(b)
If \(g\in {\textrm{co}}(M),\) there is \(f_{((\lambda _{i})_{i=1}^{n},(x_{i})_{i=1}^{n},(y_{i})_{i=1}^{n})}{\in }M\) with \(g(\mu ){\le } f_{((\lambda _{i})_{i=1}^{n},(x_{i})_{i=1}^{n},(y_{i})_{i=1}^{n})} (\mu )\) for all
-
(c)
Each \(f_{((\lambda _{i})_{i=1}^{n},(x_{i})_{i=1}^{n},(y_{i})_{i=1}^{n})}\in M\) has a value less or equal than 0.
By Ky Fan’s Lemma, there is a such that \(f(\mu )\le 0\) for all \(f\in M.\) In particular, we have
for all \(t\in {\mathbb {R}}^{+},\) \(\lambda \in {\mathbb {C}},\) \(x\in U\) and \(y\in F.\) It follows that
and, applying again the aforementioned identity, we conclude that
The case \(p=\infty \) is similarly proved but without applying the cited identity and taking \(C/p=0\) and \(p^*=1.\)
\((\text {iii})\ \Rightarrow \ (\text {i})\): Let \(u\in \Delta (U)\otimes F\) and let \( \sum _{i=1}^n\lambda _i\delta (x_i)\otimes y_i\) be a representation of u. Using (iii) and the Hölder inequality, we obtain
and taking infimum over all the representations of u, we conclude that \(\left| \varphi (u)\right| \le Cg_p(u).\) \(\square \)
We are now ready to present the announced result. Compare to [5, Theorem 2.3.1].
Theorem 5.2
(Pietsch Domination) Let \(1<p\le \infty \) and The following conditions are equivalent :
-
(i)
f is Cohen strongly p-summing holomorphic.
-
(ii)
For any \(\sum _{i=1}^n\lambda _i\delta (x_i)\otimes y^*_i\in \Delta (U)\otimes F^*,\) we have
-
(iii)
There is a constant \(C>0\) and a Borel regular probability measure \(\mu \) on \(B_{F^{**}}\) such that
$$\begin{aligned} \left| \left\langle y^*,f(x)\right\rangle \right| \le C\left\| y^*\right\| _{L_{p^*}(\mu )} \end{aligned}$$for all \(x\in U\) and \(y^*\in F^*,\) where
$$\begin{aligned} \left\| y^*\right\| _{L_{p^*}(\mu )}=\left( \int _{B_{F^{**}}}\left| y^{**}(y^*)\right| ^{p^*}d\mu (y^{**})\right) ^{\frac{1}{p^*}}. \end{aligned}$$
In this case, is the minimum of all constants \(C>0\) satisfying the preceding inequality.
Proof
\((\text {i})\ \Rightarrow \ (\text {ii})\) is immediate from Definition 2.2.
\((\text {ii})\ \Rightarrow \ (\text {iii})\): Clearly, Appealing to Corollary 3.5, consider its associate linear functional \(\Lambda _0(\kappa _F\circ f):\Delta (U)\otimes F^*\rightarrow {\mathbb {C}}.\) Given \(u=\sum _{i=1}^n\lambda _i\delta (x_i)\otimes y^*_i\in \Delta (U)\otimes F^*,\) we have
![](http://media.springernature.com/lw456/springer-static/image/art%3A10.1007%2Fs43037-023-00269-y/MediaObjects/43037_2023_269_Equ92_HTML.png)
by (ii). Since it holds for each representation of u, we deduce that
![](http://media.springernature.com/lw230/springer-static/image/art%3A10.1007%2Fs43037-023-00269-y/MediaObjects/43037_2023_269_Equ93_HTML.png)
By Theorem 5.1, there exists a Borel regular probability measure \(\mu \) on \(B_{F^{**}}\) such that
![](http://media.springernature.com/lw569/springer-static/image/art%3A10.1007%2Fs43037-023-00269-y/MediaObjects/43037_2023_269_Equ94_HTML.png)
for all \(x\in U\) and \(y^*\in F^*.\) Moreover, belongs to the set of all constants \(C>0\) satisfying the inequality in (iii).
\((\text {iii})\ \Rightarrow \ (\text {i})\): Given \(x\in U\) and \(y^*\in F^*,\) we have
by applying (iii). Now, Theorem 5.1 tells us that for any representation \(\sum _{i=1}^n\lambda _i\delta (x_i)\otimes y^*_i\) of \(u\in \Delta (U)\otimes F^*,\) we have
Hence with
This also shows the last assertion of the statement. \(\square \)
Remark 5.3
Theorem 5.2 is mainly a particular case of Theorem 4.6 in [21] since a Cohen strongly p-summing holomorphic map** \((1<p<\infty )\) is an \(R_1,R_2-S\)-abstract \((p,p^*)\)-summing map** for \(R_1:[0,1]\times U\times {\mathbb {C}}\rightarrow [0,\infty )\) defined by
\(R_2:B_{F^{**}}\times U\times F^*\rightarrow [0,\infty )\) given by
and defined by
This unified abstract version of Pietsch Domination Theorem has been used by several authors whenever trying to get a domination result in a very short way. Our proof is also short and appeals directly to Ky Fan’s Lemma as it was made to establish such an abstract version.
We now study the relationship between a Cohen strongly p-summing holomorphic map** from U to F and its associate linearization from to F.
Theorem 5.4
Let \(1<p\le \infty \) and The following conditions are equivalent :
-
(i)
\(f:U\rightarrow F\) is Cohen strongly p-summing holomorphic.
-
(ii)
is strongly p-summing.
In this case, Furthermore, the map** \(f\mapsto T_f\) is an isometric isomorphism from
onto
Proof
\((\text {i})\ \Rightarrow \ (\text {ii})\): Assume that By Theorem 5.2, there is a constant \(C>0\) and a Borel regular probability measure \(\mu \) on \(B_{F^{**}}\) such that \(\left| \left\langle y^*,f(x)\right\rangle \right| \le C\left\| y^*\right\| _{L_{p^*}(\mu )}\) for all \(x\in U\) and \(y^*\in F^*.\)
Let \(y^*\in F^*\) and By Theorem 1.2, given \(\varepsilon >0,\) we can take a representation \(\sum _{i=1}^{\infty }\lambda _{i}\delta (x_i)\) of \(\gamma \) such that \(\sum _{i=1}^{\infty }\left| \lambda _i\right| \le \left\| \gamma \right\| +\varepsilon .\) We have
As \(\varepsilon \) was arbitrary, it follows that
Taking infimum over all such constants C, we have
![](http://media.springernature.com/lw270/springer-static/image/art%3A10.1007%2Fs43037-023-00269-y/MediaObjects/43037_2023_269_Equ95_HTML.png)
by Theorem 5.2. It follows that
![](http://media.springernature.com/lw426/springer-static/image/art%3A10.1007%2Fs43037-023-00269-y/MediaObjects/43037_2023_269_Equ96_HTML.png)
for all Therefore
with
by Pietsch Domination Theorem for strongly p-summing operators [5, Theorem 2.3.1].
\((\text {ii})\ \Rightarrow \ (\text {i})\): Assume that Given \(x\in U\) and \(y^*\in F^*,\) we have
by [5, Theorem 2.3.1] for some Borel regular probability measure \(\mu \) on \(B_{F^{**}}.\) It follows that with
by Theorem 5.2.
Since for all
to prove the last assertion of the statement, it suffices to show that the map** \(f\mapsto T_{f}\) from
to
is surjective. Indeed, take
and then \(T=T_{f}\) for some
by Theorem 1.2. Hence
and thus
by the above proof. \(\square \)
The equivalence \((\text {i})\ \Leftrightarrow \ (\text {iii})\) of Theorem 5.2 admits the following reformulation.
Corollary 5.5
Let \(1<p\le \infty \) and The following conditions are equivalent :
-
(i)
\(f:U\rightarrow F\) is Cohen strongly p-summing holomorphic.
-
(ii)
There exists a complex Banach space G and an operator
such that
$$\begin{aligned} \left| \left\langle y^*,f(x)\right\rangle \right| \le \left\| S^*(y^*)\right\| \quad (x\in U,\; y^*\in F^*). \end{aligned}$$
In this case, is the infimum of all \(d_p(S)\) with S satisfying (ii), and this infimum is attained at \(T_f\) (Mujica’s linearization of f).
Proof
\((\text {i})\ \Rightarrow \ (\text {ii})\): If then
with
by Theorem 5.4. From Theorem 1.2, we infer that
for all \(x\in U\) and \(y^*\in F^*.\)
\((\text {ii})\ \Rightarrow \ (\text {i})\): Assume that (ii) holds. Then \(S^*\in \Pi _{p^*}(F^*,G^*)\) with \(\pi _{p^*}(S^*)=d_p(S)\) by [5, Theorem 2.2.2]. By Pietsch Domination Theorem for p-summing linear operators (see [6, Theorem 2.12]), there is a Borel regular probability measure \(\mu \) on \(B_{F^{**}}\) such that
for all \(y^*\in F^*.\) For any \(x\in U\) and \(y^*\in F^*,\) it follows that
Hence, with
by Theorem 5.2. \(\square \)
As a consequence of Theorem 5.4, an application of [4, Theorem 3.2] shows that the Banach ideal is generated by composition with the Banach operator ideal
but we prefer to give here a proof to complete the information.
Corollary 5.6
Let \(1<p\le \infty \) and The following conditions are equivalent :
-
(i)
\(f:U\rightarrow F\) is Cohen strongly p-summing holomorphic.
-
(ii)
There is a complex Banach space G,
and
so that \(f=T\circ g.\)
In this case, where the infimum is taken over all factorizations of f as in (ii), and this infimum is attained at \(T_f\circ g_U\) (Mujica’s factorization of f).
Proof
\((\text {i})\ \Rightarrow \ (\text {ii})\): If we have \(f=T_f\circ g_U,\) where
is a complex Banach space,
and
by Theorems 1.2 and 5.4. Moreover,
![](http://media.springernature.com/lw328/springer-static/image/art%3A10.1007%2Fs43037-023-00269-y/MediaObjects/43037_2023_269_Equ97_HTML.png)
\((\text {ii})\ \Rightarrow \ (\text {i})\): Assume \(f=T\circ g\) with G, g and T being as in (ii). Since \(g=T_g\circ g_U\) by Theorem 1.2, it follows that \(f=T\circ T_g\circ g_U\) which implies that \(T_f=T\circ T_g,\) and thus by the ideal property of
By Theorem 5.4, we obtain that
with
![](http://media.springernature.com/lw438/springer-static/image/art%3A10.1007%2Fs43037-023-00269-y/MediaObjects/43037_2023_269_Equ98_HTML.png)
and so by taking the infimum over all factorizations of f. \(\square \)
When F is reflexive, every factors through a Hilbert space as we see below.
Corollary 5.7
Let F be a reflexive complex Banach space. If then there exist a Hilbert space H, an operator
and a map**
such that \(f=T\circ g.\)
Proof
Assume that By Theorem 5.4,
Hence
by [5, Theorem 2.2.2]. By [6, Corollary 2.16 and Examples 2.9 (b)], there exist a Hilbert space H and operators \(T_1\in \Pi _2(F^*,H)\) and
such that \((T_f)^*=T_2\circ T_1.\)
On the one hand, we have \((T_f)^{**}=(T_1)^*\circ (T_2)^*,\) where by [5, Theorem 2.2.2]. On the other hand, we have
with \(\kappa _F\) being bijective (since F is reflexive). Consequently, we obtain \(f=T\circ g,\) where
and
\(\square \)
Applying Theorem 5.4 and [5, Theorem 2.4.1], we get useful inclusion relations.
Corollary 5.8
Let \(1<p_1\le p_2\le \infty .\) If then
and
\(\square \)
These inclusion relations can become coincidence relations when \(F^*\) has cotype 2 (see [6, pp. 217–221] for definitions and results on this class of spaces). Compare to [6, Corollary 11.16].
Corollary 5.9
Let \(2<p\le \infty .\) If \(F^*\) has cotype 2, then and
for all
Proof
By Corollary 5.8, we have with
for all
For the converse, let Then
with
by Theorem 5.4. Hence
with \(\pi _2((T_{f})^*)=d_2(T_f)\) by [5, Theorem 2.2.2]. Then, by [6, Corollary 11.16],
with \(\pi _1((T_{f})^*)=\pi _2((T_f)^*).\) Hence,
with \(\pi _{p^*}((T_{f})^*)\le \pi _1((T_f)^*)\) by [6, Theorem 2.8]. Then, by [5, Theorem 2.2.2],
with \(d_p(T_{f})=\pi _{p^*}((T_f)^*).\)
Finally, with
by Theorem 5.4, and therefore
\(\square \)
Given the transpose of f is the map**
defined by
It is known (see [12, Proposition 1.6]) that with \(\Vert f^t\Vert =\left\| f\right\| _{\infty }.\) Furthermore, \(f^t=J^{-1}_U\circ (T_f)^*\) with
being the identification established in Theorem 1.2.
The next result establishes the relation of a Cohen strongly p-summing holomorphic map** \(f:U\rightarrow F\) and its transpose Compare to [5, Theorem 2.2.2].
Theorem 5.10
Let \(1<p\le \infty \) and Then
if and only if
In this case,
Proof
Applying Theorem 5.4, [5, Theorem 2.2.2] and [6, 2.4 and 2.5], respectively, we have
![](http://media.springernature.com/lw418/springer-static/image/art%3A10.1007%2Fs43037-023-00269-y/MediaObjects/43037_2023_269_Equ99_HTML.png)
In this case, \(\square \)
The study of holomorphic map**s with relatively (weakly) compact range was initiated by Mujica [16] and followed in [12].
Corollary 5.11
Let \(1<p\le \infty .\)
-
(i)
Every Cohen strongly p-summing holomorphic map** \(f:U\rightarrow F\) has relatively weakly compact range.
-
(ii)
If F is reflexive, then every Cohen strongly p-summing holomorphic map** \(f:U\rightarrow F\) has relatively compact range.
Proof
If then
by Theorem 5.10. Hence the linear operator \(f^t\) is weakly compact and completely continuous by [6, 2.17]. Since \(f^t\) is weakly compact, this means that f has relatively weakly compact range by [12, Theorem 2.7]. Since \(f^t\) is completely continuous and \(F^*\) is reflexive, it is known that \(f^t\) is compact and, equivalently, f has relatively compact range by [12, Theorem 2.2]. \(\square \)
7 Pietsch factorization for Cohen strongly p-summing holomorphic map**s
We devote this section to the analogue of Pietsch Factorization Theorem for p-summing linear operators [6, Theorem 2.13] for the class of Cohen strongly p-summing holomorphic map**s. Recall that, for every Banach space F, we have the canonical isometric injections \(\kappa _F:F\rightarrow F^{**}\) and \(\iota _F:F\rightarrow C\left( B_{F^*}\right) \) defined, respectively, by
Moreover, if \(\mu \) is a regular Borel measure on \((B_{F^{**}},w^*),\) \(j_{p}\) denotes the canonical map from \(C\left( B_{F^*}\right) \) to \(L_{p}\left( \mu \right) .\)
Theorem 6.1
(Pietsch Factorization) Let \(1<p\le \infty \) and The following conditions are equivalent :
-
(i)
\(f:U\rightarrow F\) is Cohen strongly p-summing holomorphic.
-
(ii)
There exist a regular Borel probability measure \(\mu \) on \((B_{F^{**}},w^*),\) a closed subspace \(S_{p^*}\) of \(L_{p^*}(\mu )\) and a bounded holomorphic map** \(g:U\rightarrow (S_{p^*})^*\) such that the following diagram commutes :
In this case,
Proof
\((\text {i})\ \Rightarrow \ (\text {ii})\): Let Then
by Theorem 5.10. By [6, Theorem 2.13], there exist a regular Borel probability measure \(\mu \) on \((B_{F^{**}},w^*),\) a subspace \(S_{p^*}:=\overline{j_{p^*}\left( i_{F^*}\left( F^*\right) \right) }\) of \(L_{p^*}(\mu ),\) and an operator
with \(\left\| T\right\| =\Vert f^t\Vert \) such that the following diagram commutes:
![](http://media.springernature.com/lw208/springer-static/image/art%3A10.1007%2Fs43037-023-00269-y/MediaObjects/43037_2023_269_Equ101_HTML.png)
Dualizing, we obtain
![](http://media.springernature.com/lw180/springer-static/image/art%3A10.1007%2Fs43037-023-00269-y/MediaObjects/43037_2023_269_Equ102_HTML.png)
Let \(g:=T^*\circ g_U.\) Clearly, with \(\left\| g\right\| _{\infty }\le \left\| T\right\| ,\) and thus
![](http://media.springernature.com/lw240/springer-static/image/art%3A10.1007%2Fs43037-023-00269-y/MediaObjects/43037_2023_269_Equ103_HTML.png)
Moreover, since \(f^t=T\circ j_{p^*}\circ \iota _{F^*},\) we have
\((\text {ii})\ \Rightarrow \ (\text {i})\): Since \(\kappa _{F}\circ f=(\iota _{F^*})^*\circ (j_{p^*})^*\circ g,\) it follows that \(f^{t}\circ (\kappa _F)^*=((\iota _{F^*})^*\circ (j_{p^*})^*\circ g)^t.\) As \((\kappa _{F})^*\circ \kappa _{F^*}=\textrm{id}_{F^*},\) we obtain that
Since \(j_{p^*}\in \Pi _{p^*}(\iota _{F^*}(F^*),S_{p^*})\) (see [6, Examples 2.9]), then
![](http://media.springernature.com/lw221/springer-static/image/art%3A10.1007%2Fs43037-023-00269-y/MediaObjects/43037_2023_269_Equ104_HTML.png)
by [5, Theorem 2.2.2]. Hence with
![](http://media.springernature.com/lw523/springer-static/image/art%3A10.1007%2Fs43037-023-00269-y/MediaObjects/43037_2023_269_Equ105_HTML.png)
by the ideal property of Corollary 5.6 and [5, Theorem 2.2.2]. Applying Theorem 5.10 and the ideal property of \(\Pi _p,\) we deduce that
Again, Theorem 5.10 gives that
with
Moreover,
![](http://media.springernature.com/lw300/springer-static/image/art%3A10.1007%2Fs43037-023-00269-y/MediaObjects/43037_2023_269_Equ106_HTML.png)
\(\square \)
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Acknowledgements
The authors would like to thank the referees for their valuable comments that have improved considerably this paper. Research of A. Jiménez-Vargas was partially supported by project UAL-FEDER grant UAL2020-FQM-B1858, by Junta de Andalucía grants P20\(\_\)00255 and FQM194, and by grant PID2021-122126NB-C31 funded by MCIN/AEI/10.13039/501100011033 and by “ERDF A way of making Europe”. J. M. Sepulcre was also supported by PGC2018-097960-B-C22 (MCIU/AEI/ERDF, UE).
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Jiménez-Vargas, A., Saadi, K. & Sepulcre, J.M. Cohen strongly p-summing holomorphic map**s on Banach spaces. Banach J. Math. Anal. 17, 44 (2023). https://doi.org/10.1007/s43037-023-00269-y
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DOI: https://doi.org/10.1007/s43037-023-00269-y