Unified Grothendieck’s and Kwapień’s Theorems for Multilinear Operators

Abstract

Kwapień’s theorem asserts that every continuous linear operator from $\ell _{1}$ to $\ell _{p}$ is absolutely $\left( r,1\right)$-summing for $1/r=1-\left| 1/p-1/2\right| .$ When $p=2$ it recovers the famous Grothendieck’s theorem. In this paper we investigate multilinear variants of these theorems and related issues. Among other results we present a unified version of Kwapień’s and Grothendieck’s results that encompasses the cases of multiple summing and absolutely summing multilinear operators.

1 Introduction

The theory of absolutely summing operators was initially introduced and developed by Pietsch (1967), Mitiagin and Pełczyński (1966) and Lindenstrauss and Pełczyński (1968), inspired by previous works of A. Grothendieck and it plays an important role in the study of Banach Spaces and Operator Theory, with important connections with other fields of mathematics such as operator ideals, ideals of polynomials, multilinear operators, nonlinear analysis and Quantum Information Theory (see Albiac et al. 2021; Albuquerque and Rezende 2021; Araújo and Santos 2020; Belacel et al. 2023; Botelho and Wood 2022; Pellegrino and Santos 2024; Pisier 2012; Teixeira 2022 and the references therein).

Let E, F be Banach spaces and $r\ge s\ge 1$ be real numbers. A continuous linear operator $T:E\rightarrow F$ is absolutely $\left( r,s\right)$-summing if $\left( T(x_{j})\right) _{j=1}^{\infty }\in \ell _{r}(F)$ whenever $\left( x_{j}\right) _{j=1}^{\infty }\in \ell _{s}^{w}(E),$ where $\ell _{s} ^{w}(E)$ denotes the space of weakly s-summable sequences in E, i.e., the sequences $(x_{j})_{j=1}^{\infty }$ in E such that

$$\begin{aligned} \left\| (x_{j})_{j=1}^{\infty }\right\| _{w,s}:=\sup _{\varphi \in B_{E^{*}}}\left( \sum \limits _{j=1}^{\infty }\left| \varphi (x_{j})\right| ^{s}\right) ^{1/s}<\infty . \end{aligned}$$

One of the cornerstones of the theory of absolutely summing operators is Grothendieck’s theorem, which asserts that every continuous linear operator from $\ell _{1}$ to $\ell _{2}$ is absolutely $\left( 1,1\right)$-summing. Kwapień (1968) extended Grothendieck’s theorem replacing $\ell _{2}$ by $\ell _{p}$, for $p\ge 1$, as follows: every continuous linear operator from $\ell _{1}$ to $\ell _{p}$ is absolutely $\left( r,1\right)$-summing, with

$$\begin{aligned} 1/r=1-\left| 1/p-1/2\right| , \end{aligned}$$

(1)

and this result is optimal (see also Bennett 1977). In the last decades the notion of absolutely summing operators was extended to the multilinear and nonlinear setting in several different lines of research (see Albuquerque et al. 2018; Defant et al. 2010; Diestel et al. 1995 for the linear theory). In this paper we shall be interested in the notions of absolutely summing multilinear operators and multiple summing multilinear operators (for the precise definitions, see Sect. 2).

The extension of Kwapień’s theorem to multilinear operators is a natural problem to be investigated. For multiple summing operators, an immediate consequence of Botelho and Pellegrino (2009, Corollary 4.3) is that every continuous m-linear operator from $\ell _{1}$ to $\ell _{p}$ is multiple $\left( r,1\right)$-summing, with r as in (1) and this result is sharp. For absolutely summing multilinear operators, as proved in Bayart et al. (2020), every continuous m-linear operator from $\ell _{1}$ to $\ell _{p}$ is absolutely $\left( r,1\right)$-summing for

$$\begin{aligned} r=\left\{ \begin{array}{c} \frac{2p}{mp+2p-2},\text { if }1\le p\le 2\\ \frac{2p}{mp+2},\text { if }2\le p\le \infty . \end{array} \right. \end{aligned}$$

(2)

However, the optimality of the estimates (2) is not proven. Our first result shows that for $2\le p\le \infty$ the above estimate is sharp.

The following variant of Kwapień’s theorem was proved in Defant et al. (2010):

Theorem 1.1

(See Defant et al. 2010) Let $T\in \mathcal {L}\left( ^{m}\ell _{1};\ell _{p}\right)$ and $A_{k}\in \mathcal {L}\left( ^{n}\ell _{\infty };\ell _{1}\right)$ for all $k=1,...,m.$ The composition $T\left( A_{1},...,A_{m}\right)$ is multiple $\left( r,1\right)$-summing for

$$\begin{aligned} r=\left\{ \begin{array}{c} \frac{2n}{n+2-\frac{2}{p}},\text { if }1\le p\le 2\\ \frac{2n}{p+1},\text { if }2\le p\le \frac{2n}{n-1}\\ 2,\text { if }\frac{2n}{n-1}\le p\le \infty . \end{array} \right. \end{aligned}$$

This result was recently improved in Bayart et al. (2020) when $2\le p\le \infty$ and the authors also investigated the case of absolutely summing multilinear operators:

Theorem 1.2

(See Bayart et al. (2020)) Let $T\in \mathcal {L}\left( ^{m}\ell _{1};\ell _{p}\right)$ and $A_{k}\in \mathcal {L}\left( ^{n}\ell _{\infty };\ell _{1}\right)$ for all $k=1,...,m.$

(A) The composition $T\left( A_{1},...,A_{m}\right)$ is multiple $\left( r,1\right)$-summing for

$$\begin{aligned} r=\left\{ \begin{array}{c} \frac{2n}{n+2-\frac{2}{p}},\text { if }1\le p\le 2\\ \frac{2n}{n+\frac{2}{p}},\text { if }2\le p\le \infty . \end{array} \right. \end{aligned}$$

(B) Assume that $n\ge 2.$ The composition $T\left( A_{1},...,A_{m}\right)$ is absolutely $\left( r,2\right)$-summing for

$$\begin{aligned} r=\left\{ \begin{array}{c} \frac{2p}{mp+2p-2},\text { if }1\le p\le 2\\ \frac{2p}{mp+2},\text { if }2\le p\le \infty . \end{array} \right. \end{aligned}$$

Note that while (A) provides r so that $T\left( A_{1},...,A_{m}\right)$ is multiple $\left( r,1\right)$-summing, item (B) provides r so that $T\left( A_{1},...,A_{m}\right)$ is absolutely $\left( r,2\right)$-summing. However, since every continuous m-linear operator $T\in \mathcal {L}\left( ^{m}\ell _{\infty };\ell _{1}\right)$ is multiple $\left( s;s\right)$-summing for every $s\ge 2$ (see Botelho et al. 2010, Corollary 4.10) it is obvious that the composition $T\left( A_{1},...,A_{m}\right)$ is multiple $\left( s,s\right)$-summing for all $s\ge 2$ and this result is optimal in the sense that one cannot improve $\left( s,s\right)$ to $\left( r,s\right)$ for $r<s.$ Thus, in the context of multiple summing operators the nontrivial problem seems to be:

Problem 1.3

Given $1\le s<2,$ and positive integers m, n,  what is the best r so that the composition $T\left( A_{1},...,A_{m}\right)$ is multiple $\left( r,s\right)$-summing for every $T\in \mathcal {L}\left( ^{m}\ell _{1};\ell _{p}\right)$ and $A_{k}\in \mathcal {L}\left( ^{n} \ell _{\infty };\ell _{1}\right)$ for $k=1,...,m$?

The paper is organized as follows. In Sect. 2 we present some preliminary concepts and results which shall be used throughout the paper. In Sect. 3 we prove that the estimate (2) provided by Bayart, Pellegrino and Rueda in Bayart et al. (2020) is optimal when $2\le p\le \infty$. In Sect. 4 we prove Kwapień’s and Grothendieck’s inequalities for blocks and, finally, in Sect. 5 we prove variants of Theorems 1.1 and 1.2, providing a partial answer to Problem 1.3.

2 Background and Notation

Henceforth $\mathbb {K}$ represents the field of all scalars (complex or real), $E,E_{1},...,E_{m},F$ denote Banach spaces over $\mathbb {K}$ and the Banach space of all bounded m-linear operators from $E_{1}\times \cdots \times E_{m}$ to F is denoted by $\mathcal {L}(E_{1},...,E_{m};F)$ and we endow it with the classical sup norm (when $E_{1}=\cdots =E_{m}=E$ we write $\mathcal {L} (^{m}E;F)$ instead of $\mathcal {L}(E_{1},\ldots ,E_{m};F)$). The topological dual of E is denoted by $E^{*}$ and its closed unit ball is denoted by $B_{E^{*}}.$ Throughout the paper, for $p\in [1,\infty ]$, the symbol $p^{*}$ denotes the conjugate of p, that is $1/p+1/p^{*}=1$ and, as usual, $1^{*}=\infty$ and $\infty ^{*}=1.$

For the sake of completeness, we shall recall the notions of absolutely summing multilinear operators and multiple summing multilinear operators.

If $\left( r,s\right) \in (0,\infty )\times [1,\infty ]$ and $1/r\le m/s,$ an operator $T\in \mathcal {L}(E_{1},...,E_{m};F)$ is absolutely (r; s)-summing is there is a constant $C>0$ be such that

$$\begin{aligned} \left( \sum \limits _{j=1}^{\infty }\left\| T(x_{j}^{(1)},...,x_{j} ^{(m)})\right\| _{F}^{r}\right) ^{\frac{1}{r}}\le C\prod _{k=1} ^{m}\left\| (x_{j_{k}}^{(k)})_{j_{k}=1}^{\infty }\right\| _{w,s} \end{aligned}$$

for every $\left( x_{j_{k}}^{(k)}\right) _{j_{k}=1}^{\infty }\in \ell _{s} ^{w}(E_{k}).$

The class of absolutely (r; s)-summing operators is denoted by $\mathcal {L} _{as,(r;s)}(E_{1},...,E_{m};F)$ and the infimum taken over all possible constants $C>0$ satisfying the previous inequality defines a norm in $\mathcal {L}_{as,(r;s)}(E_{1},...,E_{m};F)$, which is denoted by $\pi _{as\left( r;s\right) }^{m}$ (see Alencar and Matos 1989).

If $1\le s\le r<\infty ,$ an operator $T:E_{1}\times \cdots \times E_{m}\rightarrow F$ is multiple (r; s)-summing if there is a constant $C>0$ be such that

$$\begin{aligned} \left( \sum \limits _{j_{1},...,j_{m}=1}^{\infty }\left\| T(x_{j_{1}} ^{(1)},...,x_{j_{m}}^{(m)})\right\| _{F}^{r}\right) ^{\frac{1}{r}}\le C\prod _{k=1}^{m}\left\| (x_{j_{k}}^{(k)})_{j_{k}=1}^{\infty }\right\| _{w,s} \end{aligned}$$

for every $\left( x_{j_{k}}^{(k)}\right) _{j_{k}=1}^{\infty }\in \ell _{s} ^{w}(E_{k}).$

The class of multiple (r; s)-summing operators is denoted by $\Pi _{mult(r,s)}(E_{1},...,E_{m};F)$ (see Matos 2003; Pérez-García and Villanueva 2003). We recall the following inclusion theorem (see Pellegrino et al. 2017, Proposition 3.4) and Albuquerque and Rezende 2018; Bayart 2018; Raposo and Serrano-Rodríguez 2023 for extended versions and applications) that will be useful later:

Theorem 2.1

(See Pellegrino et al. (2017)) Let m be a positive integer and $1\le s\le u<\frac{mrs}{mr-s}.$ Then, for any Banach spaces $E_{1},...,E_{m},F$ we have

$$\begin{aligned} \Pi _{mult(r;s)}\left( E_{1},\dots ,E_{m};F\right) \subset \Pi _{mult\bigg (\frac{rsu}{su+mrs-mru};u\bigg )}\left( E_{1},\dots ,E_{m};F\right) \end{aligned}$$

and the inclusion has norm 1.

3 Optimality of Kwapień’s Inequality for Multilinear Operators

In this section we show that (2) is optimal for $2\le p\le \infty .$ Let $0<r<\frac{2p}{mp+2}$. The proof is an adaptation or an argument used in the proof of Pellegrino and Seoane-Sepúlveda (2015, Theorem 1.1). Let $n\in \mathbb {N}$ and $x_{1},...,x_{n}\in \ell _{1}$ be non null vectors. Consider $x_{1}^{*},...,x_{n}^{*}\in B_{\ell _{\infty }}$ so that $x_{j}^{*}(x_{j})=\Vert x_{j}\Vert$ for every $j=1,...,n$. Let $a_{1},...,a_{n}$ be scalars such that $\sum _{j=1}^{n}|a_{j}|^{p/r}=1$ and define the following m-linear operator

$$\begin{aligned} T_{n}:\ell _{1}\times \cdots \times \ell _{1}\longrightarrow \ell _{p},\ \ T_{n} (x^{(1)},...,x^{(m)})= {\textstyle \sum \limits _{j=1}^{n}} |a_{j}|^{\frac{1}{r}}x_{j}^{*}(x^{(1)})\cdots x_{j}^{*}(x^{(m)})e_{j} \end{aligned}$$

where $e_{j}$ is the jth canonical vector of $\ell _{p}$. Note that, for every $(x^{(1)},...,x^{(m)})\in \ell _{1}\times \cdots \times \ell _{1}$, we have

$$\begin{aligned} \Vert T_{n}(x^{(1)},...,x^{(m)})\Vert&=\left( {\textstyle \sum \limits _{j=1}^{n}} \left| |a_{j}|^{\frac{1}{r}}x_{j}^{*}(x^{(1)})\cdots x_{j}^{*}(x^{(m)})\right| ^{p}\right) ^{\frac{1}{p}}\\&\le \left( {\textstyle \sum \limits _{j=1}^{n}} |a_{j}|^{\frac{p}{r}}\right) ^{\frac{1}{p}}\Vert x^{(1)}\Vert \cdots \Vert x^{(m)}\Vert \\&=\Vert x^{(1)}\Vert \cdots \Vert x^{(m)}\Vert . \end{aligned}$$

It is plain that $T_{n}$ is absolutely (r; 1)-summing. Note that for $k=1,\ldots ,n$, we have

$$\begin{aligned} \Vert T_{n}(x_{k},....,x_{k})\Vert =\left\| \sum _{j=1}^{n}|a_{j}|^{\frac{1}{r}}x_{j}^{*}(x_{k})^{m}e_{j}\right\| \ge |a_{k}|^{\frac{1}{r}} x_{k}^{*}(x_{k})^{m}=|a_{k}|^{\frac{1}{r}}\Vert x_{k}\Vert ^{m}. \end{aligned}$$

Hence

$$\begin{aligned} \left( {\textstyle \sum \limits _{j=1}^{n}} \Vert x_{j}\Vert ^{mr}|a_{j}|\right) ^{\frac{1}{r}}&=\left( {\textstyle \sum \limits _{j=1}^{n}} \left( \Vert x_{j}\Vert ^{m}|a_{j}|^{\frac{1}{r}}\right) ^{r}\right) ^{\frac{1}{r}}\\&\le \left( {\textstyle \sum \limits _{j=1}^{n}} \Vert T_{n}(x_{j},...,x_{j})\Vert ^{r}\right) ^{\frac{1}{r}}\\&\le \pi _{r,1}(T_{n})\Vert (x_{j})_{j=1}^{n}\Vert _{w,1}^{m}. \end{aligned}$$

Since this last inequality holds whenever $\sum _{j=1}^{n}|a_{j}|^{\frac{p}{r} }=1$, if $\left( p/r\right) ^{*}$ is the conjugate of p/r,  we obtain

$$\begin{aligned} \left( \sum _{j=1}^{n}\Vert x_{j}\Vert ^{mr\left( p/r\right) ^{*}}\right) ^{\frac{1}{\left( p/r\right) ^{*}}}&=\sup \left\{ \sum _{j=1} ^{n}|a_{j}|\Vert x_{j}\Vert ^{mr};\sum _{j=1}^{n}|a_{j}|^{\frac{p}{r}}=1\right\} \\&\le \left( \pi _{r,1}(T_{n})\Vert (x_{j})_{j=1}^{n}\Vert _{w,1}^{m}\right) ^{r} \end{aligned}$$

and, then,

$$\begin{aligned} \frac{\left( {\textstyle \sum _{j=1}^{n}} \Vert x_{j}\Vert ^{mr\left( \frac{p}{r}\right) ^{*}}\right) ^{\frac{1}{r\left( \frac{p}{r}\right) ^{*}}}}{\Vert (x_{j})_{j=1}^{n}\Vert _{w,1}^{m}}\le \pi _{r,1}(T_{n}). \end{aligned}$$

(3)

Since $0<r<\frac{2p}{mp+2}$ we have $mr\left( p/r\right) ^{*}<2$ and by the Dvoretzky–Rogers Theorem (see Diestel et al. 1995, Theorem 10.5), we know that $id_{\ell _{1}}$ is not $\left( mr\left( p/r\right) ^{*};1\right)$-summing. Hence

$$\begin{aligned} \lim _{n\rightarrow \infty }\pi _{r,1}(T_{n})=\infty ~\text {and}~\Vert T_{n}\Vert =1 \end{aligned}$$

(4)

and we conclude that the space of all absolutely (r; 1)-summing m-linear operators from $\ell _{1}$ to $\ell _{p}$ is not closed in $\mathcal {L}(^{m} \ell _{1};\ell _{p})$.

4 Kwapień’s Theorem for Blocks of Sequences

We shall need to introduce some terminology on tensor products. The product

$$\begin{aligned} \widehat{\otimes }_{j\in \{1,\ldots ,n\}}^{\pi }E_{j}=E_{1}\widehat{\otimes }^{\pi }\cdots \widehat{\otimes }^{\pi }E_{n} \end{aligned}$$

denotes the completed projective n-fold tensor product of $E_{1},\ldots ,E_{n}$. The tensor $x_{1}\otimes \cdots \otimes x_{n}$ is denoted for short by $\otimes _{j\in \{1,\ldots ,n\}}x_{j}$, whereas $\otimes _{n}x$ denotes the tensor $x\otimes \cdots \otimes x$. In a similar way, $\times _{j\in \{1,\ldots ,n\}}E_{j}$ denotes the product space $E_{1}\times \cdots \times E_{n}$.

We need the following result (see Albuquerque et al. 2018):

Proposition 4.1

(Albuquerque et al. 2018, Proposition 2.3) Let m be a positive integer and let $E_{1},\dots ,E_{m},F$ be Banach spaces. Let $1\le k\le m$ and $I_{1},\ldots ,I_{k}$ be pairwise disjoint non-void subsets of $\{1,\ldots ,m\}$ such that $\cup _{j=1}^{k}I_{j}=\{1,\ldots ,m\}$. Then given $T\in {\mathcal {L} }(E_{1},\ldots ,E_{m};F)$, there is a unique $\widehat{T}\in {\mathcal {L} }(\widehat{\otimes }_{j\in I_{1}}^{\pi }E_{j},\ldots ,\widehat{\otimes }_{j\in I_{k}}^{\pi }E_{j};F)$ such that

$$\begin{aligned} \widehat{T}(\otimes _{j\in I_{1}}x_{j},\dots ,\otimes _{j\in I_{k}}x_{j} )=T(x_{1},\dots ,x_{m}) \end{aligned}$$

and $\Vert \widehat{T}\Vert =\Vert T\Vert$. The correspondence $T\leftrightarrow \widehat{T}$ determines an isometric isomorphism between the spaces ${\mathcal {L}}(E_{1},\ldots ,E_{m};F)$ and ${\mathcal {L}}(\widehat{\otimes }_{j\in I_{1}}^{\pi }E_{j},\ldots ,\widehat{\otimes }_{j\in I_{k}}^{\pi }E_{j};F)$.

Let m be a positive integer and let $1\le k\le m$ and $\mathcal {I} =\left\{ I_{1},\ldots ,I_{k}\right\}$ be a family of pairwise disjoint non-void subsets of $\{1,\ldots ,m\}$ such that $\cup _{j=1}^{k}I_{j} =\{1,\ldots ,m\}$. Let $\Omega _{I_{j}}\subset \mathbb {N}^{\left| I_{j}\right| }$ defined by

$$\begin{aligned} \Omega _{I_{j}}=Diag\left( \mathbb {N}^{\left| I_{j}\right| }\right) =\left\{ \left( i,\ldots ,i\right) \in \mathbb {N}^{\left| I_{j} \right| }\right\} \end{aligned}$$

for all $j\in \{1,...,k\}$. Let us also define $\Omega _{\mathcal {I}} \subset \mathbb {N}^{m}$ as the product of diagonals

$$\begin{aligned} \Omega _{\mathcal {I}}=\Omega _{I_{1}}\times \cdots \times \Omega _{I_{k}}. \end{aligned}$$

We will denote by

$$\begin{aligned} \mathcal {L}_{\mathcal {I},as\left( r;1\right) }\left( ^{m}\ell _{1};F\right) \end{aligned}$$

the space of all $T\in \mathcal {L}\left( ^{m}\ell _{1};F\right)$, for which there is a constant $C>0$ be such that

$$\begin{aligned} {\textstyle \sum \limits _{\left( j_{1},...,j_{m}\right) \in \Omega _{\mathcal {I}}}} \left\| T\left( x_{j_{1}}^{(1)},...,x_{j_{m}}^{(m)}\right) \right\| _{F}^{r}\le C\prod _{i=1}^{m}\left\| \left( x_{j_{i}}^{(i)}\right) _{j_{i}=1}^{\infty }\right\| _{w,1} \end{aligned}$$

for every

$$\begin{aligned} \left( x_{j_{i}}^{(i)}\right) _{j_{i}=1}^{\infty }\in \ell _{1}^{w}\left( \ell _{1}\right) . \end{aligned}$$

Obviously

$$\begin{aligned} \mathcal {L}_{\left\{ \left\{ 1\right\} ,\ldots ,\left\{ m\right\} \right\} ,as\left( r;1\right) }\left( ^{m}\ell _{1};F\right) =\Pi _{mult(r;1)}\left( ^{m}\ell _{1};F\right) \end{aligned}$$

and

$$\begin{aligned} \mathcal {L}_{\left\{ 1,\ldots ,m\right\} ,as\left( r;1\right) }\left( ^{m}\ell _{1};F\right) =\mathcal {L}_{as\left( r;1\right) }\left( ^{m} \ell _{1};F\right) . \end{aligned}$$

The following lemma plays a crucial role in the proof of the main result of this section:

Lemma 4.2

Let F be a Banach space and let $\mathcal {I}=\left\{ I_{1},\ldots ,I_{k}\right\}$ be a family of pairwise disjoint non-void subsets of $\{1,\ldots ,m\}$ such that $\cup _{j=1}^{k}I_{j}=\{1,\ldots ,m\}$. If $n=\min \left\{ \left| I_{1}\right| ,\ldots ,\left| I_{k} \right| \right\}$, then the following assertions are equivalent:

• (a) $\mathcal {L}\left( ^{n}\ell _{1};F\right) =\mathcal {L}_{as\left( r;1\right) }\left( ^{n}\ell _{1};F\right)$.

• (b) $\mathcal {L}\left( ^{m}\ell _{1};F\right) =\mathcal {L}_{\mathcal {I},as\left( r;1\right) }\left( ^{m}\ell _{1};F\right)$.

Proof

Let us fix in each set $I_{i}$ an order. We define the sets $J_{1},...,J_{n}$ in the following way: for every $j\in \left\{ 1,...,n\right\} ,$

$$\begin{aligned} \left( J_{j}\right) _{i}=\left( I_{i}\right) _{j},\text { for }i\le k, \end{aligned}$$

where $\left( J_{j}\right) _{i}$ means the ith element of $J_{j}$ (in the same way for $\left( I_{i}\right) _{j}$). The rest of elements that are not yet in any $J_{j}$ are included in $J_{1}.$ Thus, $J_{1}$ has $m-\left( n-1\right) k$ elements and $J_{j}$, for $j\in \left\{ 2,...,n\right\} ,$ has k elements.

Obviously $\left\{ J_{1},...,J_{n}\right\}$ is a family of non–void pairwise disjoint subsets of $\{1,\ldots ,m\}$ such that $\cup _{j=1}^{n} J_{j}=\{1,\ldots ,m\}$.

(a) implies (b). Let $A\in \mathcal {L}(^{m}\ell _{1};F)$ and by Proposition 4.1, let $\widehat{A}\in {\mathcal {L}}(\widehat{\otimes }_{j\in J_{1} }^{\pi }\ell _{1},\ldots ,\widehat{\otimes }_{j\in J_{n}}^{\pi }\ell _{1};F)$ be such that

$$\begin{aligned} \widehat{A}(\otimes _{j\in J_{1}}x^{\left( j\right) },\dots ,\otimes _{j\in J_{n}}x^{\left( j\right) })=A(x^{\left( 1\right) },\dots ,x^{\left( m\right) }) \end{aligned}$$

for every $x^{\left( j\right) }\in \ell _{1}$. Since $\widehat{\otimes }_{j\in J_{j}}^{\pi }\ell _{1}$ is isometrically isomorphic to $\ell _{1}$, for all $j\in \left\{ 1,...,n\right\}$, and by assumption we have that $\widehat{A}\in \mathcal {L}_{as\left( r;1\right) }\left( ^{n}\ell _{1};F\right)$. Then $\pi _{as\left( r;1\right) }^{n}$ $(\widehat{A})\le M\Vert \widehat{A}\Vert =M\Vert A\Vert$, where M is a constant independent of A. We get

$$\begin{aligned}&\left( {\textstyle \sum \limits _{\left( j_{1},...,j_{m}\right) \in \Omega _{\mathcal {I}}}} \left\| A(x_{j_{1}}^{\left( 1\right) },\dots ,x_{j_{m}}^{\left( m\right) })\right\| _{F}^{r}\right) ^{\frac{1}{r}}\\&\quad =\left( {\textstyle \sum \limits _{\left( j_{1},...,j_{m}\right) \in \Omega _{\mathcal {I}}}} \left\| \widehat{A}(\otimes _{i_{1}\in J_{1}}x_{j_{i_{1}}}^{\left( i_{1}\right) },\dots ,\otimes _{i_{n}\in J_{n}}x_{j_{i_{n}}}^{\left( i_{n}\right) })\right\| _{F}^{r}\right) ^{\frac{1}{r}}\\&\quad \le \pi _{as\left( r;1\right) }^{n}(\widehat{A})\left\| \left( \otimes _{i_{1}\in J_{1}}x_{j_{i_{1}}}^{\left( i_{1}\right) }\right) _{\left( J_{1}\right) _{1},...,\left( J_{1}\right) _{m-\left( n-1\right) k}=1}^{\infty }\right\| _{w,1}\prod _{s=2}^{n}\left\| \left( \otimes _{i_{s}\in J_{s}}x_{j_{i_{s}}}^{\left( i_{s}\right) }\right) _{\left( J_{s}\right) _{1},...,\left( J_{s}\right) _{k}=1}^{\infty }\right\| _{w,1}\\&\quad \le M\Vert A\Vert K_{G}^{2\left( m-\left( n-1\right) k\right) -2} \prod _{i_{1}\in J_{1}}\left\| \left( x_{j}^{\left( i_{1}\right) }\right) _{j=1}^{\infty }\right\| _{w,1}\prod _{s=2}^{n}K_{G}^{2k-2} \prod _{i_{s}\in J_{s}}\left\| \left( x_{j}^{\left( i_{s}\right) }\right) _{j=1}^{\infty }\right\| _{w,1},\hspace{25em} \end{aligned}$$

where $K_{G}$ stands for Grothendieck’s constant [see Botelho and Pellegrino (2009, pg. 1420)]. We thus conclude that $A\in \mathcal {L}_{\mathcal {I},as\left( r;1\right) }\left( ^{m}\ell _{1};F\right)$.

(b) implies (a). Without loss of generality, we will assume that $I_{1}=\{1,\ldots ,n\}$. Let $A:\times _{j\in \{1,\ldots ,n\}}\ell _{1}\rightarrow F$ be a bounded n-linear operator. Define the m-linear map $T_{A}:\times _{j\in \{1,\ldots ,m\}}\ell _{1}\rightarrow F$ by

$$\begin{aligned} T_{A}(x^{(1)},\ldots ,x^{(m)}):=A(x^{(1)},\ldots ,x^{(n)})x_{1}^{\left( n+1\right) }x_{1}^{\left( n+2\right) }\cdots x_{1}^{\left( m\right) } \end{aligned}$$

for every $(x^{(1)},\ldots ,x^{(m)})\in \times _{j\in \{1,\ldots ,m\}}\ell _{1}$.

Since $\mathcal {L}\left( ^{m}\ell _{1};F\right) =\mathcal {L}_{\mathcal {I},as\left( r;1\right) }\left( ^{m}\ell _{1};F\right)$, we have

$$\begin{aligned} \left( \sum \limits _{j=1}^{\infty }\left\| A(x_{j}^{(1)},...,x_{j} ^{(n)})\right\| _{F}^{r}\right) ^{\frac{1}{r}}&\le \left( {\textstyle \sum \limits _{\left( j_{1},...,j_{m}\right) \in \Omega _{\mathcal {I}}}} \left\| T_{A}(x_{j_{1}}^{\left( 1\right) },\dots ,x_{j_{n}}^{\left( n\right) },e_{1},\dots ,e_{1})\right\| _{F}^{r}\right) ^{\frac{1}{r}}\\&\le \prod _{k=1}^{n}\left\| \left( x_{j_{i}}^{(i)}\right) _{j_{i} =1}^{\infty }\right\| _{w,1}. \end{aligned}$$

We thus conclude that $A\in \mathcal {L}_{as\left( r;1\right) }\left( ^{n}\ell _{1};F\right)$. $\square$

Theorem 4.3

(Kwapień’s Theorem for blocks) Let $\mathcal {I}=\left\{ I_{1},\ldots ,I_{k}\right\}$ be a family of pairwise disjoint non-void subsets of $\{1,\ldots ,m\}$ such that $\cup _{j=1}^{k}I_{j}=\{1,\ldots ,m\}$. If $n=\min \left\{ \left| I_{1}\right| ,\ldots ,\left| I_{k} \right| \right\} ,$ then

$$\begin{aligned} \mathcal {L}\left( ^{m}\ell _{1};\ell _{p}\right) =\mathcal {L}_{\mathcal {I},as\left( t;1\right) }\left( ^{m}\ell _{1};\ell _{p}\right) . \end{aligned}$$

with

$$\begin{aligned} t=\left\{ \begin{array}{c} \frac{2p}{np+2p-2},\text { if }1\le p\le 2\\ \frac{2p}{np+2},\text { if }2\le p\le \infty . \end{array} \right. \end{aligned}$$

Moreover, the parameter t is optimal when $2\le p\le \infty$.

Proof

The result follows from a combination of (2) and Lemma 4.2. The optimality follows from what we have just proved in Sect. 3. $\square$

When $k=1$ we recover Kwapień’s Theorem for absolutely summing multilinear operators and when $k=m$ we recover Kwapień’s Theorem for multiple summing operators. In the special case $p=2$ we obtain a unified Grothendieck’s theorem:

Corollary 4.4

(Unified Grothendieck’s Theorem) Let $\mathcal {I}=\left\{ I_{1},\ldots ,I_{k}\right\}$ be a family of pairwise disjoint non-void subsets of $\{1,\ldots ,m\}$ such that $\cup _{j=1}^{k}I_{j}=\{1,\ldots ,m\}$. If $n=\min \left\{ \left| I_{1}\right| ,\ldots ,\left| I_{k} \right| \right\}$ then we have

$$\begin{aligned} \mathcal {L}\left( ^{m}\ell _{1};\ell _{2}\right) =\mathcal {L}_{\mathcal {I},as\left( \frac{2}{n+1};1\right) }\left( ^{m}\ell _{1};\ell _{2}\right) \end{aligned}$$

and the result is optimal.

5 Other Variants of Kwapień’s Theorem

In this final section we present partial answers to Problem 1.3. Of course, using the Inclusion Theorem (Theorem 2.1), provided that $1\le u<\frac{2mn}{2mn-1}$, we can prove that $T\left( A_{1},...,A_{m}\right)$ is multiple $\left( t,u\right)$-summing for a certain t. However, the following result provides better estimates for other choices of u : 

Theorem 5.1

Let $T\in \mathcal {L}\left( ^{m}\ell _{1};\ell _{p}\right)$ and $A_{k} \in \mathcal {L}\left( ^{n}\ell _{\infty };\ell _{1}\right)$ for all $k=1,...,m.$ Then the composition $T\left( A_{1},...,A_{m}\right)$ is multiple $\left( t,u\right)$-summing in the following cases:

(i) For $\left( p,u\right) \in [1,2]\times [\frac{2n}{n+1},2]$ and

$$\begin{aligned} t=\frac{2pu}{4p+2u-pu-4}; \end{aligned}$$

(ii) For $\left( p,u\right) \in [2,\infty ]\times [\frac{2n}{n+1},2]$ and

$$\begin{aligned} t=\frac{2pu}{pu-2u+4}; \end{aligned}$$

(iii) For $\left( p,u\right) \in [1,2]\times [1,\frac{2n}{n+1}]$ and

$$\begin{aligned} t=\frac{2np}{2p+np-2}; \end{aligned}$$

(iv) For $\left( p,u\right) \in [2,\infty ]\times [1,\frac{2n}{n+1}]$ and

$$\begin{aligned} t=\frac{2np}{np+2}. \end{aligned}$$

Proof

(i) We proceed as in the proof of Bayart et al. (2020, Theorem 2.3).

Note that

$$\begin{aligned} \widehat{T}\left( \widehat{A_{1}}\left( \ell _{u}^{w}(\ell _{\infty })\right) ,...,\widehat{A_{n}}\left( \ell _{u}^{w}(\ell _{\infty })\right) \right)&\in \ell _{2}\left( \ell _{1}\right) \\ \widehat{T}\left( \widehat{A_{1}}\left( \ell _{u}^{w}(\ell _{\infty })\right) ,...,\widehat{A_{n}}\left( \ell _{u}^{w}(\ell _{\infty })\right) \right)&\in \ell _{u}\left( \ell _{2}\right) . \end{aligned}$$

Let us consider $p=1.$ The operators $A_{1},...,A_{n}\in \mathcal {L}\left( ^{n}\ell _{\infty };\ell _{1}\right)$ are multiple $\left( 2,2\right)$-summing (see Araújo and Pellegrino 2017) and thus $T(A_{1},...,A_{m})$ is multiple $\left( 2,2\right)$-summing when $T\in \mathcal {L}\left( ^{m}\ell _{1};\ell _{1}\right)$ and, a fortiori, $T(A_{1},...,A_{m})$ is multiple $\left( 2,u\right)$-summing.

If $p=2,$ as $\frac{2n}{n+1}\le u,$ the operators $A_{1},...,A_{n} \in \mathcal {L}\left( ^{n}\ell _{\infty };\ell _{1}\right)$ are weakly multiple $\left( u,u\right)$-summing (see Araújo and Pellegrino 2017) and, since $u\le 2,$ it is well-known that $T\in \mathcal {L}\left( ^{m}\ell _{1};\ell _{2}\right)$ is multiple $\left( u,u\right)$-summing. Hence $T(A_{1},...,A_{m})$ is multiple $\left( u,u\right)$-summing.

Proceeding as in Bayart et al. (2020, Theorem 2.3) we have that

$$\begin{aligned} \widehat{T}\left( \widehat{A_{1}}\left( \ell _{u}^{w}(\ell _{\infty })\right) ,...,\widehat{A_{n}}\left( \ell _{u}^{w}(\ell _{\infty })\right) \right) \in \ell _{t}\left( \ell _{p}\right) \end{aligned}$$

for

$$\begin{aligned} \frac{1}{t}=\frac{\frac{2-p}{p}}{2}+\frac{1-\frac{2-p}{p}}{_{u}}. \end{aligned}$$

We thus have

$$\begin{aligned} t=\frac{2pu}{4p+2u-pu-4}. \end{aligned}$$

(ii) Note that, as in the first case,

$$\begin{aligned} \widehat{T}\left( \widehat{A_{1}}\left( \ell _{u}^{w}(\ell _{\infty })\right) ,...,\widehat{A_{n}}\left( \ell _{u}^{w}(\ell _{\infty })\right) \right)&\in \ell _{2}\left( \ell _{\infty }\right) \\ \widehat{T}\left( \widehat{A_{1}}\left( \ell _{u}^{w}(\ell _{\infty })\right) ,...,\widehat{A_{n}}\left( \ell _{u}^{w}(\ell _{\infty })\right) \right)&\in \ell _{u}\left( \ell _{2}\right) . \end{aligned}$$

Thus

$$\begin{aligned} \widehat{T}\left( \widehat{A_{1}}\left( \ell _{u}^{w}(\ell _{\infty })\right) ,...,\widehat{A_{n}}\left( \ell _{u}^{w}(\ell _{\infty })\right) \right) \in \ell _{t}\left( \ell _{p}\right) \end{aligned}$$

for

$$\begin{aligned} \frac{1}{t}=\frac{1-\frac{2}{p}}{2}+\frac{\frac{2}{p}}{_{u}}, \end{aligned}$$

and thus

$$\begin{aligned} t=\frac{2pu}{pu-2u+4}. \end{aligned}$$

(iii) If $p=1,$ $T(A_{1},...,A_{m})$ is multiple $\left( 2,u\right)$-summing (Botelho and Pellegrino 2009, Proposition 3.3). If $p=2,$ the operators $A_{1},...,A_{n}\in \mathcal {L}\left( ^{n}\ell _{\infty };\ell _{1}\right)$ are weakly multiple $\left( \frac{2n}{n+1},u\right)$-summing (see Araújo and Pellegrino 2017) and $T\in \mathcal {L}\left( ^{m}\ell _{1};\ell _{2}\right)$ is multiple $\left( \frac{2n}{n+1},\frac{2n}{n+1}\right)$-summing. Thus $T(A_{1},...,A_{m})$ is multiple $\left( \frac{2n}{n+1},u\right)$-summing.

Proceeding as in Bayart et al. (2020), for $T\in \mathcal {L}\left( ^{m}\ell _{1};\ell _{p}\right)$ we have that $T(A_{1},...,A_{m})$ is multiple $\left( t,u\right)$-summing for

$$\begin{aligned} t=\frac{2np}{2p+np-2}. \end{aligned}$$

(iv) If $p=\infty ,$ the operator $T(A_{1},...,A_{m})$ is multiple $\left( 2,2\right)$-summing, because $A_{1},...,A_{m}$ are multiple $\left( 2,2\right)$-summing (Botelho and Pellegrino 2009, Proposition 3.3). Thus $T(A_{1},...,A_{m})$ is multiple $\left( 2,u\right)$-summing

If $p=2,$ as in the previous case we know that $T(A_{1},...,A_{m})$ is multiple $\left( \frac{2n}{n+1},u\right)$-summing.

Proceeding as in Bayart et al. (2020), for $T\in \mathcal {L}\left( ^{m}\ell _{1};\ell _{p}\right)$ we have that $T(A_{1},...,A_{m})$ is multiple $\left( t,u\right)$-summing for

$$\begin{aligned} t=\frac{2np}{np+2}. \end{aligned}$$

$\square$

Our final result extends (2) of Theorem 1.2:

Theorem 5.2

Let $T\in \mathcal {L}\left( ^{m}\ell _{1};\ell _{p}\right)$ and $A_{k} \in \mathcal {L}\left( ^{n}\ell _{\infty };\ell _{1}\right)$ for all $k=1,...,m.$ Assume that $n\ge 2.$ If $s\ge 1,$ the composition $T\left( A_{1},...,A_{m}\right)$ is absolutely $\left( r;2\,s\right)$-summing for

$$\begin{aligned} r=\left\{ \begin{array}{c} \frac{2ps}{2mp-2s+2ps-mps}\text { if }1\le p\le 2\\ \frac{2ps}{2mp+2s-mps}\text { if }2\le p\le \infty . \end{array} \right. \end{aligned}$$

Proof

By Botelho et al. (2010, Theorem 3.15) we know that every operator in $\mathcal {L} \left( ^{n}\ell _{\infty };\mathbb {K}\right)$ is absolutely $\left( s;2\,s\right)$-summing for all $s\ge 1$ and hence every $A_{k}$ is weakly absolutely $\left( s;2s\right)$-summing.

Let us suppose $1\le p\le 2.$ Since every $T\in \mathcal {L}\left( ^{m} \ell _{1};\ell _{p}\right)$ is absolutely $\left( \frac{2p}{mp+2p-2};1\right)$-summing, by the Inclusion Theorem we conclude that every $T\in \mathcal {L}\left( ^{m}\ell _{1};\ell _{p}\right)$ is absolutely $\left( \frac{2ps}{2mp-2\,s+2ps-mps};s\right)$-summing. Thus, $T\left( A_{1},...,A_{m}\right)$ is absolutely $\left( \frac{2ps}{2mp-2\,s+2ps-mps};2\,s\right)$-summing.

If $2\le p\le \infty ,$ since every $T\in \mathcal {L}\left( ^{m}\ell _{1};\ell _{p}\right)$ is absolutely $\left( \frac{2p}{mp+2};1\right)$-summing, by the Inclusion Theorem we conclude that every $T\in \mathcal {L}\left( ^{m}\ell _{1};\ell _{p}\right)$ is absolutely $\left( \frac{2ps}{2mp-2\,s+mps};s\right)$-summing. Thus, $T\left( A_{1},...,A_{m}\right)$ is absolutely $\left( \frac{2ps}{2mp-2\,s+mps};2\,s\right)$-summing. $\square$