Pietsch-Maurey-Rosenthal factorization of summing multilinear operators

TL;DR

This paper introduces a new class of summing multilinear operators on Banach lattices, establishing a factorization theorem under weaker convexity conditions and extending operator classes with applications to $q$-concavity.

Contribution

It develops a novel Pietsch-Maurey-Rosenthal type factorization theorem for multilinear operators with relaxed convexity assumptions, extending the theory to special Banach lattices and tensor products.

Findings

Established a weaker convexity-based factorization theorem.

Extended multilinear operators to lattices with order continuity and $p$-convexity.

Provided new factorization results for $q$-dominated operators.

Abstract

The main purpose of this paper is the study of a~new class of summing multilinear operators acting from the product of Banach lattices with some nontrivial lattice convexity. A~mixed Pietsch-Maurey-Rosenthal type factorization theorem for these operators is proved under weaker convexity requirements than the ones that are needed in the Maurey-Rosenthal factorization through products of $L^q$-spaces. A~by-product of our factorization is an extension of multilinear operators defined by a~$q$-concavity type property to a~product of special Banach function lattices which inherit some lattice-geometric properties of the domain spaces, as order continuity and $p$-convexity. Factorization through Fremlin's tensor products is also analyzed. Applications are presented to study a~special class of linear operators between Banach function lattices that can be characterized by a strong version of $q$-concavity. This class contains $q$-dominated operators, and so the obtained results provide a~new factorization theorem for operators from this class.

Equations

$$\sup_{x^{*}\in B_{X^{*}}}\Big{(}\sum_{k=1}^{n}|\langle x_{k},x^{*}\rangle|^{q}\Big{)}^{1/q}\leq\sup_{x^{*}\in B_{X^{*}}}\sup_{(\beta_{k})\in B_{\ell^{r}}}\Big{(}\sum_{k=1}^{n}|\beta_{k}\langle x_{k},x^{*}\rangle|^{p}\Big{)}^{1/p}$$

$$\leq\sup_{(\beta_{k})\in B_{\ell^{r}}}\Big{\|}\Big{(}\sum_{k=1}^{n}|\beta_{k}x_{k}|^{p}\Big{)}^{1/p}\Big{\|}_{X}\leq\Big{\|}\Big{(}\sum_{k=1}^{n}|x_{k}|^{q}\Big{)}^{1/q}\,\Big{\|}_{X}$$

$$\text{p-summing }\Rightarrow \text{p-strongly q-concave}\Rightarrow \text{q-concave}$$

$$⟨x,x^{\ast }⟩={\int }_{\Omega }x{x}^{\prime }d\mu ,x\in X.$$

$$\|x\|_{p}=\Big{(}\sum_{k=1}^{n}|t_{k}|^{p}\Big{)}^{1/p},$$

$$\Vert x{\Vert }_{\infty }=\underset{1\le k\le n}{\max }|t_k|.$$

$$\Big{(}\sum_{k=1}^{n}\|Tx_{k}\|_{Y}^{q}\Big{)}^{1/q}\leq C_{(q)}\Big{\|}\Big{(}\sum_{k=1}^{n}|x_{k}|^{q}\Big{)}^{1/q}\,\Big{\|}_{X}$$

$$\Big{\|}\Big{(}\sum_{k=1}^{n}|x_{k}|^{p}\Big{)}^{1/p}\,\Big{\|}_{X}\leq C^{(p)}\Big{(}\sum_{k=1}^{n}\|x_{k}\|_{X}^{p}\Big{)}^{1/p},$$

$$\Big{(}\sum_{k=1}^{n}\|x_{k}\|_{X}^{q}\Big{)}^{1/q}\leq C_{(q)}\Big{\|}\Big{(}\sum_{k=1}^{n}|x_{k}|^{q}\Big{)}^{1/q}\Big{\|}_{X}$$

$$B:X_1\times \cdot \cdot \cdot \times X_n\to Y$$

$$\|x\|_{|\pi|}=\inf\bigg{\{}\sum_{i=1}^{n}\sum_{j=1}^{m}\|x_{i,j}\|_{X_{j}};\,x_{i,j}\in X_{j}^{+},\,\,|x|\leq\sum_{i=1}^{n}x_{i,1}\otimes\cdot\cdot\cdot\otimes x_{i,m}\bigg{\}}.$$

$$\{x_1\odot \cdot \cdot \cdot \odot x_n;x_j\in X_j,1\le j\le n\}$$

$$(x_1\odot \cdot \cdot \cdot \odot x_n)({\omega }_1,\dots ,{\omega }_n):=x_1({\omega }_1)\cdot \cdot \cdot x_n({\omega }_n)$$

$$\Big{(}\sum_{k=1}^{n}\|Tx_{k}\|_{E}^{q}\Big{)}^{1/q}\leq C\sup_{(\beta_{k})\in B_{r}^{n}}\Big{\|}\Big{(}\sum_{k=1}^{n}|\beta_{k}x_{k}|^{p}\Big{)}^{1/p}\,\Big{\|}_{X}$$

$$\sup_{(\beta_{k})\in B_{r}^{n}}\Big{\|}\Big{(}\sum_{k=1}^{n}|\beta_{k}x_{k}|^{p}\Big{)}^{1/p}\,\Big{\|}_{X}\leq\Big{\|}\Big{(}\sum_{k=1}^{n}|x_{k}|^{q}\Big{)}^{1/q}\,\Big{\|}_{X},$$

$$\displaystyle\Big{(}\sum_{k=1}^{n}\big{\|}x_{k}\big{\|}^{q}_{X}\Big{)}^{1/q}$$

$$\displaystyle\leq\sup_{(\beta_{k})\in B_{r}^{n}}\Big{\|}\Big{(}\sum_{k=1}^{n}|\beta_{k}\,x_{k}|^{p}\Big{)}^{1/p}\Big{\|}^{p/q}\Big{(}\sum_{k=1}^{n}\|x_{k}\|^{q}\Big{)}^{p/(rq)}.$$

$$\Big{(}\sum_{k=1}^{n}\big{\|}x_{k}\big{\|}^{q}_{X}\Big{)}^{p/q^{2}}=\Big{(}\sum_{k=1}^{n}\big{\|}x_{k}\big{\|}^{q}_{X}\Big{)}^{1/q-p/(rq)}\leq\sup_{(\beta_{k})\in B_{r}^{n}}\Big{\|}\Big{(}\sum_{k=1}^{n}|\beta_{k}\,x_{k}|^{p}\Big{)}^{1/p}\Big{\|}^{p/q},$$

$$Y:=\Big{\{}f\in L^{0}(\mu);\,(f\xi_{A_{k}})_{k=1}^{\infty}\in\big{(}\oplus L^{p}(\mu|_{A_{k}}\big{)}\big{)}_{\ell_{q}}\Big{\}}$$

$$\|f\|_{Y}:=\|(f\xi_{A_{k}})\|_{(\oplus L^{p}(\mu|_{A_{k}}))_{\ell_{q}}}=\Big{(}\sum_{k=1}^{\infty}\Big{(}\int_{A_{k}}|f|^{p}\,d\mu\Big{)}^{q/p}\Big{)}^{1/q},\quad f\in Y.$$

$$\Vert {\chi }_{A_k}{\Vert }_{(Y_p{)}^{\prime }}=1,k\in \mathbb{N}.$$

$$x_k^{\ast }(f)={\int }_{A_k}fd\mu ,f\in Y_p.$$

$$R(f)=\Big{(}\frac{1}{2^{k/q}}\int_{A_{k}}f\,d\mu\Big{)}_{k},\,\quad\,f\in Y$$

$$\|f\|_{p,q;\nu}:=\Big{(}\int_{B_{(Y_{p})^{*}}^{+}}\big{|}\langle|f|^{p},y^{*}\rangle\big{|}^{q/p}\,d\nu(y^{*})\Big{)}^{1/q}<\infty.$$

$$\|f\|_{p,q,\nu}=\Big{(}\sum_{k=1}^{\infty}\frac{1}{2^{k}}\Big{(}\int_{A_{k}}|f|^{p}\,d\mu\Big{)}^{q/p}\Big{)}^{1/q},\quad f\in Y.$$

$$\|R(f)\|_{\ell^{q}}=\Big{\|}\Big{(}\frac{1}{2^{k/q}}\int_{A_{k}}f\,d\mu\Big{)}_{k}\Big{\|}_{\ell^{q}}\leq\Big{(}\sum_{k=1}^{\infty}\frac{1}{2^{k}}\Big{(}\int_{A_{k}}|f|^{p}\,d\mu\Big{)}^{q/p}\,\mu(A_{k})^{q/p^{\prime}}\Big{)}^{1/q}=\|f\|_{Y}.$$

$$\sum _{k=1}^n\Vert T(f_k){\Vert }_{{\ell }^q}^q$$

$$\displaystyle\leq\sup_{y^{*}\in{B_{(Y_{p})^{*}}^{+}}}\sum_{k=1}^{n}\big{|}\langle|f_{k}|^{p},y^{*}\rangle\big{|}^{q/p}$$

$$\displaystyle=\sup_{(\beta_{k})\in B_{r}^{n}}\Big{\|}\Big{(}\sum_{k=1}^{n}|\beta_{k}f_{k}|^{p}\Big{)}^{1/p}\,\Big{\|}_{Y}^{q},$$

$$\Big{(}\sum_{k=1}^{n}\|T(x^{1}_{k},\ldots,x^{m}_{k})\|_{Y}^{q}\Big{)}^{1/q}\leq C\prod_{j=1}^{m}\sup_{(\beta^{j}_{k})\in B_{\ell^{r_{j}}}}\Big{\|}\Big{(}\sum_{k=1}^{n}|\beta^{j}_{k}x_{k}^{j}|^{p_{j}}\Big{)}^{1/{p_{j}}}\,\Big{\|}_{X_{j}}.$$

$$\sup_{(\beta_{k})\in B_{r}^{n}}\Big{\|}\Big{(}\sum_{k=1}^{n}|\beta_{k}x_{k}|^{p}\Big{)}^{1/p}\,\Big{\|}_{E}=\sup_{x^{*}\in B_{{(E_{p})}^{*}}^{+}}\Big{(}\sum_{k=1}^{n}\langle|x_{k}|^{p},x^{*}\rangle^{q/p}\,\Big{)}^{1/q}$$

$$\displaystyle\sup_{(\beta_{k})\in B_{r}^{n}}\Big{\|}\Big{(}\sum_{k=1}^{n}|\beta_{k}x_{k}|^{p}\Big{)}^{1/p}\,\Big{\|}_{E}^{p}$$

Full text

The first named author was supported by National Science Center, Poland, project no. 2015/17/B/ST1/00064. The second named author was supported by the Ministerio de Economía y Competitividad (Spain) under project MTM2016-77054-C2-1-P

1. Introduction

Domination inequalities for multilinear operators are of interest in applications to factorization of various types of operators (see [2, 4, 5, 11]). In the case of operators defined on products of Banach lattices, these dominations are deeply related to Banach lattice geometric notions, as $q$-convexity or $q$-concavity. It should be pointed out that domination does not lead in general to a nice factorization in the multilinear case. However, in some situations the relation between domination and factorization works as in the linear case. We recall that the famous Pietsch’s factorization theorem is given by a domination result associated to summability properties also in the multilinear case, in which $L^{p}$-spaces are involved. We also point out that under the assumption of some variants of convexity properties of the involved lattices, the Maurey-Rosenthal multilinear theorem allows to link a $q$-concavity type domination inequality with a  factorization/extension of the multilinear operator.

In this paper we are concerned with the analysis of some new lattice geometric properties that we call $p$-strong $q$-concavity (see Section 2). The motivation for this is to prove domination/factorization characterizations for multilinear operators from the product of Banach lattices that satisfy a certain vector norm inequality. Recall that in the case of linear operators acting in Banach lattices, if an operator $T\colon X\to Y$ is $q$-summing then it is also $q$-concave. This is the main lattice-type property that is normally used when a summability property for an operator among Banach lattices is considered. Indeed, this implies —using the Maurey-Rosenthal factorization and under the assumption of $q$-convexity of the domain lattice—, that the operator factors through an $L^{q}$-space. Now take an index $1\leq p<q$ and write $r$ for the real number satisfying that $1/q+1/r=1/p$. Then we can easily see that

[TABLE]

[TABLE]

for every finite sequence $(x_{k})_{k=1}^{n}$ in the Banach lattice $X$. A look to the definitions (see Section 2) shows that the implications

[TABLE]

hold for operators acting in Banach lattices.

The main advantage in using this new lattice property —$p$-strong $q$-concavity— is that the requirement on the $q$-convexity of the original space can be relaxed and still obtain an standard factorization theorem. Indeed, Maurey-Rosenthal theorem implies that $q$-summability of the operator plus $q$-convexity of the domain space allows a strong factorization through an $L^{q}$-space. In the preliminary paper [6], it is shown that for $1\leq p<q$, every $p$-strongly $q$-concave operator —and so every $q$-summing operator— acting in a $p$-convex space factors strongly through a Banach function lattice space of the new class $S_{X_{p}}^{q}(\xi)$, that admit an easy description and whose lattice properties are naturally associated to $p$-strongly $q$-concave operators. The aim of this paper is to draw the complete picture for this class of lattice dominations/factorizations of operators by analyzing their multilinear variants. By applying them to the linear case we will show new factorization theorems for the classical $q$-dominated (linear) operators among Banach lattices.

In Section 2 we sketch some background from the theory of general Banach lattices, Fremlin’s tensor product of Banach lattices, and also summing operators. We also provide examples which motivates our study.

In Section 3 we study a new class of summing multilinear operators acting from the product of Banach lattices nontrivial lattice convexity. We prove an extension theorem for these operators acting in products of Banach lattices with some nontrivial convexity. We give a mixed Pietsch-Maurey-Rosenthal type factorization theorem for the multilinear case. We show that a particular class of multilinear operators defined by a $q$-concavity type property can be extended to a product of Banach lattice satisfying some lattice-geometric properties, as order continuity and $p$-convexity.

In section 4 we show the relation among summability of multilinear operators from suitable products of Banach function lattices and Fremlin tensor product. Factorization theorems are proved.

In Section 5 we center our attention on the factorization of the linear dominated operators associated to a new geometric definition introduced in the paper. Indeed, the lattice-geometric domination inequality appearing in the definition of the $p$-strongly $q$-concave operators motivates the definition of the dual notion.

2. Notation and background material

The purpose of this section is to sketch some background from the theory of general Banach lattices and summing operators. We shall also take the opportunity to establish some notation. For a given dual pair $\langle X,Y,(\cdot,\cdot)\rangle$ the evaluation map $(x,y)$ is denoted by $\langle x,y\rangle$ for all $x\in X$, $y\in Y$.

For notations concerning vector lattices we follow [1, 10], and tensor products of Banach lattices we follow [8, 9]. Let $(E,\leq)$ be a vector lattice (called also a Riesz space). If $A\subset E$, then $A^{+}:=\{x\in A;\,x\geq 0\}$. Let us recall, that if $A$ is a subset of a Banach lattice $E$, then a functional $x^{*}\in E^{*}$ satisfying the condition $\langle x,x^{*}\rangle>0$ whenever $0<x\in A$ is called strictly positive on $A$. It is known that strictly positive functionals on $E$ exist when $E$ has the order continuous norm and a weak unit (see [1, Theorem 12.43] or [10, Proposition 1.b.15]).

We also recall that a Banach lattice possessing order continuous norm and a weak unit is order isomorphic to a Banach function lattice on a finite measure space.

We recall that if $(\Omega,\Sigma,\mu)$ is a $\sigma$-finite measure space $L^{0}(\mu)$ denotes the space of $\mu$-a.e equal equivalence classes of functions. A Banach function lattice is a Banach space $X\subset L^{0}(\mu)$ with norm $\|\cdot\|_{X}$ such that if $f\in L^{0}(\mu)$, $g\in X$ and $|f|\leq|g|$ $\mu$-a.e. then $f\in X$ and $\|f\|_{X}\leq\|g\|_{X}$. Every Banach function space is a Banach lattice with the pointwise $\mu$-a.e. order. The Köthe dual $X^{\prime}$ of a Banach function space $X$ is the subspace of the dual space $X^{*}$ of the functionals that has an integral representation, that is, $x^{*}\in X^{*}$ for which there exists $x^{\prime}\in L^{0}(\mu)$ such that

[TABLE]

In what follows we consider a dual pair $\langle X,X^{\prime}\rangle$ with the evaluation map $(x,x^{\prime})\mapsto\langle x,x^{\prime}\rangle:=\int_{\Omega}xx^{\prime}\,d\mu$ for all $(x,x^{\prime})\in X\times X^{\prime}$.

A Banach function lattice is said to have the Fatou property if for every sequence $(f_{n})$ in $X$ such that $0\leq f_{n}\uparrow f$ a.e. and $\sup_{n}\|f_{n}\|_{X}<\infty$, it follows that $f\in X$ and $\|f_{n}\|_{X}\uparrow\|f\|_{X}$. This is equivalent to the fact that $X=X^{\prime\prime}$ with equality of norms.

We use $\mathcal{M}(K)$ to denote the space of regular Borel probability spaces on a compact Hausdorff space. We recall that the weak*∗* topology on the dual $E^{*}$ a Banach space $E$ is the topology pointwise convergence. Then the unit ball is compact, by the Banach-Alaoglu theorem.

The normed space $(\mathbb{R}^{n},\|\cdot\|_{p})$ is denoted by $\ell_{p}^{n}$ for $1\leq p\leq\infty$, where as usual for any $x=(t_{1},\ldots,t_{n})\in\mathbb{R}^{n}$,

[TABLE]

and

[TABLE]

In what follows the unit ball $B_{\ell_{p}^{n}}$ is denoted by $B_{p}^{n}$ for short.

Given a Banach lattice $X$, a Banach space $Y$, and numbers $1\leq p$, $q<\infty$. An operator $T\colon X\to Y$ is said to be $q$-concave if there exists $C_{(q)}>0$ such that

[TABLE]

for every choice of elements $x_{1},...,x_{n}$ in $X$. The infimum of the values $C_{(q)}$ for which the inequality above is satisfied will be denoted by $M_{(q)}(T)$.

A Banach lattice $X$ is said to be $p$-convex, $1\leq p<\infty$, respectively $q$-concave, $1\leq q<\infty$, if there are positive constants $C^{(p)}$ and $C_{(q)}$ such that

[TABLE]

respectively,

[TABLE]

for every finite sequence $(x_{k})_{k=1}^{n}$ in $X$. The least such $C^{(p)}$ (respectively, $C_{(q)})$) is denoted by $M^{(p)}(X)$ (respectively, $M_{(q)}(X)$). It is well-known that a $p$-convex Banach ($q$-concave) lattice can always be renormed with a lattice norm in such a way that $M^{(p)}(X)=1$ ($M_{(q)}(X)=1$). Henceforth, throughout the paper we shall always assume that $M^{(p)}(X)=1$. We refer to [10, Ch. 1.d] or [12, Ch.2] for information about the classical geometric concepts of (lattice) $p$-convexity and $q$-concavity.

If a Banach lattice $X$ is $p$-convex with $1\leq p<\infty$, then its $p$-concavification is a Banach lattice $X_{p}$ (see [10, p. 54] for details). Note that in the case of a Banach function lattice $X$ on $(\Omega,\Sigma,\mu)$, $X_{p}$ is identified with the space of all $f\in L^{0}(\mu)$ so that $|f|^{1/p}\in X$ and equipped with the norm $\|f\|_{X_{p}}=\||f|^{1/p}\|_{X}^{p}$.

We will use Fremlin tensor products of Banach lattices. Let $X_{1},\ldots,X_{n}$ and $Y$ be Archimedean Riesz spaces. An $n$-linear map

[TABLE]

is called positive if $B(x_{1},\ldots,x_{n})\in Y^{+}$ whenever $x_{k}\in X_{k}^{+}$, $1\leq k\leq n$; it is called a Riesz $n$-morphism if $B(|x_{1}|,\ldots,|x_{n}|)=|B(x_{1},\ldots,x_{n})|$ for all $x_{k}\in X_{k}$, $1\leq k\leq n$.

Following [9] (see also [13]) one can construct an Archimedean Riesz space $X_{1}\bar{\otimes}\cdot\cdot\cdot\bar{\otimes}X_{n}$ and a Riesz morphism (called the Fremlin map) $\bigotimes$.

We recall fundamental properties of this construction;

(a) $X_{1}\otimes\cdot\cdot\cdot\otimes X_{n}$ is dense in $X_{1}\bar{\otimes}\cdot\cdot\cdot\bar{\otimes}X_{n}$, i.e., for any $u\in X_{1}\bar{\otimes}\cdot\cdot\cdot\bar{\otimes}X_{n}$ there exist $x_{k}\in X_{k}^{+}$ ($1\leq k\leq n$) such that for all $\varepsilon>0$ there is a $v\in X_{1}\bar{\otimes}\cdot\cdot\cdot\bar{\otimes}X_{n}$ with $|u-v|\leq\varepsilon(x_{1}\otimes\cdot\cdot\cdot\otimes x_{k}$).

(b) If $u\in X_{1}\bar{\otimes}\cdot\cdot\cdot\bar{\otimes}X_{n}$, then there exist $x_{k}\in X_{k}^{+}$ ($1\leq k\leq n$) such that $|u|\leq x_{1}\otimes\cdot\cdot\cdot\otimes x_{n}$.

If $X_{1},\ldots,X_{n}$ are Banach lattices, then we can define the positive-projective norm $\|\cdot\|_{|\pi|}$ on $X_{1}\bar{\otimes}\cdot\cdot\cdot\bar{\otimes}X_{n}$ by

[TABLE]

We define the Fremlin tensor product to be the Banach lattice $X_{1}\otimes_{|\pi|}\cdot\cdot\cdot\otimes_{|\pi|}X_{m}$ given by the completion of $X_{1}\bar{\otimes}\cdot\cdot\cdot\bar{\otimes}X_{n}$ with respect to $\|\cdot\|_{|\pi|}$.

We note that in the case of Banach function lattices $X_{1},\ldots,X_{m}$ on measure spaces $(\Omega_{1},\Sigma_{1},\mu_{1}),\ldots,(\Omega_{n},\Sigma_{n},\mu_{n})$, respectively, we can define the Riesz space $X_{1}\bar{\otimes}\ldots\bar{\otimes}X_{n}$ generated by

[TABLE]

in $L^{0}(\mu_{1}\times\cdot\cdot\cdot\times\mu_{n})$, where

[TABLE]

for all $(x_{1},\ldots,x_{n})\in X_{1}\times\cdots\times X_{n}$ and $(\omega_{1},\ldots,\omega_{n})\in\Omega_{1}\times\cdots\times\Omega_{n}$.

Let us introduce now the notion that motivates the multilinear definition given in this paper. Let $1\leq p\leq q<\infty$. Consider $T\colon X\to E$ a linear operator from a Banach lattice $X$ into a Banach space $E$. We will say that $T$ is $p$-strongly $q$-concave if there exists $C>0$ such that

[TABLE]

for every finite sequence $(x_{k})_{k=1}^{n}$ in $X$, where $1<r\leq\infty$ is such that $1/r=1/p-1/q$.

We present some examples showing the nature of linear $p$-strongly $q$-concave operators. The reader can find more examples in [6].

Fix $1\leq p<q<\infty$ and let $1/r=1/p-1/q$. Clearly that $r/p$ and $q/p$ are conjugate exponents, that is, $1/(r/p)+1/(q/p)=1$. Since

[TABLE]

it follows that a $p$-strongly $q$-concave operator is always $q$-concave.

Now observe that if $1\leq p\leq q<\infty$ and $X$ is a $p$-concave Banach lattice, then the identity map $\iota\colon X\to X$ is $p$-strongly $q$-concave. To see this we fix a finite sequence $(x_{k})_{k=1}^{n}$ in a $p$-concave Banach function lattice $X$. Without loss of generality we may assume that $M_{(p)}(X)=1$. Let $\alpha_{k}=\|x_{k}\|^{q/r}/(\sum_{k=1}^{n}\|x_{k}\|^{q})^{1/r}$ for each $1\leq k\leq n$. Since $q=(pq/r)+p$, $\sum_{k=1}^{n}\alpha_{k}^{r}=1$ and so

[TABLE]

Hence

[TABLE]

and this gives the above mentioned statement.

We note that the above observation shows that all $L^{p}$-spaces are $p$-strongly $q$-concave, thus an operator acting in $L^{p}$-space is so. However, there are of course other situations.

We show an example of a $p$-strongly $q$-concave operator acting in a $p$-convex Banach function lattice that is not an $L^{p}$ space. To see this we need to define special spaces and show some preliminary results.

Let $1<p<q<\infty$ and let $p^{\prime}$ be the conjugate number given by $1/p^{\prime}=1-1/p$. Assume that $(\Omega,\Sigma,\mu)$ is a $\sigma$-finite measure space such that there is a measurable partition $(A_{k})_{k=1}^{\infty}$ of $\Omega$ with $\mu(A_{k})=1$ for each $k$. Consider the sequence of characteristic functions $(\chi_{A_{k}})_{k}$ and define an order continuous Banach function lattice

[TABLE]

equipped with the norm

[TABLE]

It is easy to check that

[TABLE]

This implies that for each $k\in\mathbb{N}$ we have a functional $x_{k}^{*}\in B_{(Y_{p})^{*}}$ given by

[TABLE]

Now observe that the linear map $R$ defined by

[TABLE]

is bounded from $Y$ to $\ell_{q}$.

For the Borel regular measure on $B_{(Y_{p})^{*}}$ given by $\nu=\sum_{k=1}^{\infty}2^{-k}\,\delta_{x_{k}^{*}}$, we denote by $S^{q}_{Y_{p}}(\nu)$ the space of all $f\in L^{0}(\mu)$ such that

[TABLE]

A direct computation shows that

[TABLE]

Therefore, $\|\cdot\|_{p,q,\nu}\leq\|\cdot\|_{Y}$ and the operator $R$ can be extended to the space $S^{q}_{Y_{p}}(\nu)$, since

[TABLE]

Now observe that for any $f_{1},\ldots,f_{n}\in Y$ we have

[TABLE]

where we have used Lemma 3.3 (see Section 3 below).

3. Summing multilinear operators on products of Banach lattices

In what follows we assume that the $m$-tuples $(p_{1},\ldots,p_{m})$, $(q_{1},\ldots,q_{m})$ and $(r_{1},\ldots,r_{m})$ of real numbers satisfy $1\leq p_{j}\leq q_{j}$, $1/r_{j}=1/p_{j}-1/q_{j}$ for each $1\leq j\leq m$. We also define $q$ by $1/q:=1/q_{1}+\ldots+1/q_{m}$.

A multilinear operator $T\colon X_{1}\times\cdots\times X_{m}\to Y$ —where $X_{j}$ is a Banach lattice and $Y$ is a Banach space—, is said to be $(p_{1},\ldots,p_{m})$-strongly $(q_{1},\ldots,q_{m})$-concave whenever there exists a constant $C>0$ such that for any finite sequence $(x^{j}_{k})_{k=1}^{n}$ in $X_{j}$, $1\leq j\leq m$, we have that

[TABLE]

We will use a lemma which is a general version of Lemma 2 in [6].

Lemma 3.1**.**

Let $1\leq p<\infty$ and let $E$ be a $p$-convex Banach lattice. Then

[TABLE]

for every choice of $(x_{k})_{k=1}^{n}$ in $E$ where $1/r=1/p-1/q$.

Proof.

Fix a finite set $\{x_{1},\ldots,x_{n}\}$ of $E$, and note that $(\ell^{q/p})^{*}=\ell^{r/p}$, by $r/p+q/p=1$. Then we have that

[TABLE]

∎

Now we state our first main theorem.

Theorem 3.2**.**

Let $X_{j}$ be $p_{j}$-convex Banach lattices and let $1\leq q_{j}<\infty$ for each $1\leq j\leq m$. If $1/q=1/q_{1}+\ldots+1/q_{m}$, then the following are equivalent statements about a multilinear operator $T$ from $X_{1}\times\cdots\times X_{m}$ to a Banach space $E$.

• (i)

$T$* is $(p_{1},\ldots,p_{m})$-strongly $(q_{1},\ldots,q_{m})$-concave.*

• (ii)

There is a constant $C>0$ such that for every $(x_{1},\ldots,x_{m})\in X_{1}\times\cdot\cdot\cdot\times X_{m}$,

[TABLE]

where $\nu_{j}$ is a probability Borel measure on the weak*∗** compact set $B_{{((X_{j})_{p})}^{*}}^{+}$ for each $1\leq j\leq m$.*

Proof.

(i) $\Rightarrow$ (ii). Fix finite sequences $(x^{j}_{i})_{i=1}^{n}$ in $X_{j}$ for each $1\leq j\leq m$.

First consider Lemma 3.1 (with $m=1$ and $E=X_{j}$ for each $j$) for all the factors in the product of the left hand side of the inequality that provides the definition of $(p_{1},...,p_{m})$-strongly $(q_{1},...,q_{m})$-concave $m$-linear operator. We obtain that the following inequality is equivalent to the one in this definition

[TABLE]

From this on, the proof uses some methods from [3, 11]. We only sketch the main arguments for the convenience of the reader; using Young’s inequality, we obtain that the inequality above implies

[TABLE]

We now define a convex set of continuous real functions

[TABLE]

each one associated to each finite sets of finite sequences as the ones at the beginning of the proof, and given by the formula

[TABLE]

Note that $\mathcal{M}(B_{{((X_{j})_{p_{j}})}^{*}}^{+})$ is a compact set with the product topology defined by means of the weak*∗* topology of the dual of each Banach space ${(X_{j})}_{p_{j}}$ for each $1\leq j\leq m$. Recall that these spaces are Banach as a consequence of the requirement that each of them is $p_{i}$-convex. The functions are continuous with respect to the product topology and satisfy all the properties needed for applying Ky Fan’s Lemma (see, e.g., [7, Lemma 9.10]). This gives an element

[TABLE]

satisfying

[TABLE]

Now, the multilinearity of $T$ allows to use a direct argument for choosing the right constants for getting from this sum domination the product domination that is written in (ii) (see [3, Theorem 1] for the details).

To conclude it is enough to observe that the converse inequality is obvious by using Lemma 3.1. ∎

The next result deals with factorization of $(p_{1},\ldots,p_{m})$-strongly $(q_{1},\ldots,q_{m})$-concave multilinear operators. Motivated by the above result we define Banach lattices which are connected with the obtained characterization of these operators.

Let $1\leq p\leq q<\infty$. For a given $p$-convex Banach function lattice $X$ and a regular Borel probability measure $\nu$ on $B_{(X_{p})^{*}}$ equipped with the weak*∗*-topology we define on $X$ a functional by

[TABLE]

We put $F_{X_{p}}^{q}(\nu):=(X,\|\cdot\|_{p,q,\nu})$. Clearly that $\rho(\cdot):=\|\cdot\|_{p,q,\nu}$ defines a lattice seminorm on $X$. If $N$ is the null ideal of $\rho$, i.e., $N=\{x\in X;\,\rho(x)=0\}$, then $X/N$ is a normed lattice (under the natural order) equipped with the norm

[TABLE]

The norm completion $\widetilde{F}_{X_{p}}^{q}(\nu)$ of $X/N$ with respect to the above lattice norm is a Banach lattice. Note that $\|x\|_{p,q,\nu}\leq\|x\|_{X}$ for all $x\in X$ implies that the map $i_{X}$ defined by

[TABLE]

is bounded from $X$ to $\widetilde{F}_{X_{p}}^{q}(\nu)$.

We have the following useful lemma.

Lemma 3.3**.**

Let $1\leq p\leq q<\infty$ and let $X$ be a $p$-convex Banach lattice.

• (i)

If there exists a strictly positive functional on $X_{p}$, then for every $\nu\in\mathcal{M}(B_{(X_{p})^{*}}^{+})$ there exists $\widetilde{\nu}\in\mathcal{M}(B_{(X_{p})^{*}}^{+})$ such that $F_{X_{p}}(\widetilde{\nu})=(X,\|\cdot\|_{p,q,\widetilde{\nu}})$ is a normed lattice such that

[TABLE]

• (ii)

If $X$ is an order continuous Banach function lattice on $(\Omega,\Sigma,\mu)$, then for every $\nu\in\mathcal{M}(B_{(X_{p})^{*}}^{+})$ there exists a probability Borel measure $\xi\in\mathcal{M}(B_{(X_{p})^{{}^{\prime}}}^{+})$ such that the completion of $F_{X_{p}}(\widetilde{\nu})$ —for $\widetilde{\nu}$ as in (ii)— is an order continuous Banach function lattice $(S_{X_{p}}^{q}(\xi),\|\cdot\|)$ on $(\Omega,\Sigma,\mu)$ given by

[TABLE]

and satisfying

[TABLE]

Proof.

(i). Let $x^{*}\in(X_{p})^{*}$ be a norm one strictly positive functional on $X_{p}$ and let $\delta_{x^{*}}\in\mathcal{M}(B_{(X_{p})^{*}}^{+})$ be the associated Dirac measure. For a given $\nu\in\mathcal{M}(B_{(X_{p})^{*}}^{+})$, we define $\widetilde{\nu}:=1/2(\nu+\delta_{x^{*}})\in\mathcal{M}(B_{(X_{p})^{*}}^{+})$. It is obvious that $\widetilde{\nu}$ satisfies the required properties.

(ii). Our hypothesis that $X$ (and so $X_{p}$) is order continuous implies that for every $x^{*}\in(X_{p})^{*}$ there exists unique $h=h_{x^{*}}\in(X_{p})^{\prime}$ such that

[TABLE]

with $\|x^{*}\|_{(X_{p})^{*}}=\|h\|_{(X_{p})^{\prime}}$, and moreover the map $(X_{p})^{*}\ni x^{*}\mapsto h_{x^{*}}$ is an order isometrical isomorphism. We denote the restriction of this map to $B_{(X_{p})^{*}}$ by $\varphi$. Clearly, $\varphi$ is a topological homeomorphism of $B_{(X_{p})^{*}}$ equipped with the weak*∗* topology onto $B_{(X_{p})^{\prime}}$ equipped with the pointwise topology induced by $\sigma(X^{\prime},X)$.

For a fixed $\nu\in\mathcal{M}(B_{(X_{p})^{*}}^{+})$, we define $\nu_{\varphi}\in\mathcal{M}(B_{(X_{p})^{\prime}}^{+})$ by $\nu_{\varphi}(A):=\nu(\varphi^{-1}(A))$ for any Borel subset of $B_{(X_{p})^{\prime}}^{+}$. Then for every $x\in X$, we get that

[TABLE]

Since $(X_{p})^{\prime}$ is a Banach function lattice on $(\Omega,\Sigma,\mu)$ there exists $h\in B_{(X_{p})^{\prime}}$ with $h>0$ on $\Omega$. Then $\xi:=1/2(\nu_{\varphi}+\delta_{h})\in\mathcal{M}(B_{(X_{p})^{\prime}})$, where $\delta_{h}$ is a Dirac measure generated by $h$.

Combining the above formula with [6, Proposition 1], we conclude that

[TABLE]

is the desired Banach function lattice. ∎

We are now ready to state the following factorization theorem.

Theorem 3.4**.**

Let $X_{j}$ be $p_{j}$-convex Banach function lattices and let $1\leq q_{j}<\infty$ for each $1\leq j\leq m$. If $1/q=1/q_{1}+\ldots+1/q_{m}$, then the following are equivalent statements about a multilinear operator $T$ from $X_{1}\times\cdots\times X_{m}$ to a Banach space $Y$.

• (i)

$T$* is $(p_{1},\ldots,p_{m})$-strongly $(q_{1},\ldots,q_{m})$-concave.*

• (ii)

There are probability Borel measures $\nu_{j}$ in $\mathcal{M}(B_{((X_{j})_{p_{j}})^{*}})$ for each $1\leq j\leq m$, and a multilinear operator $S$ such that $T$ factors as

[TABLE]

where $\iota_{j}=\iota_{X_{j}}$ for each $1\leq j\leq m$.

Proof.

(i) $\Rightarrow$ (ii). From Theorem 3.2, it follows that there is a constant $C>0$ such that for every $(x_{1},\ldots,x_{m})\in X_{1}\times\cdot\cdot\cdot\times X_{m}$,

[TABLE]

where $\nu_{j}$ is a probability Borel measure on the weak*∗*- compact set $B_{{((X_{j})_{p})}^{*}}^{+}$ for each $1\leq j\leq m$. This implies that

[TABLE]

holds for all $(x_{1},\ldots,x_{m})\in X_{1}\times\cdot\cdot\cdot\times X_{m}$ with $\rho_{j}(\cdot)=\|\cdot\|_{P_{j},q_{j},\nu_{j}}$ for each $1\leq j\leq m$. In particular, this implies that the formula

[TABLE]

defines a bounded multilinear operator from $X_{1}/N_{1}\times\cdot\cdot\cdot\times X_{m}/N_{m}$ to $Y$, where $N_{j}=\{x\in X_{j};\,\rho_{j}(x)=0\}$ for each $1\leq j\leq m$. Denote by $S$ the unique multilinear continuous extension of $T_{0}$ to $\widetilde{F}_{X_{p_{1}}}^{q_{1}}\times\cdot\cdot\cdot\times\widetilde{F}_{X_{p_{m}}}^{q_{m}}$. Clearly we have that $\iota_{1}\times\cdot\cdot\cdot\times\iota_{m}$ given by

[TABLE]

for all $(x_{1},\ldots,x_{m})\in X_{1}\times\cdot\cdot\cdot\times X_{m}$ is a bounded linear operator from $X_{1}\times\cdot\cdot\cdot\times X_{m}$ to $\widetilde{F}_{X_{p_{1}}}^{q_{1}}(\nu_{1})\times\cdot\cdot\cdot\times\widetilde{F}_{X_{p_{m}}}^{q_{m}}(\nu_{m})$ and so we have the required factorization

[TABLE]

The implication (ii) $\Rightarrow$ (i) is obvious. ∎

Combing the above corollary with Lemma 3.3 we obtain the following result for the case of order continuous Banach function lattices.

Corollary 3.5**.**

Let $X_{j}$ be order continuous $p_{j}$-convex Banach function lattices on measure spaces $(\Omega_{j},\Sigma_{j},\mu_{j})$ and let $1\leq q_{j}<\infty$, $1\leq j\leq m$. If $1/q=1/q_{1}+\ldots+1/q_{m}$, then the following are equivalent statements about an $m$-linear operator $T$ from $X_{1}\times\cdots\times X_{m}$ to a Banach space $Y$.

• (i)

$T$* is $(p_{1},\ldots,p_{m})$-strongly $(q_{1},\ldots,q_{m})$-concave.*

• (ii)

There are probability measures $\nu_{j}$ in $\mathcal{M}(B_{((X_{j})_{p_{j}})^{{}^{\prime}}})$ for each $1\leq j\leq m$, and a multilinear operator $S$ such that $T$ factors through the product of Banach function lattices $S^{q_{j}}_{X_{p_{j}}}(\nu_{j})$ on the corresponding measure spaces $(\Omega_{j},\Sigma_{j},\mu_{j})$ as

[TABLE]

where $\iota_{j}\colon X_{j}\to S^{q_{j}}_{X_{p_{j}}}(\nu_{j})$ are continuous inclusions for $1\leq j\leq m$.

4. Domination and the Fremlin tensor product

In this section we show the relation among summability of multilinear operators from suitable products of Banach function lattices and Fremlin tensor products. This will provide a class of multilinear operators which is different of the one analyzed in the previous section. The main difference is that the factorization is in the present case defined by a multilinear operator with values in a tensor product structure and a linear map, in an opposite way as what happens with the class of $(p_{1},\ldots,p_{m})$-strongly $(q_{1},\ldots,q_{m})$-concave operators.

Theorem 4.1**.**

Let $T\colon X_{1}\times\cdots\times X_{m}\to Y$ be a Banach space valued multilinear operator, where $X_{j}$, $1\leq j\leq m$, are Banach lattices. Suppose that $X_{1}\otimes_{|\pi|}\cdots\otimes_{|\pi|}X_{m}$ is embedded in the $p$-convex Banach lattice $E$. Then the following statements are equivalent.

• (i)

There is a constant $C>0$ such that for each $1\leq j\leq m$ and every choice of $(x^{j}_{i})_{i=1}^{n}$ in $X_{j}$,

[TABLE]

• (ii)

There is a constant $C>0$ such that for every $(x_{1},\ldots,x_{m})\in X_{1}\times\cdots\times X_{m}$,

[TABLE]

where $\nu$ is a probability Borel measure on the weak compact set $B_{{({E}_{p})}^{*}}^{+}$.*

Proof.

The argument follows the lines of the one given for Theorem 3.2. The $p$-convexity of $E$ implies that $E_{p}$ is a Banach lattice. By the inclusion of the Fremlin tensor product $X_{1}\otimes_{|\pi|}\cdot\cdot\cdot\otimes_{|\pi|}X_{m}\hookrightarrow E$ we have that all the tensors $x_{1}\otimes\cdot\cdot\cdot\otimes x_{m}$ are in $E$, and so $|x_{1}\otimes\cdot\cdot\cdot\otimes x_{m}|^{p}$ define a continuous function in $C(B_{(E_{p})^{*}})$, where $B_{(E_{p})^{*}}$ is equipped with the induced topology by the weak∗ topology of $(E_{p})^{*}$. From this point on, the proof using Ky Fan’s Lemma is similar to the one of Theorem 3.2, using Lemma 3.1 for defining the right set of functions $\phi\colon\mathcal{M}(B_{(E_{p})^{*}})\to\mathbb{R}$, where only functions as $|x_{1}\otimes\cdot\cdot\cdot\otimes x_{m}|^{p}$ are considered.

The converse implication is easily obtained by a direct calculation. ∎

In the case of Banach function lattices on measure spaces we obtain the following results on factorization of multilinear operators.

Corollary 4.2**.**

Let $T\colon X_{1}\times\cdots\times X_{m}\to Y$ be a Banach space valued multilinear operator, where $X_{j}$ are Banach function lattices on $(\Omega_{j},\Sigma_{j},\mu_{j})$, $1\leq j\leq m$. Suppose that $X_{1}\otimes_{|\pi|}\cdots\otimes_{|\pi|}X_{m}$ is continuously embedded in $E$, where $E=E(\mu_{1}\times\cdots\times\mu_{m})$ is a $p$-convex Banach function lattice on the product measure space. Then the following statements are equivalent.

• (i)

There is a constant $C>0$ such that for each $1\leq j\leq m$ and for every choice of sequences $(x^{j}_{i})_{i=1}^{n}$ in $X_{j}$,

[TABLE]

• (ii)

There is a constant $C>0$ such that for every $(x_{1},\ldots,x_{m})\in X_{1}\times\cdots\times X_{m}$,

[TABLE]

where $\nu$ is a probability Borel measure on the weak compact set $B_{{({E}_{p})}^{*}}^{+}$.*

Using the same proof but changing single tensors $x_{1}\otimes\cdot\cdot\cdot\otimes x_{m}$ by finite combinations of these products, we obtain the corresponding factorization theorem.

Corollary 4.3**.**

Under the assumptions of Theorem 4.1 on the spaces $X_{1},\ldots,X_{m}$, $E$ and the multilinear operator $T\colon X_{1}\times\cdot\cdot\cdot\times X_{m}\to Y$, the following statements are equivalent.

• (i)

There is a constant $C>0$ such that for each $1\leq j\leq m$ and for every choice of matrices $(x^{j}_{i,k})_{i=1,k=1}^{N,M}$ in $X_{j}$, $(\lambda_{i,k})_{i=1,k=1}^{N,M}$ in $\mathbb{R}$,

[TABLE]

• (ii)

The operator $T$ admits the following factorization

[TABLE]

where $\nu$ is a probability Borel measure on the weak compact set $B_{{({E}_{p})}^{*}}^{+}$ and $\bigotimes$ is the Fremlin map.*

5. Factorization of $p$-strongly $q$-dominated operators.

In this section we prove a factorization theorem for a special class of linear operators between Banach lattices. We start with the following definition. Let $1\leq q\leq\infty$ and $1/q+1/q^{\prime}=1$. Let $1\leq p_{1}\leq q_{1}=q$, $1\leq p_{2}\leq q_{2}=q^{\prime}$ and $1/r_{1}=1/p_{1}-1/q_{1}$, $1/r_{2}=1/p_{2}-1/q_{2}$. An operator $T\colon X\to Y$ between Banach lattices is said to be $(p_{1},p_{2})$-strongly $(q_{1},q_{2})$-concave whenever

[TABLE]

for every choice of sequences $(x_{k})_{k=1}^{n}$ in $X$ and $(y_{k}^{*})_{k=1}^{n}$ in $Y^{*}$.

We note that general examples of $(p_{1},p_{2})$-strongly $(q_{1},q_{2})$-concave operators are given by the classical $q$-dominated operators. Indeed, an operator $T\colon X\to Y$ is said to be $q$-dominated ($1\leq q<\infty$) if

[TABLE]

for every choice of $(x_{k})_{k=1}^{n}$ in $X$ and $(y_{k}^{*})_{k=1}^{n}$ in $Y^{*}$.

Since $1/q=1-1/q^{\prime}$, $1/q^{\prime}=1-1/q$ and

[TABLE]

we conclude that $T\colon X\to Y$ is $(1,1)$-strongly $(q,q^{\prime})$-concave operator.

Before going on to results, let us observe the following fact. Suppose that $T\colon X\to Y$ is an operator between Banach lattices such that $X$ is $p_{1}$-convex and $Y$ is $p_{2}^{\prime}$-concave with $1/p_{2}+1/p_{2}^{\prime}=1$. Then it follows from Theorem 3.2 that $T$ is $(p_{1},p_{2})$-strongly $(q_{1},q_{2})$-concave operator if and only if there exist $C>0$ and probability measures $\nu_{1}\in\mathcal{M}(B_{(X_{p_{1}})^{*}})$ and $\nu_{2}\in\mathcal{M}(B_{((Y^{*})_{p_{2}})^{*}})$ such that for every $(x,y^{*})\in X\times Y^{*}$,

[TABLE]

We need the following lattice formula (see [14, Proposition 12.6]).

Proposition 5.1**.**

Let $E$ be a Banach lattice, $x_{k}\in E$ $(1\leq k\leq n)$, and $1\leq p\leq\infty$. Then

[TABLE]

An application of the above proposition is the following corollary.

Corollary 5.2**.**

Let $1\leq p_{1}<q$, $1\leq p_{2}<q^{\prime}$, and let $r_{1}$ and $r_{2}$ be given by $1/r_{1}=1/p_{1}-1/q$ and $1/r_{2}=1/p_{2}-1/q^{\prime}$. Assume that $T\colon X\to Y$ is an operator between Banach lattices such that $X$ is $p_{1}$-convex and $Y$ be $p^{\prime}_{2}$-concave. If there exists a constant $C>0$ such that for every sequence $(x_{k})_{k=1}^{n}$,

[TABLE]

then $T$ is $(p_{1},p_{2})$-strongly $(q,q^{\prime})$-concave.

Proof.

From Proposition 5.1, it follows that it is enough to show that

[TABLE]

for every choice of finite sequences $(x_{k})_{k=1}^{n}$ in $X$ and $(y_{k}^{*})_{k=1}^{n}$ in $Y^{*}$.

Assume that $p_{2}>1$; the proof for $p_{2}=1$ is the same with the obvious changes in the computations. Fix now $(\alpha_{k})$ in $B_{r_{2}}^{n}$. We have

[TABLE]

Thus, we get that

[TABLE]

Combining with Proposition 5.1, this completes the proof. ∎

Note that Corollary 5.2 allows to show that some classical operators are $(p_{1},p_{2})$-strongly $(q_{1},q_{2})$-concave. Consider the following example. Let $([0,1],\mathcal{B},\mu)$ be Lebesgue measure space, and let $(A_{k})_{k=1}^{\infty}$ be the decreasing sequence of the intervals $A_{k}:=[0,1/2^{k-1}]$ for each $k\in\mathbb{N}$. Consider the integral evaluation operator $T\colon L^{1}[0,1]\to\ell^{\infty}$ given by

[TABLE]

We claim that $T$ satisfies the assumptions of Corollary 5.2 with $q=2=q^{\prime}$, $p_{1}=p_{2}=1$ and $r_{1}=r_{2}=2$. To see this fix a finite set $\{x_{1},...,x_{n}\}$ in $L^{1}[0,1]$ and define the following constants,

[TABLE]

Note that \big{(}\sum_{i=1}^{n}\alpha_{0,i}^{2}\big{)}^{1/2}=1. Then

[TABLE]

Thus, Corollary 5.2 applies and so $T$ is $(1,1)$-strongly $(2,2)$-concave.

Theorem 5.3**.**

Let $1\leq p_{1},p_{2},q_{1},q_{2}$ be real numbers such that $p_{1}\leq q_{1}$ and $p_{2}\leq q_{2}=q^{\prime}_{1}$. Let $(\Omega,\Sigma,\mu)$ be a finite measure space. Let $X$ be an order continuous $p_{1}$-convex Banach function space and $Y$ a $p_{2}^{\prime}$-concave order continuous Banach function lattice with the Fatou property, where $1/p_{2}+1/p^{\prime}_{2}=1$. Assume that $Y^{\prime}$ is also order continuous. The following statements about an operator $T\colon X\to Y$ are equivalent.

• (i)

$T$* is $(p_{1},p_{2})$-strongly $(q_{1},q_{2})$-concave.*

• (ii)

There is a constant $C>0$ such that for every choice of $(x_{k})_{k=1}^{n}$ in $X$ and $(y_{k}^{*})_{k=1}^{n}$ in $Y^{*}$,

[TABLE]

[TABLE]

• (iii)

$T$* admits the factorization*

[TABLE]

Proof.

The equivalence between (i) and (ii) is just given by Corollary 5.2. Let us show the equivalence of (i) and (iii). Applying Corollary 3.4, we conclude that (i) implies that the bounded bilinear form on $X\times Y^{\prime}$ given by

[TABLE]

admits a bilinear continuous extension from the product $S^{q_{1}}_{X_{p_{1}}}(\nu_{1})\times S^{q^{\prime}_{2}}_{(Y^{\prime})_{p^{\prime}_{2}}}(\nu_{2})$ of Banach function lattices for some probability Borel measure spaces, i.e., there exists a continuous bilinear form $S\colon E\times F\to\mathbb{R}$ such that

[TABLE]

where $E:=S^{q_{1}}_{X_{p_{1}}}(\nu_{1})$, $F:=S^{q^{\prime}_{2}}_{(Y^{\prime})_{p^{\prime}_{2}}}(\nu_{2})$ and

[TABLE]

are continuous inclusions.

The required factorization follows then by using standard arguments. At first we observe that for any fixed $f\in E$ the formula $\langle T_{0}(f),\cdot\rangle:=S(f,\cdot)$ defines a continuous functional on $F$ with

[TABLE]

This clearly implies that $T_{0}\colon E\to F$ is a bounded linear operator with $\|T_{0}\|\leq\|S\|$.

Since $Y$ is $p^{\prime}_{2}$-concave, $Y^{\prime}$ is $p_{2}$-convex. Our assumption on $Y^{\prime}$ yields that of $(Y^{\prime})_{p_{2}}$ is also order continuous. Consequently, we have that the Köthe adjoint of the inclusion $\iota_{Y^{\prime}}\colon Y^{\prime}\to F$ appearing in the factorization given by Corollary 3.4 for the bilinear map can be considered,

[TABLE]

Combining the Köthe duality with $Y^{\prime\prime}=Y$ (by the Fatou property) yields the required factorization shown in (iii). The converse is obvious. ∎