Banach actions preserving unconditional convergence

TL;DR

This paper characterizes Banach actions that preserve unconditional convergence, linking them to absolutely summing operators, and applies these results to sequence space multiplications, revealing precise conditions for preservation.

Contribution

It provides a new characterization of Banach actions preserving unconditional convergence via absolutely summing operators and applies this to sequence space multiplications.

Findings

Characterization of Banach actions preserving unconditional convergence.

Connection to absolutely summing operators and Grothendieck theorem.

Conditions for coordinatewise multiplication on sequence spaces to preserve unconditional convergence.

Abstract

Let $A,X,Y$ be Banach spaces and $A\times X\to Y$, $(a,x)\mapsto ax$, be a continuous bilinear function, called a *Banach action*. We say that this action *preserves unconditional convergence* if for every bounded sequence $(a_n)_{n\in\omega}$ in $A$ and unconditionally convergent series $\sum_{n\in\omega}x_n$ in $X$ the series $\sum_{n\in\omega}a_nx_n$ is unconditionally convergent. We prove that a Banach action $A\times X\to Y$ preserves unconditional convergence if and only if for any linear functional $y^*\in Y^*$ the operator $D_{y^*}:X\to A^*$, $D_{y^*}(x)(a)=y^*(ax)$, is absolutely summing. Combining this characterization with the famous Grothendieck theorem on the absolute summability of operators from $\ell_1$ to $\ell_2$, we prove that a Banach action $A\times X\to Y$ preserves unconditional convergence if $A$ is a Hilbert space possessing an orthonormal basis $(e_n)_{n\in\omega}$ such that for every $x\in X$ the series $\sum_{n\in\omega}e_nx$ is weakly absolutely convergent. Applying known results of Garling on the absolute summability of diagonal operators between sequence spaces, we prove that for (finite or infinite) numbers $p,q,r\in[1,\infty]$ with $\frac1r\le\frac1p+\frac1q$, the coordinatewise multplication $\ell_p\times\ell_q\to\ell_r$ preserves unconditional convergence if and only if one of the following conditions holds: (i) $p\le 2$ and $q\le r$, (ii) $2<p<q\le r$, (iii) $2<p=q<r$, (iv) $r=\infty$, (v) $2\le q<p\le r$, (vi) $q<2<p$ and $\frac1p+\frac1q\ge\frac1r+\frac12$.