On weak compactness in projective tensor products

TL;DR

This paper investigates the preservation of strong weak compactness in the projective tensor products of Banach spaces with specific structural properties, establishing conditions under which the property is maintained.

Contribution

It provides new conditions involving unconditional Schauder decompositions and lower p-estimates that ensure the strong weak compactness property is preserved in tensor products.

Findings

Strong weak compactness is preserved under tensor products with certain Banach space conditions.

The main result applies when 1/p + 1/q ≥ 1 and spaces have disjoint lower p- and q-estimates.

The paper extends understanding of compactness properties in tensor product spaces.

Abstract

We study the property of being strongly weakly compactly generated (and some relatives) in projective tensor products of Banach spaces. Our main result is as follows. Let $1<p,q<\infty$ be such that $1/p+1/q\geq 1$. Let $X$ (resp., $Y$) be a Banach space with a countable unconditional finite-dimensional Schauder decomposition having a disjoint lower $p$-estimate (resp., $q$-estimate). If $X$ and $Y$ are strongly weakly compactly generated, then so is its projective tensor product $X \widehat{\otimes}_\pi Y$.