On norm-attainment in (symmetric) tensor products

TL;DR

This paper explores norm-attainment in symmetric tensor products of Banach spaces, establishing conditions for density of norm-attaining elements and providing new examples that answer open questions in the field.

Contribution

It introduces a new concept of norm-attainment in symmetric tensor products and proves density results for various classes of Banach spaces, including new examples addressing open problems.

Findings

Norm-attaining symmetric tensors are dense in many Banach spaces.

Existence of symmetric tensors that do not attain their norms.

New examples of Banach spaces with dense norm-attaining tensors in projective tensor products.

Abstract

In this paper, we introduce a concept of norm-attainment in the projective symmetric tensor product $\widehat{\otimes}_{\pi,s,N} X$ of a Banach space $X$, which turns out to be naturally related to the classical norm-attainment of $N$-homogeneous polynomials on $X$. Due to this relation, we can prove that there exist symmetric tensors that do not attain their norms, which allows us to study the problem of when the set of norm-attaining elements in $\widehat{\otimes}_{\pi,s,N} X$ is dense. We show that the set of all norm-attaining symmetric tensors is dense in $\widehat{\otimes}_{\pi,s,N} X$ for a large set of Banach spaces as $L_p$-spaces, isometric $L_1$-predual spaces or Banach spaces with monotone Schauder basis, among others. Next, we prove that if $X^*$ satisfies the Radon-Nikod\'ym and the approximation property, then the set of all norm-attaining symmetric tensors in $\widehat{\otimes}_{\pi,s,N} X^*$ is dense. From these techniques, we can present new examples of Banach spaces $X$ and $Y$ such that the set of all norm-attaining tensors in the projective tensor product $X \widehat{\otimes}_\pi Y$ is dense, answering positively an open question from the paper by S. Dantas, M. Jung, \'O. Rold\'an and A. Rueda Zoca.