Norm-attaining tensors and nuclear operators

TL;DR

This paper introduces and studies the concept of norm-attainment in nuclear operators and tensor products, exploring when such elements are dense and providing examples and counterexamples across various Banach spaces.

Contribution

It defines norm-attainment in nuclear operators and tensor products, investigates density conditions, and presents new examples and counterexamples in Banach space theory.

Findings

Density of norm-attaining elements holds for many classical Banach spaces.

Counterexamples show failure of density in spaces lacking the approximation property.

Connections to classical norm-attaining operator theory are discussed.

Abstract

Given two Banach spaces $X$ and $Y$, we introduce and study a concept of norm-attainment in the space of nuclear operators $\mathcal{N}(X,Y)$ and in the projective tensor product space $X \widehat{\otimes}_\pi Y$. We exhibit positive and negative examples where both previous norm-attainment hold. We also study the problem of whether the class of elements which attain their norms in $\mathcal{N}(X,Y)$ and in $X\widehat{\otimes}_\pi Y$ is dense or not. We prove that, for both concepts, the density of norm-attaining elements holds for a large class of Banach spaces $X$ and $Y$ which, in particular, covers all classical Banach spaces. Nevertheless, we present Banach spaces $X$ and $Y$ failing the approximation property in such a way that the class of elements in $X\widehat{\otimes}_\pi Y$ which attain their projective norms is not dense. We also discuss some relations and applications of our work to the classical theory of norm-attaining operators throughout the paper.