Rank-one perturbations and norm-attaining operators

TL;DR

This paper constructs specific operators on Banach spaces demonstrating that small rank-one perturbations can prevent norm attainment, answering a previously posed question and exploring related properties.

Contribution

It shows the existence of operators with rank-one perturbations that do not attain their norm, addressing an open problem and analyzing related geometric properties.

Findings

Existence of operators with rank-one perturbations that do not attain their norm

Relationship between the V-property, weak maximizing property, and norm-attaining perturbations

Counterexamples in reflexive Banach spaces

Abstract

The main goal of this article is to show that for every (reflexive) infinite-dimensional Banach space $X$ there exists a reflexive Banach space $Y$ and $T, R \in \mathcal{L}(X,Y)$ such that $R$ is a rank-one operator, $\|T+R\|>\|T\|$ but $T+R$ does not attain its norm. This answers a question posed by S. Dantas and the first two authors. Furthermore, motivated by the parallelism exhibited in the literature between the $V$-property introduced by V.A. Khatskevich, M.I. Ostrovskii and V.S. Shulman and the weak maximizing property introduced by R.M. Aron, D. Garc\'ia, D. Pellegrino and E.V. Teixeira, we also study the relationship between these two properties and norm-attaining perturbations of operators.