Norm attaining operators and pseudospectrum

TL;DR

This paper investigates conditions under which operators on certain Banach spaces attain their norm, providing new results for subspaces of $ ext{ell}_p$ sums and constructing counterexamples in reflexive spaces.

Contribution

It establishes norm attainment for operators on specific Banach spaces and constructs examples where the sum operator does not attain its norm, highlighting nuanced behaviors.

Findings

Operators on certain Banach spaces attain their norm when perturbed by compact operators.

Counterexamples show norm attainment can fail in reflexive Banach spaces with rank-one operators.

Provides conditions linking space structure and norm attainment properties.

Abstract

It is shown that if $1<p<\infty$ and $X$ is a subspace or a quotient of an $\ell_p$-direct sum of finite dimensional Banach spaces, then for any compact operator $T$ on $X$ such that $\|I+T\|>1$, the operator $I+T$ attains its norm. A reflexive Banach space $X$ and a bounded rank one operator $T$ on $X$ are constructed such that $\|I+T\|>1$ and $I+T$ does not attain its norm.