Perturbations by norm attaining operators

Abstract

In this note, we develop a connection between Fredholm operators and norm attaining operators in the set L(H) of bounded linear operators on a Hilbert space. We use this connection to extend an earlier result about the stability of norm attaining operators under perturbations. Specifically, we show that an element A of the proper subset of the norm attaining operators, A _d = {A : ∥A∥ ∈ σ_disc(|A|)}, remains in A _d under sufficiently small perturbations. We apply this result to show that the complement of the set A _d in L(H) is nowhere dense (and porous). This extends an earlier result that the complement of the norm attaining operators in L(H) is nowhere dense (and porous).