Norm-attaining compact operators

TL;DR

This paper demonstrates that certain compact operators between Banach spaces cannot be approximated by norm-attaining operators, providing a negative answer to a long-standing open question from the 1970s.

Contribution

It constructs examples of compact operators that defy approximation by norm-attaining operators, especially involving strictly convex Banach spaces without the approximation property.

Findings

Counterexamples in Banach spaces show non-approximability

Strictly convex Banach spaces without the approximation property serve as range spaces

Examples exist with domain spaces having a Schauder basis

Abstract

We show examples of compact linear operators between Banach spaces which cannot be approximated by norm attaining operators. This is the negative answer to an open question posed in the 1970's. Actually, any strictly convex Banach space failing the approximation property serves as the range space. On the other hand, there are examples in which the domain space has a Schauder basis.