Operators that achieve the norm

TL;DR

This paper investigates operators on complex Hilbert spaces that attain their norm on the unit sphere, providing key characterizations of AN operators and including compact operators within this class.

Contribution

It offers new theoretical characterizations of AN operators, expanding understanding of their structure and relation to compact operators.

Findings

Characterization of AN operators on complex Hilbert spaces

Proof that the algebra of compact operators is contained in AN operators

Development of criteria to identify norm-attaining operators

Abstract

In this paper we study the theory of operators on complex Hilbert spaces, which achieve the norm in the unit sphere. We prove important results concerning the characterization of the AN operators, see Definition 1.2. The class of AN operators contains the algebra of the compact ones.