A Spectral Characterization of $\mathcal{AN}$ Operators

TL;DR

This paper provides a spectral characterization of a class of operators on complex Hilbert spaces that attain their norm on every closed subspace, revealing structural properties and cone formation within hermitian operators.

Contribution

It introduces a spectral characterization theorem for norm-attaining operators on Hilbert spaces and analyzes their algebraic and geometric structure, including cone properties.

Findings

Operators attain their norm on every closed subspace

The class is not closed under addition

Intersection with positive operators forms a proper cone

Abstract

We establish a spectral characterization theorem for the operators on complex Hilbert spaces of arbitrary dimensions that attain their norm on every closed subspace. The class of these operators is not closed under addition. Nevertheless, we prove that the intersection of these operators with the positive operators forms a proper cone in the real Banach space of hermitian operators.