On Absolutely norm (minimum) attaining $2\times 2$ block operator matrix

TL;DR

This paper investigates absolutely norm attaining and minimum attaining properties of positive 2x2 block operator matrices on Hilbert spaces, providing characterizations, closure conditions, and examples to deepen understanding of these classes.

Contribution

It offers new characterizations and closure conditions for absolutely norm attaining and minimum attaining operators, especially for positive block matrices, and explores their relationships.

Findings

Characterization of idempotent operators in these classes

Conditions for operators to be in the norm closure of these classes

Concrete examples illustrating the theoretical results

Abstract

In this article, we study absolutely norm attaining operators ($\mathcal{AN}$-operators, in short), that is, operators that attain their norm on every non-zero closed subspace of a Hilbert space. Our focus is primarily on positive $2\times2$ block operator matrices in Hilbert spaces. Subsequently, we examine the analogous problem for operators that attain their minimum modulus on every nonzero closed subspace; these are referred to as absolutely minimum attaining operators (or $\mathcal{AM}$-operators, in short). We provide conditions under which these operators belong to the operator norm closure of the above two classes. In addition, we give a characterization of idempotent operators that fall into these three classes. Finally, we illustrate our results through examples that involve concrete operators.