On the closure of Absolutely Norm attaining Operators

TL;DR

This paper characterizes the norm closure of absolutely norm attaining operators on Hilbert spaces, providing spectral descriptions and representations for positive and normal operators, and establishing the equivalence of closures for minimum attaining operators.

Contribution

It offers a complete spectral characterization of the norm closure of $ ext{AN}$-operators and shows the closure of $ ext{AM}$-operators coincides with that of $ ext{AN}$-operators, including their properties.

Findings

Spectral characterization of positive operators in the closure.

Representation results for normal operators in this class.

Closure of $ ext{AM}$-operators equals that of $ ext{AN}$-operators.

Abstract

Let $H_1$ and $H_2$ be complex Hilbert spaces and $T:H_1\rightarrow H_2$ be a bounded linear operator. We say $T$ to be norm attaining, if there exists $x\in H_1$ with $\|x\|=1$ such that $\|Tx\|=\|T\|$. If for every closed subspace $M$ of $H_1$, the restriction $T|_{M}:M\rightarrow H_2$ is norm attaining then, $T$ is called absolutely norm attaining operator or $\mathcal{AN}$-operator. If we replace the norm of the operator by the minimum modulus $m(T)=\inf{\{\|Tx\|:x\in H_1,\; \|x\|=1}\}$, then $T$ is called the minimum attaining and the absolutely minimum attaining operator (or $\mathcal{AM}$-operator) respectively. In this article, we discuss about the operator norm closure of the $\mathcal{AN}$-operators. We completely characterize operators in this closure and study several important properties. We mainly give the spectral characterization of the positive operators in this class and give the representation when the operator is normal. Later we also study the analogous properties for $\mathcal{AM}$-operators and prove that the closure of $\mathcal{AM}$-operators is same as that of the closure of $\mathcal{AN}$-operators. As a consequence, we prove similar results for operators in the norm closure of $\mathcal{AM}$-operators.