Representation and normality of Hyponormal operators in the closure of $\mathcal{AN}$-operators

TL;DR

This paper investigates the structure and properties of hyponormal operators that are limits of absolutely norm attaining operators, providing new representations and conditions for normality.

Contribution

It introduces representations of quasinormal and related operators within the closure of $\\mathcal{AN}$-operators and extends these results to hyponormal operators, exploring conditions for normality.

Findings

Representations of quasinormal $\mathcal{AN}$ and $\mathcal{AM}$-operators.

Extension of results to hyponormal operators in the closure of $\mathcal{AN}$-operators.

Sufficient conditions for hyponormal operators to be normal.

Abstract

Let $H_1$, $H_2$ be complex Hilbert spaces. A bounded linear operator $T : H_1 \to H_2$ is said to be norm attaining if there exists a unit vector $x \in H_1$ such that $\|Tx\| = \|T\|$. If $T|_{M} : M \to H_2$ is norm attaining for every closed subspace $M$ of $H_1$, then we say that $T$ is an absolutely norm attaining ($\mathcal{AN}$-operator). If the norm of the operator is replaced by the minimum modulus $m(T) = \inf\{\|Tx\| : x \in H_1, \|x\| =1\}$, then $T$ is said to be a minimum attaining and an absolutely minimum attaining operator ($\mathcal{AM}$-operator), respectively. In this article, we give representations of quasinormal $\mathcal{AN}$, $\mathcal{AM}$-operators and the operators in the closure of these two classes. Later we extend these results to the class of hyponormal operators in the closure of $\mathcal{AN}$-operators and a further look at some sufficient conditions under which these operators become normal.