Conditions implying the normality of $\ast$-paranormal operators in the closure of $\mathcal{AN}$-operators

TL;DR

This paper investigates conditions under which $ ext{ extsterling}$-paranormal operators become normal, focusing on their invariant subspaces, representations in closures of norm-attaining operators, and specific classes like Toeplitz and Hankel operators.

Contribution

It provides new criteria for the normality of $ ext{ extsterling}$-paranormal operators within the closure of absolutely norm-attaining operators, including representations and special cases.

Findings

Existence of invariant subspaces for norm-attaining $ ext{ extsterling}$-paranormal operators

Representation of $ ext{ extsterling}$-paranormal operators in the closure of absolutely norm-attaining operators

Sufficient conditions for the normality of such operators, including Toeplitz and Hankel cases

Abstract

In this article, we first prove the existence of an invariant subspace for a norm-attaining $\ast$-paranormal operator. Then give a representation for $\ast$-paranormal operators in the closure of absolutely norm-attaining operators and further study a few sufficient conditions for the normality of such operators. Finally, we discuss Toeplitz and Hankel $\ast$-paranormal operators in the closure of absolutely norm-attaining operators on the Hardy space.