Representation and normality of $\ast$-paranormal absolutely norm attaining operators

TL;DR

This paper provides a detailed representation of $ ext{ extsterling}$-paranormal absolutely norm attaining operators, showing their structure and conditions under which they are normal, expanding understanding of their properties.

Contribution

It introduces a decomposition for $ ext{ extsterling}$-paranormal absolutely norm attaining operators and compares their class with normal operators, highlighting new structural insights.

Findings

Decomposition of $ ext{ extsterling}$-paranormal $ ext{ extsterling}$-AN operators as $U igoplus D$.

$ ext{ extsterling}$-paranormal $ ext{ extsterling}$-AN operators are larger than normal $ ext{ extsterling}$-AN operators.

Such operators are normal if invertible or null spaces of the operator and its adjoint have equal dimension.

Abstract

In this article, we give a representation of $\ast$-paranormal absolutely norm attaining operator. Explicitly saying, every $\ast$-paranormal absolutely norm attaining ($\mathcal{AN}$ in short) $T$ can be decomposed as $U\oplus D$, where $U$ is a direct sum of scalar multiple of unitary operators and $D$ is a $2\times 2$ upper diagonal operator matrix. By the representation it is clear that the class of $\ast$-paranormal $\mathcal{AN}$-operators is bigger than the class of normal $\mathcal{AN}$-operators but here we observe that a $\ast$-paranormal $\mathcal{AN}$-operator is normal if either it is invertible or dimension of its null space is same as dimension of null space of its adjoint.