On $(n,k)$-quasi-*-paranormal operators

TL;DR

This paper introduces a new class of operators called $(n,k)$-quasi-*-paranormal operators, explores their properties, and examines their spectral characteristics, extending known classes like $n$-*-paranormal and $k$-quasi-*-class A operators.

Contribution

The paper defines the $(n,k)$-quasi-*-paranormal class, studies their properties, and relates them to existing operator classes, providing new insights and examples.

Findings

Includes the class of $n$-*-paranormal operators

Contains the class of $k$-quasi-*-class A operators

Analyzes spectral properties and matrix representations

Abstract

For nonnegative integers $n$ and $k$, we introduce in this paper a new class of $(n,k)$-quasi-*-paranormal operators satisfying $$||T^{1+n}(T^{k}x)||^{1/(1+n)}||T^{k}x||^{n/(1+n)} \geq ||T^*(T^{k}x)|| \makebox{\ for all} x \in H.$$ This class includes the class of $n$-*-paranormal operators and the class of $(1,k)$-quasi-*-paranormal operators contains the class of $k$-quasi-*-class $A$ operators. We study basic properties of $(n,k)$-quasi-*-paranormal operators: (1) inclusion relations and examples; (2) a matrix representation; (3) joint (approximate) point spectrum and single valued extension property.