Reduced commutativity of moduli of operators

TL;DR

This paper explores conditions under which certain algebraic equations involving powers of operators imply their quasinormality or normality, providing new characterizations based on specific finite sets of exponents.

Contribution

It establishes novel conditions involving finite sets of exponents that guarantee quasinormality or normality of operators, extending previous operator theory results.

Findings

Operators are quasinormal if certain power equations hold for specific exponent sets.

Invertible operators are normal under particular exponent conditions.

The paper discusses inequalities involving operator powers and their implications.

Abstract

In this paper, we investigate the question of when the equations $A^{*s}A^{s}=(A^{*}A)^{s}$ for all $s \in S$, where $S$ is a finite set of positive integers, imply the quasinormality or normality of $A$. In particular, it is proved that if $S=\{p,m,m+p,n,n+p\}$, where $2\leq m < n$, then $A$ is quasinormal. Moreover, if $A$ is invertible and $S=\{m,n,n+m\}$, where $m \leq n$, then $A$ is normal. Furthermore, the case when $S=\{m,m+n\}$ and $A^{*n}A^n \leq (A^*A)^n$ is discussed.