Reduced commutativity of moduli of operators

Abstract

In this paper, we investigate the question of when the equations $A^{\ast s}{A}^s=(A^{\ast }A{)}^s$ 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 $2\le m<n$, then $A$ is quasinormal. Moreover, if $A$ is invertible and $S=\{m,n,n+m\}$, where $m\le n$, then $A$ is normal. Furthermore, the case when $S=\{m,m+n\}$ and $A^{\ast n}{A}^n\le (A^{\ast }A{)}^n$ is discussed.

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Cited by in corpus (4)

• Subnormal $n$th roots of quasinormal operators are quasinormal
• On $n$th roots of bounded and unbounded quasinormal operators
• A Shimorin-type analytic model on an annulus for left-invertible operators and applications
• Generalized multipliers for left-invertible operators and applications