Reduced commutativity of moduli of operators

TL;DR

This work addresses when a finite set $S$ of exponents suffices to deduce quasinormality or normality from the operator identities $A^{*s}A^{s}=(A^{*}A)^{s}$ for all $s\in S$. It builds a framework based on operator monotone/convex functions and the Davis-Choi-Jensen inequality to derive finite-set characterizations, avoiding separability assumptions in key cases. The main contributions are a quasinormal characterization for $S=\{p,m,m+p,n,n+p\}$ with $2\le m<n$ (and a variant for $S=\{p,q,p+q,2p,2p+q\}$), and a parallel normality criterion, including the invertible case with $S=\{m,n,m+n\}$, that mirrors the quasinormal results. Additionally, the paper develops several operator-inequality results, establishing power-inequality chains under finite-set hypotheses and employing tools like Löwner-Heinz and Jensen-type inequalities to illuminate when partial information about $A$ enforces normal or quasinormal structure.

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.

Theorems & Definitions (52)

• Theorem 1.1
• Theorem 2.1
• Example 2.2
• Theorem 2.3
• Theorem 2.4
• Theorem 2.5
• Lemma 3.1
• proof
• Theorem 3.2
• proof