The single equality $A^{*n}A^n = (A^*A)^n$ does not imply the quasinormality of weighted shifts on rootless directed trees

TL;DR

The paper addresses whether the single operator identity $A^{*n}A^{n}=(A^{*}A)^{n}$ for $n\ge2$ forces quasinormality. It shows a sharp contrast: for bounded injective bilateral weighted shifts this identity implies quasinormality, while for more general settings it does not, by constructing for every $n\ge2$ bounded non-quasinormal weighted shifts on a rootless directed tree with one branching vertex that satisfy the identity. It further demonstrates that such shifts yield non-quasinormal composition operators in $L^2$-spaces, and uses a transcendence argument to facilitate the delicate parameter choices. The results highlight a nuanced boundary between bilateral and non-bilateral cases and provide explicit counterexamples alongside a positive bilateral-shift criterion. Overall, the work clarifies when the equality influences quasinormality and when it does not, with implications for operator theory on directed trees and composition operators.

Abstract

It is proved that each bounded injective bilateral weighted shift $W$ satisfying the equality $W^{*n}W^{n}=(W^{*}W)^{n}$ for some integer $n\geq 2$ is quasinormal. For any integer $n\geq 2$, an example of a bounded non-quasinormal weighted shift $A$ on a rootless directed tree with one branching vertex which satisfies the equality $A^{*n}A^{n}=(A^{*}A)^{n}$ is constructed. It is also shown that such an example can be constructed in the class of composition operators in $L^2$-spaces over $σ$-finite measure spaces.

Theorems & Definitions (18)

• Proposition 2.1: memo
• Proposition 2.2: memo
• Proposition 2.3: memo
• Theorem 3.1
• Corollary 3.2
• proof
• Lemma 4.1
• proof
• Lemma 4.2
• proof