The Aluthge and the mean transforms of $m$-isometries

TL;DR

This paper investigates the properties of the Aluthge and mean transforms applied to $m$-isometries, showing through examples that these transforms do not preserve the class of $m$-isometries.

Contribution

The paper provides the first analysis demonstrating that the Aluthge and mean transforms do not preserve $m$-isometries, using weighted shift operators as examples.

Findings

Aluthge and mean transforms do not preserve $m$-isometries.

Examples with weighted shift operators illustrate the non-preservation.

The results clarify limitations of these transforms in operator theory.

Abstract

Let $T\in B(H)$ be a bounded linear operator on a Hilbert space $H$, let $T = V|T|$ be its polar decomposition of $T$ and let $\lambda\in [0,1]$. The $\lambda$-Aluthge transform $\Delta_{\lambda}(T)$ and the mean transforms $M(T)$ are defined respectively by: \[\Delta_{\lambda}(T):=|T|^{\lambda}V|T|^{1-\lambda} \;\; \text{and} \;\; M(T):=\frac12(|T|V+V|T|).\] In this paper, we use several examples of weighted shift operators to prove that the Aluthge and mean transforms do not preserve the class of $m-$isometries in any directions.