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

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: \[Δ_λ(T):=|T|^λV|T|^{1-λ} \;\; \text{and} \;\; M(T):=\frac12(|T|V+V|T|).\] 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.