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

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 $λ\in [0,1]$. The $λ$-Aluthge transform $Δ_λ(T)$ 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.

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