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arXiv · 2203.15658

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

Fadil Chabbabi·Maëva Ostermann
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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.

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Fadil Chabbabi, Maëva Ostermann. 2022-03-29. The Aluthge and the mean transforms of $m$-isometries. https://arxiv.org/abs/2203.15658

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