Power bounded $m$-left invertible operators

TL;DR

This paper investigates power bounded operators with specific invertibility and isometry properties, showing that such operators are similar to isometries under certain conditions, extending classical results to the $m$-isometric setting.

Contribution

It establishes that power bounded $m$-left invertible operators are similar to isometries, generalizing known results for isometric and $m$-isometric operators.

Findings

Power bounded $m$-left invertible operators are similar to isometries.

$m$-isometric and $(m,C)$-isometric operators with power boundedness are 1-isometric.

The results extend classical invertibility and isometry properties to the $m$-operator setting.

Abstract

A Hilbert space operator $S\in\B$ is left $m$-invertible by $T\in\B$ if $$\sum_{j=0}^m{(-1)^{m-j}\left(\begin{array}{clcr}m\\j\end{array}\right)T^jS^j}=0,$$ $S$ is $m$-isometric if $$\sum_{j=0}^m{(-1)^{m-j}\left(\begin{array}{clcr}m\\j\end{array}\right){S^*}^jS^j}=0$$ and $S$ is $(m,C)$-isometric for some conjugation $C$ of $\H$ if $$\sum_{j=0}^m{(-1)^{m-j}\left(\begin{array}{clcr}m\\j\end{array}\right){S^*}^jCS^jC}=0.$$ If a power bounded operator $S$ is left invertible by a power bounded operator $T$, then $S$ (also, $T^*$) is similar to an isometry. Translated to $m$-isometric and $(m,C)$-isometric operators $S$ this implies that $S$ is $1$-isometric, equivalently isometric, and (respectively) $(1,C)$-isometric.