Power bounded $m$-left invertible operators

Abstract

A Hilbert space operator $S\in\B$ is left $m$-invertible by $T\in\B$ if $$\sum _{j=0}^m(-1{)}^{m-j}\left(\begin{array}{c}m\\ j\end{array}\right)T^j{S}^j=0,$$ $S$ is $m$-isometric if $$\sum _{j=0}^m(-1{)}^{m-j}\left(\begin{array}{c}m\\ j\end{array}\right){S^{\ast }}^j{S}^j=0$$ and $S$ is $(m,C)$-isometric for some conjugation $C$ of $\H$ if $$\sum _{j=0}^m(-1{)}^{m-j}\left(\begin{array}{c}m\\ j\end{array}\right){S^{\ast }}^{j}C{S}^{j}C=0.$$ If a power bounded operator $S$ is left invertible by a power bounded operator $T$, then $S$ (also, $T^{\ast }$) 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.

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