Power bounded $m$-left invertible operators

A Hilbert space operator $S\in \text{B}$ is left $m$-invertible by $T\in \text{B}$ 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.