  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.
