Power bounded m-left invertible operators

A Hilbert space operator S is an element of B(H) is left m-invertible by T is an element of B(H) if Sigma(m)(j=0) (-1)(m-j) ((m) (j)) (TSj)-S-j = 0, S is m-isometric if Sigma(m)(j=0) (-1)(m-j) ((m) (j)) S*S-j(j) = 0 and S is (m, C)-isometric for some conjugation C of H if Sigma(m)(j=0) (-1)(m-j) ((m) (j)) S*(CSC)-C-j-C-j = 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.