Power bounded m-left invertible operators

Abstract

A Hilbert space operator $S\in B\left(\mathcal{H}\right)$ is left m-invertible by $T\in B\left(\mathcal{H}\right)$ 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 $\mathcal{H}$ 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.

Keywords:

• Hilbert space
• power bounded operator
• m-left invertible operator
• m-isometric operator
• $(m,C)$-isometric operator
• similar to an isometry

AMS(MOS) subject classification (2010):

• Primary 47A05
• 47A55
• Secondary 47A80
• 47A10

Disclosure statement

No potential conflict of interest was reported by the authors.