Perturbation of m-Isometries by Nilpotent Operators

Abstract

We prove that if $T$ is an $m$-isometry on a Hilbert space and $Q$ an $n$-nilpotent operator commuting with $T$, then $T+Q$ is a $\left(2n+m-2\right)$-isometry. Moreover, we show that a similar result for $\left(m, q\right)$-isometries on Banach spaces is not true.