Powers of $m$-isometries
Volume 208 / 2012
Studia Mathematica 208 (2012), 249-255
MSC: Primary 47B99.
DOI: 10.4064/sm208-3-4
Abstract
A bounded linear operator $T$ on a Banach space $X$ is called an $(m,p)$-isometry for a positive integer $m$ and a real number $p \geq 1$ if, for any vector $x \in X$, $$ \sum _{k=0}^m (-1)^{k} \left({m\atop k}\right ) \|T^k x\| ^p =0 . $$We prove that any power of an $(m,p)$-isometry is also an $(m,p)$-isometry. In general the converse is not true. However, we prove that if $T^r$ and $T^{r+1}$ are $(m,p)$-isometries for a positive integer $r$, then $T$ is an $(m,p)$-isometry. More precisely, if $T^r$ is an $(m,p)$-isometry and $T^s$ is an $(l,p)$-isometry, then $T^t$ is an $(h,p)$-isometry, where $t={\rm gcd}(r, s)$ and $h={\rm min}(m,l)$.
