Powers of $m$-isometries

Teresa Bermúdez¹ ; Carlos Díaz Mendoza¹ ; Antonio Martinón¹

¹ Departamento de Análisis Matemático Universidad de La Laguna 38271 La Laguna (Tenerife), Spain

Studia Mathematica, Tome 208 (2012) no. 3, pp. 249-255

Résumé

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 $h={\rm min}(m,l)$.

Détail

DOI : 10.4064/sm208-3-4

Keywords: bounded linear operator banach space called isometry positive integer real number geq vector sum atop right prove power isometry isometry general converse however prove isometries positive integer isometry precisely isometry isometry isometry where gcd min