Powers of m-isometries

Teresa Bermúdez ; Carlos Díaz Mendoza ; Antonio Martinón

Studia Mathematica, Tome 209 (2012), p. 249-255 / Harvested from The Polish Digital Mathematics Library

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 ≥ 1 if, for any vector x ∈ X, $∑_{k=0}^{m} (-1)^{k} ({m \atop k}) ||T^{k}x||^{p} = 0$. 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 = gcd(r,s) and h = min(m,l).

Détail

Publié le : 2012-01-01

Zbl 1256.47023

EUDML-ID : urn:eudml:doc:285510