Powers of m-isometries

Teresa Bermúdez, Carlos Díaz Mendoza, and Antonio Martinón. "Powers of m-isometries." Studia Mathematica 208.3 (2012): 249-255. <http://eudml.org/doc/285510>.

@article{TeresaBermúdez2012,
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 ≥ 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).},
author = {Teresa Bermúdez, Carlos Díaz Mendoza, Antonio Martinón},
journal = {Studia Mathematica},
keywords = {isometry; (m,p)-isometry; recursive equation},
language = {eng},
number = {3},
pages = {249-255},
title = {Powers of m-isometries},
url = {http://eudml.org/doc/285510},
volume = {208},
year = {2012},
}

TY - JOUR
AU - Teresa Bermúdez
AU - Carlos Díaz Mendoza
AU - Antonio Martinón
TI - Powers of m-isometries
JO - Studia Mathematica
PY - 2012
VL - 208
IS - 3
SP - 249
EP - 255
AB - 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).
LA - eng
KW - isometry; (m,p)-isometry; recursive equation
UR - http://eudml.org/doc/285510
ER -