Growth orders of Cesàro and Abel means of uniformly continuous operator semi-groups and cosine functions

Sato, Ryotaro. "Growth orders of Cesàro and Abel means of uniformly continuous operator semi-groups and cosine functions." Commentationes Mathematicae Universitatis Carolinae 51.3 (2010): 441-451. <http://eudml.org/doc/38140>.

@article{Sato2010,
abstract = {It will be proved that if $N$ is a bounded nilpotent operator on a Banach space $X$ of order $k+1$, where $k\ge 1$ is an integer, then the $\gamma $-th order Cesàro mean $C_\{t\}^\{\gamma \}:=\gamma t^\{-\gamma \}\int _\{0\}^\{t\}(t-s)^\{\gamma -1\}T(s)\,ds$ and Abel mean $A_\{\lambda \}:=\lambda \int _\{0\}^\{\infty \}e^\{-\lambda s\}T(s)\,ds$ of the uniformly continuous semigroup $(T(t))_\{t\ge 0\}$ of bounded linear operators on $X$ generated by $iaI+N$, where $0\ne a\in \mathbb \{R\}$, satisfy that (a) $\Vert C_\{t\}^\{\gamma \}\Vert \sim t^\{k-\gamma \}\;(t\rightarrow \infty )$ for all $0< \gamma \le k+1$; (b) $\Vert C_\{t\}^\{\gamma \}\Vert \sim t^\{-1\}\;(t\rightarrow \infty )$ for all $\gamma \ge k+1$; (c) $\Vert A_\{\lambda \}\Vert \sim \lambda \;(\lambda \downarrow 0)$. A similar result will be also proved for the uniformly continuous cosine function $(C(t))_\{t\ge 0\}$ of bounded linear operators on $X$ generated by $(iaI+N)^\{2\}$.},
author = {Sato, Ryotaro},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {Cesàro mean; Abel mean; growth order; uniformly continuous operator semi-group and cosine function; Cesàro mean; Abel mean; operator semigroup},
language = {eng},
number = {3},
pages = {441-451},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {Growth orders of Cesàro and Abel means of uniformly continuous operator semi-groups and cosine functions},
url = {http://eudml.org/doc/38140},
volume = {51},
year = {2010},
}

TY - JOUR
AU - Sato, Ryotaro
TI - Growth orders of Cesàro and Abel means of uniformly continuous operator semi-groups and cosine functions
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 2010
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 51
IS - 3
SP - 441
EP - 451
AB - It will be proved that if $N$ is a bounded nilpotent operator on a Banach space $X$ of order $k+1$, where $k\ge 1$ is an integer, then the $\gamma $-th order Cesàro mean $C_{t}^{\gamma }:=\gamma t^{-\gamma }\int _{0}^{t}(t-s)^{\gamma -1}T(s)\,ds$ and Abel mean $A_{\lambda }:=\lambda \int _{0}^{\infty }e^{-\lambda s}T(s)\,ds$ of the uniformly continuous semigroup $(T(t))_{t\ge 0}$ of bounded linear operators on $X$ generated by $iaI+N$, where $0\ne a\in \mathbb {R}$, satisfy that (a) $\Vert C_{t}^{\gamma }\Vert \sim t^{k-\gamma }\;(t\rightarrow \infty )$ for all $0< \gamma \le k+1$; (b) $\Vert C_{t}^{\gamma }\Vert \sim t^{-1}\;(t\rightarrow \infty )$ for all $\gamma \ge k+1$; (c) $\Vert A_{\lambda }\Vert \sim \lambda \;(\lambda \downarrow 0)$. A similar result will be also proved for the uniformly continuous cosine function $(C(t))_{t\ge 0}$ of bounded linear operators on $X$ generated by $(iaI+N)^{2}$.
LA - eng
KW - Cesàro mean; Abel mean; growth order; uniformly continuous operator semi-group and cosine function; Cesàro mean; Abel mean; operator semigroup
UR - http://eudml.org/doc/38140
ER -