Ryotaro Sato
Growth orders of Ces\`aro and Abel means of uniformly continuous operator semi-groups and cosine functions

Comment.Math.Univ.Carolin. 51,3 (2010) 441-451.

Abstract:It will be proved that if $N$ is a bounded nilpotent operator on a Banach space $X$ of order $k+1$, where $k\geq 1$ is an integer, then the $\gamma$-th order Ces\`aro 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 $(T(t))_{t\geq 0}$ of bounded linear operators on $X$ generated by $iaI+N$, where $0\neq a\in \mathbb{R}$, satisfy that (a) $\|C_{t}^{\gamma}\|\sim t^{k-\gamma}\;(t\to\infty)$ for all $0

Keywords: Ces\`aro mean, Abel mean, growth order, uniformly continuous operator semi-group and cosine function
AMS Subject Classification: 47D06 47D09 47A35