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Title: Growth orders of Cesàro and Abel means of uniformly continuous operator semi-groups and cosine functions (English)
Author: Sato, Ryotaro
Language: English
Journal: Commentationes Mathematicae Universitatis Carolinae
ISSN: 0010-2628 (print)
ISSN: 1213-7243 (online)
Volume: 51
Issue: 3
Year: 2010
Pages: 441-451
Summary lang: English
Category: math
Summary: 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 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 $(T(t))_{t\geq 0}$ 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 $0< \gamma\leq k+1$; (b) $\|C_{t}^{\gamma}\|\sim t^{-1}\;(t\to\infty)$ for all $\gamma\geq k+1$; (c) $\|A_{\lambda}\|\sim \lambda\;(\lambda\downarrow 0)$. A similar result will be also proved for the uniformly continuous cosine function $(C(t))_{t\geq 0}$ of bounded linear operators on $X$ generated by $(iaI+N)^{2}$. (English)
Keyword: Cesàro mean
Keyword: Abel mean
Keyword: growth order
Keyword: uniformly continuous operator semi-group and cosine function
MSC: 47A35
MSC: 47D06
MSC: 47D09
idZBL: Zbl 1222.47068
idMR: MR2741877
Date available: 2010-09-02T14:15:27Z
Last updated: 2013-09-22
Stable URL:
Reference: [1] Chen J.-C., Sato R., Shaw S.-Y.: Growth orders of Cesàro and Abel means of functions in Banach spaces.Taiwanese J. Math.(to appear). MR 2674604
Reference: [2] Li Y.-C., Sato R., Shaw S.-Y.: Boundedness and growth orders of means of discrete and continuous semigroups of operators.Studia Math. 187 (2008), 1–35. Zbl 1151.47048, MR 2410881, 10.4064/sm187-1-1
Reference: [3] Sato R.: On ergodic averages and the range of a closed operator.Taiwanese J. Math. 10 (2006), 1193–1223. Zbl 1124.47008, MR 2253374
Reference: [4] Sova M.: Cosine operator functions,.Rozprawy Math. 49 (1966), 1–47. Zbl 0156.15404, MR 0193525
Reference: [5] Tomilov Y., Zemànek J.: A new way of constructing examples in operator ergodic theory.Math. Proc. Cambridge Philos. Soc. 137 (2004), 209–225. MR 2075049, 10.1017/S0305004103007436


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