# Article

 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: http://hdl.handle.net/10338.dmlcz/140720 . 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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