| 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 |
| . |
