Article
Full entry |
PDF
(0.2 MB)
Feedback
PDF
(0.2 MB)
Feedback
Keywords:
Cesàro mean; Abel mean; growth order; uniformly continuous operator semi-group and cosine function
Cesàro mean; Abel mean; growth order; uniformly continuous operator semi-group and cosine function
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 $(iaI+N)^{2}$.
References:
[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
[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. DOI 10.4064/sm187-1-1 | MR 2410881 | Zbl 1151.47048
[3] Sato R.: On ergodic averages and the range of a closed operator. Taiwanese J. Math. 10 (2006), 1193–1223. MR 2253374 | Zbl 1124.47008
[4] Sova M.: Cosine operator functions. Rozprawy Math. 49 (1966), 1–47. MR 0193525 | Zbl 0156.15404
[5] Tomilov Y., Zemànek J.: A new way of constructing examples in operator ergodic theory. Math. Proc. Cambridge Philos. Soc. 137 (2004), 209–225. DOI 10.1017/S0305004103007436 | MR 2075049
Search
Partner of
