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2010 GROWTH ORDERS OF MEANS OF DISCRETE SEMIGROUPS OF OPERATORS IN BANACH SPACES
Ryotaro Sato
Taiwanese J. Math. 14(3B): 1111-1116 (2010). DOI: 10.11650/twjm/1500405908

Abstract

We study the growth orders of $\gamma$-th order Cesàro means $C_{n}^{\gamma}(T)$ ($\gamma \geq 0$) and Abel means $A_{r}(T)$ of the discrete semigroup $\{T^{n}: n \geq 0\}$ generated by a bounded linear operator $T$ on a Banach space. Let $T$ be of the form $T = -(I+N)$, where $N$ is a nilpotent operator of order $k+1$ with $k \in \mathbb{N}$. Thus $N^{k+1} = 0$ and $N^{k} \not = 0$. Then we prove that (a) $\|C_{n}^{\gamma}(T)\|\sim n^{k-\gamma}\; (n \to \infty)$ if $0 \leq \gamma \leq k+1$, and $\|C_{n}^{\gamma}(T)\|\sim n^{-1}\; (n \to \infty)$ if $\gamma \geq k+1$; (b) $\|A_{r}(T)\|\sim 1-r\; (r \uparrow 1)$. Here $a(n) \sim b(n)\; (n \to \infty)$ [resp. $a(r) \sim b(r)\; (r \uparrow 1)$] means that $0 \lt \liminf_{n \to \infty} a(n)/b(n) \leq \limsup_{n \to \infty} a(n)/b(n) \lt \infty$ [resp. $0 \lt \liminf_{r \uparrow 1} a(r)/b(r) \leq \limsup_{r \uparrow 1} a(r)/b(r) \lt \infty$].

Citation

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Ryotaro Sato. "GROWTH ORDERS OF MEANS OF DISCRETE SEMIGROUPS OF OPERATORS IN BANACH SPACES." Taiwanese J. Math. 14 (3B) 1111 - 1116, 2010. https://doi.org/10.11650/twjm/1500405908

Information

Published: 2010
First available in Project Euclid: 18 July 2017

zbMATH: 1218.47061
MathSciNet: MR2674599
Digital Object Identifier: 10.11650/twjm/1500405908

Subjects:
Primary: 47A35

Rights: Copyright © 2010 The Mathematical Society of the Republic of China

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Vol.14 • No. 3B • 2010
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