GROWTH ORDERS OF MEANS OF DISCRETE SEMIGROUPS OF OPERATORS IN BANACH SPACES

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. $0 \lt \liminf_{r \uparrow 1} a(r)/b(r) \leq \limsup_{r \uparrow 1} a(r)/b(r) \lt \infty$].