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Cesàro mean; Abel mean; growth order; uniformly continuous operator semi-group and cosine function
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 $(iaI+N)^{2}$.
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