ACTA issues

## Remarks on uniform ergodic theorems

Michael Lin, David Shoikhet, Laurian Suciu

Acta Sci. Math. (Szeged) 81:1-2(2015), 251-283
57/2012

 Abstract. Let $T$ be a bounded linear operator on a Banach space ${\cal X}$. In this paper we study uniform Ces?ro ergodicity when $T$ is not necessarily power-bounded, and relate it to the uniform convergence of the Abel averages. When ${\cal X}$ is over the complex field, we show that uniform Abel ergodicity is equivalent to the uniform convergence of the powers of all (one of) the Abel averages $A_\alpha$, $\alpha\in (0,1)$. This is equivalent to uniform Ces?ro ergodicity of $T$ when $\|T^n\|/n \to0$. For positive operators on real or complex Banach lattices, uniform Abel ergodicity is equivalent to uniform Ces?ro ergodicity. An example shows that this is not true in general. For a $C_0$-semi-group $\{T_t\}_{t\ge0}$ on ${\cal X}$ complex satisfying $\lim_{t\to\infty } \|T_t\|/t=0$, we show that uniform ergodicity is equivalent to uniform convergence of $(\lambda R_\lambda )^n$ for every (one) $\lambda >0$, where $R_\lambda$ is the resolvent family of the generator of the semi-group. DOI: 10.14232/actasm-012-307-4 AMS Subject Classification (1991): 47A35, 47B65; 47B20 Keyword(s): uniform ergodic theorem, Ces?ro bounded operators, Abel convergence, one-point peripheral spectrum Received August 2, 2012, and in revised form August 13, 2014. (Registered under 57/2012.)