Abstract. New growth conditions on the Ces?ro means of higher order are investigated for Banach space operators with peripheral spectrum reduced to $\{1\} $. Certain consequences concerning the powers of such operators are derived. The uniform and strong convergence of the differences of consecutive Ces?ro means are studied, and several examples are presented. These topics are related to the boundedness and convergence of Ces?ro means of higher order, and also to Gelfand--Hille and Esterle--Katznelson--Tzafriri type theorems. In particular, if $V$ denotes the classical Volterra operator, then our results provide a simultaneous conceptual proof showing that the operator $I-V$ is Ces?ro ergodic on $L^p(0,1)$ for $1\le p< \infty $, completing the known cases $p=1$ and $p=2$. Even every power of the latter operator is Ces?ro ergodic, though the operator itself is not power-bounded if $p\not=2$. Analogous examples, with respect to uniform ergodicity, are given as well. We also obtain improvements on the general 1939 Lorch theorem, within the above spectral picture.

AMS Subject Classification (1991): 47A10, 47A35

Keyword(s): Abel mean, Ces?ro mean, Kreiss resolvent condition, spectrum, power-dominated operator, nilpotent operator, ascent, descent

Received March 7, 2013, and in revised form July 29, 2013. (Registered under 20/2013.)