Ergodic Characterizations of Reflexivity of Banach Spaces

Let X be a Banach space with a basis. We prove the following characterizations: (i) X is finite-dimensional if and only if every power-bounded operator is uniformly ergodic. (ii) X is reflexive if and only if every power-bounded operator is mean ergodic. (iii) X is quasi-reflexive of order one if and only if for every power-bounded operator T, T or T* is mean ergodic.