Poisson’s equation for mean ergodic operators

Let T be a bounded linear operator on a Banach space χ. For T power-bounded, mean ergodicity and weak mean ergodicity are equivalent, but in general, even in Hilbert spaces, this is not so. For T weakly mean ergodic, Poisson’s equation (I − T)x = y can be solved for a given y if and only if (formula presented) converges weakly. In this paper we study, for T weakly mean ergodic, the set S(T) := {y ∈ X : sup_n ||A_n(T)_y|| < ∞}. We show that for T Cesàro bounded with Tⁿ/n → 0 weakly, S(T) is closed if and only if (I −T)χ is closed, and then T is weakly mean ergodic; if ||Tⁿ||/n → 0, then T is uniformly ergodic. As an application of our study in reflexive spaces, we supplement the Fonf-Lin-Wojtaszczyk characterization of reflexivity of Banach spaces with a basis: Such a space χ is reflexive if and only if every weakly mean ergodic T satisfies (I − T)χ = S(T), if and only if every power-bounded mean ergodic T satisfies (I − T)7 = S(T).