Poisson’s Equation for Mean Ergodic Operators
Michael Lin, Laurian Suciu · Contemporary mathematics - American Mathematical Society · 2015
Let T T be a bounded linear operator on a Banach space X \mathcal {X} . For T T power-bounded, mean ergodicity and weak mean ergodicity are equivalent, but in general, even in Hilbert spaces, this is not so. For T T weakly mean ergodic, Poisson’s equation ( I − T ) x = y (I-T)x=y can be solved for a given y y if and only if A n ( T ) y := 1 n ∑ k = 1 n ∑ j = 0 k − 1 T j y A_n(T)y:= \frac 1n\sum _{k=1}^n\sum _{j=0}^{k-1}T^j y converges weakly. In this paper we study, for T T weakly mean ergodic, the set S ( T ) := { y ∈ X : sup n ‖ A n ( T ) y ‖