Fractional Poisson equations and ergodic theorems for fractional coboundaries

Abstract

For a given contractionT in a Banach spaceX and 0 0 anda j >0 and Σ j=1 ∞ a j =1. The operator calculus justifies the notation(I−T) α :=I−T α (e.g., (I−T 1/2)2=I−T). A vectory∈X is called an, α-fractional coboundary for T if there is anx∈X such that(I−T) α x=y, i.e.,y is a coboundary forT α . The fractional Poisson equation forT is the Poisson equation forT α . We show that if(I−T)X is not closed, then(I−T) α X strictly contains(I−T)X (but has the same closure).