Fractional Poisson equations and ergodic theorems for fractional coboundaries

For a given contraction T in a Banach space X and 0 < alpha < 1, we define the contraction T-alpha = Sigma (infinity)(j-1), a(j)T(j), where {a(j)) are the coefficients in the power series expansion (1 - t)(alpha) = 1 - Sigma (infinity)(j=1) a(j)t(j) in the open unit disk, which satisfy a(j) > 0 and Sigma (infinity)(j=1) a(j) = 1. The operator calculus justifies the notation (I - T)(alpha) := I - T-alpha (e.g., (I - T-1/2)(2) = I - T). A vector y is an element of X is called an a-fractional coboundary for T if there is an x is an element of X such that (I - T)(alpha)x = y, i.e., y is a coboundary for T-alpha. The fractional Poisson equation for T is the Poisson equation for T-alpha. We show that if (I - T)X is not closed, then (I - T)X-alpha strictly contains (I - T)X (but has the same closure). For T mean ergodic, we obtain a series solution (converging in norm) to the fractional Poisson equation. We prove that y is an element of X is an alpha -fractional coboundary if and only if Sigma (infinity)(k=1) T(k)y/k(1-alpha) converges in norm, and conclude that lim(n) \\(1/n(1-alpha)) Sigma (n)(k=1) T(k)y\\ = 0 for such y. For a Dunford-Schwartz operator T on L-1 of a probability space, we consider also a.e. convergence. We prove that if f is an element of (I - T)L-alpha(1) for some 0 < alpha < 1, then the one-sided Hilbert transform Sigma (infinity)(k=1) T-k f/k converges a.e. For 1 < p < infinity, we prove that if f is an element of (I - T)L-alpha(p) with alpha > 1 - 1/p = 1/q, then Sigma (infinity)(k=1) T(k)f/k(1/p) converges a.e., and thus (1/n(1/p)) Sigma (k=1) T-k f converges a.e. to zero. When f is an element of (I - T)(1/q) L-p (the case alpha = 1/q), we prove that (1/n(1/p)(log n)(1/q)) Sigma (k=1) T-k f converges a.e. to zero.