## Abstract

For a given contraction T in a Banach space X and 0 < α < 1, we define the contraction T_{α} = Σ^{∞}_{j=1} a_{j}T^{j} where {a_{j}} are the coefficients in the power series expansion (1 - t)^{α} = 1 - Σ^{∞}_{j=1} a_{j}t^{j} in the open unit disk, which satisfy a_{j} > 0 and Σ^{∞}_{j=1} = 1. The operator calculus justifies the notation (I - T)^{α} := I - T_{α} (e.g., (I - T_{1/2})^{2} = I - T). A vector y ∈ X is called an α-fractional coboundary for T if there is an x ∈ X such that (I -T)^{α}x = y, i.e., y is a coboundary for T_{α}. The fractional Poisson equation for T is the Poisson equation for T_{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). For T mean ergodic, we obtain a series solution (converging in norm) to the fractional Poisson equation. We prove that y ∈ X is an α-fractional coboundary if and only if Σ^{∞}_{k=1} T^{k}y/k^{1-α} converges in norm, and conclude that lim_{n} ∥(1/n^{1-α}) Σ^{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 ∈ (I - T)^{α} L_{1} for some 0 < α < 1, then the one-sided Hilbert transform Σ^{∞}_{k=1} T^{k} f/k converges a.e. For 1 < p < ∞, we prove that if ∈ 6 (I -T)^{α} L_{p} with α > 1 - 1/p = 1/9, then Σ^{∞}_{k=1} T^{k} f/k^{1/p} converges a.e., and thus (1/n^{1/p}) Σ^{n}_{k=1} T^{k} f converges a.e. to zero. When f ∈ (I - T)^{1/q} L_{p} (the case α = 1/q), we prove that (1/n^{1/p}(log n)^{1/q}) Σ^{n}_{k=1} T^{k} f converges a.e. to zero.

Original language | English |
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Pages (from-to) | 93-130 |

Number of pages | 38 |

Journal | Israel Journal of Mathematics |

Volume | 123 |

DOIs | |

State | Published - 1 Jan 2001 |

## ASJC Scopus subject areas

- General Mathematics