Norm convergence of some power series of operators in L-P with applications in ergodic theory

摘要

Let X be a closed subspace of L-P(mu), where mu is an arbitrary measure and 1 < p < infinity. Let U be an invertible operator on X such that sup(n epsilon Z) parallel to U-n parallel to < infinity. Motivated by applications in ergodic theory, we obtain (optimal) conditions for the convergence of series like Sigma(n >= 1) (U-n f)(1-alpha), 0 <= alpha < 1, in terms of parallel to f + ... + Un-1 f parallel to(p), generalizing results for unitary (or normal) operators in L-2(mu). The proofs make use of the spectral integration initiated by Berkson and Gillespie and, more particularly, of results from a paper by Berkson-Bourgain-Gillespie.