Norm convergence of some power series of operators in Lpwith applications in ergodic theory

Christophe Cuny
2010 Studia Mathematica  
Let X be a closed subspace of L p (µ), where µ is an arbitrary measure and 1 < p < ∞. Let U be an invertible operator on X such that sup n∈Z U n < ∞. Motivated by applications in ergodic theory, we obtain (optimal) conditions for the convergence of series like P n≥1 (U n f )/n 1−α , 0 ≤ α < 1, in terms of f + · · · + U n−1 f p, generalizing results for unitary (or normal) operators in L 2 (µ). The proofs make use of the spectral integration initiated by Berkson and Gillespie and, more
more » ... and, more particularly, of results from a paper by Berkson-Bourgain-Gillespie. 2010 Mathematics Subject Classification: Primary 47B40, 37A30; Secondary 42B25, 42A45.
doi:10.4064/sm200-1-1 fatcat:v6yn4peqwvdifjyrne4gfzq5oe