Dirichlet series and uniform ergodic theorems for linear operators in Banach spaces
Volume 141 / 2000
Studia Mathematica 141 (2000), 69-83
DOI: 10.4064/sm-141-1-69-83
Abstract
We study the convergence properties of Dirichlet series for a bounded linear operator T in a Banach space X. For an increasing sequence $μ = {μ_n}$ of positive numbers and a sequence $f = {f_n}$ of functions analytic in neighborhoods of the spectrum σ(T), the Dirichlet series for ${f_n(T)}$ is defined by D[f,μ;z](T) = ∑_{n=0}^{∞} e^{-μ_nz} f_n(T), z∈ ℂ. Moreover, we introduce a family of summation methods called Dirichlet methods and study the ergodic properties of Dirichlet averages for T in the uniform operator topology.
