Dirichlet series and uniform ergodic theorems for linear operators in Banach spaces

We study the convergence properties of Dirichlet series for a bounded linear operator T in a Banach space X. For an increasing sequence ${\displaystyle \mu =\left\{{\mu }_n\right\}}$ of positive numbers and a sequence ${\displaystyle f=\left\{f_n\right\}}$ of functions analytic in neighborhoods of the spectrum σ(T), the Dirichlet series for ${\displaystyle \left\{f_n\left(T\right)\right\}}$ 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.