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Dirichlet series and uniform ergodic theorems for linear operators in Banach spaces. (English) Zbl 0969.47007

The author studies the convergence properties of the Dirichlet series for a bounded operator \(T\) on a Banach space \(X\). For an increasing sequence \(\mu=\{\mu_n\}\) of positive numbers and for a sequence \(f=\{f_n\}\) of functions analytic in a neighbourhood of the spectrum \(\sigma(T)\) the Dirichlet series is defined by \[ D[f,\mu,z](T)=\mathop{\sum}\limits_{n=0}^{\infty}e^{-\mu_nz}f_n(T), \;\;\;z\in C. \] The author discusses uniform convergence of this series and the abscissa of convergence, i.e. finds a lower bound for \(\text{ Re} z\) such that the series converges.

MSC:

47A35 Ergodic theory of linear operators
47A10 Spectrum, resolvent
11F66 Langlands \(L\)-functions; one variable Dirichlet series and functional equations
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