On the uniform ergodic theorem in Banach spaces that do not contain duals

Vladimir Fonf; Michael Lin; Alexander Rubinov

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Displaying similar documents to “On the uniform ergodic theorem in Banach spaces that do not contain duals”

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Let (X, F, μ) be a finite measure space. Let T: X → X be a measure preserving transformation and let Af denote the average of Tf, k = 0, ..., n. Given a real positive function v on X, we prove that {Af} converges in the a.e. sense for every f in L(v dμ) if and only if inf v(Tx) > 0 a.e., and the same condition is equivalent to the finiteness of a related ergodic power function Pf for every f in L(v dμ). We apply this result to characterize, being T null-preserving, the finite...

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For a Cesàro bounded operator in a Hilbert space or a reflexive Banach space the mean ergodic theorem does not hold in general. We give an additional geometrical assumption which is sufficient to imply the validity of that theorem. Our result yields the mean ergodic theorem for positive Cesàro bounded operators in $L^{p}$ (1 < p < ∞). We do not use the tauberian theorem of Hardy and Littlewood, which was the main tool of previous authors. Some new examples, interesting for summability...

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

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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-μnz fn(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. ...