Averages of unitary representations and weak mixing of random walks

Michael Lin; Rainer Wittmann

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Displaying similar documents to “Averages of unitary representations and weak mixing of random walks”

Ergodic properties of an operator obtained from a continuous representation

Michael Lin (1977)

Annales de l'I.H.P. Probabilités et statistiques

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Weak almost periodicity of $L_{1}$ contractions and coboundaries of non-singular transformations

Isaac Kornfeld, Michael Lin (2000)

Studia Mathematica

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It is well known that a weakly almost periodic operator T in a Banach space is mean ergodic, and in the complex case, also λT is mean ergodic for every |λ|=1. We prove that a positive contraction on $L_{1}$ is weakly almost periodic if (and only if) it is mean ergodic. An example shows that without positivity the result is false. In order to construct a contraction T on a complex $L_{1}$ such that λT is mean ergodic whenever |λ|=1, but T is not weakly almost periodic, we prove the following: Let...

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

Vladimir Fonf, Michael Lin, Alexander Rubinov (1996)

Studia Mathematica

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Let T be a power-bounded linear operator in a real Banach space X. We study the equality (*) $(I - T) X = z \in X : s u p_{n} ∥ \sum_{k = 0}^{n} T^{k} z ∥ < \infty$ . For X separable, we show that if T satisfies and is not uniformly ergodic, then $\bar{(I - T) X}$ contains an isomorphic copy of an infinite-dimensional dual Banach space. Consequently, if X is separable and does not contain isomorphic copies of infinite-dimensional dual Banach spaces, then (*) is equivalent to uniform ergodicity. As an application, sufficient conditions for uniform ergodicity of irreducible...

Vector-valued transference and maximal ergodic theory in UMD-valued function spaces.

Naklé H. Asmar, Brian P. Kelly (1996)

Collectanea Mathematica

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On the pointwise ergodic theorems in $L_{p}$ (1

R. Emilion (1985)

Studia Mathematica

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On the mean ergodic theorem for Cesàro bounded operators

Yves Derriennic (2000)

Colloquium Mathematicae

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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...

Remarks on the tightness of cocycles

Jon Aaronson, Benjamin Weiss (2000)

Colloquium Mathematicae

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We prove a generalised tightness theorem for cocycles over an ergodic probability preserving transformation with values in Polish topological groups. We also show that subsequence tightness of cocycles over a mixing probability preserving transformation implies tightness. An example shows that this latter result may fail for cocycles over a mildly mixing probability preserving transformation.

Convergence of the averages and finiteness of ergodic power funtions in weighted L spaces.

Pedro Ortega Salvador (1991)

Publicacions Matemàtiques

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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...

Ergodic properties of skew products with Lasota-Yorke type maps in the base

Zbigniew Kowalski (1993)

Studia Mathematica

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We consider skew products $T (x, y) = (f (x), T_{e (x)} y)$ preserving a measure which is absolutely continuous with respect to the product measure. Here f is a 1-sided Markov shift with a finite set of states or a Lasota-Yorke type transformation and $T_{i}$ , i = 1,..., max e, are nonsingular transformations of some probability space. We obtain the description of the set of eigenfunctions of the Frobenius-Perron operator for T and consequently we get the conditions ensuring the ergodicity, weak mixing and exactness of T....

Vector-valued ergodic theorems for multiparameter additive processes

Ryotaro Sato (1999)

Colloquium Mathematicae

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Let X be a reflexive Banach space and (Ω,Σ,μ) be a σ-finite measure space. Let d ≥ 1 be an integer and T=T(u):u=( $u_{1}$ , ... , $u_{d})$ , $u_{i}$ ≥ 0, 1 ≤ i ≤ d be a strongly measurable d-parameter semigroup of linear contractions on $L_{1}$ ((Ω,Σ,μ);X). We assume that to each T(u) there corresponds a positive linear contraction P(u) defined on $L_{1}$ ((Ω,Σ,μ);ℝ) with the property that ∥ T(u)f(ω)∥ ≤ P(u)∥f(·)∥(ω) almost everywhere on Ω for all f ∈ $L_{1}$ ((Ω,Σ,μ);X). We then prove stochastic and pointwise ergodic theorems...