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Support overlapping L 1 contractions and exact non-singular transformations

Michael Lin — 2000

Colloquium Mathematicae

Let T be a positive linear contraction of L 1 of a σ-finite measure space (X,Σ,μ) which overlaps supports. In general, T need not be completely mixing, but it is in the following cases: (i) T is the Frobenius-Perron operator of a non-singular transformation ϕ (in which case complete mixing is equivalent to exactness of ϕ). (ii) T is a Harris recurrent operator. (iii) T is a convolution operator on a compact group. (iv) T is a convolution operator on a LCA group.

Averages of unitary representations and weak mixing of random walks

Michael LinRainer Wittmann — 1995

Studia Mathematica

Let S be a locally compact (σ-compact) group or semigroup, and let T(t) be a continuous representation of S by contractions in a Banach space X. For a regular probability μ on S, we study the convergence of the powers of the μ-average Ux = ʃ T(t)xdμ(t). Our main results for random walks on a group G are: (i) The following are equivalent for an adapted regular probability on G: μ is strictly aperiodic; U n converges weakly for every continuous unitary representation of G; U is weakly mixing for any...

Weak almost periodicity of L 1 contractions and coboundaries of non-singular transformations

Isaac KornfeldMichael Lin — 2000

Studia Mathematica

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 τ be an invertible...

Pointwise ergodic theorems with rate and application to the CLT for Markov chains

Christophe CunyMichael Lin — 2009

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

Let be Dunford–Schwartz operator on a probability space (, ). For ∈ (), >1, we obtain growth conditions on ‖∑ ‖ which imply that (1/ )∑ →0 -a.e. In the particular case that =2 and is the isometry induced by a probability preserving transformation we get better results than in the general case; these are used to obtain a quenched central...

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

Vladimir FonfMichael LinAlexander Rubinov — 1996

Studia Mathematica

Let T be a power-bounded linear operator in a real Banach space X. We study the equality (*) ( I - T ) X = z X : s u p n k = 0 n T k z < . For X separable, we show that if T satisfies and is not uniformly ergodic, then ( 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 Markov chains...

Consistency of the LSE in Linear regression with stationary noise

Guy CohenMichael LinArkady Tempelman — 2004

Colloquium Mathematicae

We obtain conditions for L₂ and strong consistency of the least square estimators of the coefficients in a multi-linear regression model with a stationary random noise. For given non-random regressors, we obtain conditions which ensure L₂-consistency for all wide sense stationary noise sequences with spectral measure in a given class. The condition for the class of all noises with continuous (i.e., atomless) spectral measures yields also L p -consistency when the noise is strict sense stationary with...

The one-sided ergodic Hilbert transform in Banach spaces

Guy CohenChristophe CunyMichael Lin — 2010

Studia Mathematica

Let T be a power-bounded operator on a (real or complex) Banach space. We study the convergence of the one-sided ergodic Hilbert transform l i m n k = 1 n ( T k x ) / k . We prove that weak and strong convergence are equivalent, and in a reflexive space also s u p n | | k = 1 n ( T k x ) / k | | < is equivalent to the convergence. We also show that - k = 1 ( T k ) / k (which converges on (I-T)X) is precisely the infinitesimal generator of the semigroup ( I - T ) | ( I - T ) X ¯ r .

Almost everywhere convergence of convolution powers on compact abelian groups

Jean-Pierre ConzeMichael Lin — 2013

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

It is well-known that a probability measure μ on the circle 𝕋 satisfies μ n * f - f d m p 0 for every f L p , every (some) p [ 1 , ) , if and only if | μ ^ ( n ) | l t ; 1 for every non-zero n ( μ is strictly aperiodic). In this paper we study the a.e. convergence of μ n * f for every f L p whenever p g t ; 1 . We prove a necessary and sufficient condition, in terms of the Fourier–Stieltjes coefficients of μ , for the strong sweeping out property (existence of a Borel set B with lim sup μ n * 1 B = 1 a.e. and lim inf μ n * 1 B = 0 a.e.). The results are extended to general compact Abelian groups G with Haar...

Poisson's equation and characterizations of reflexivity of Banach spaces

Vladimir P. FonfMichael LinPrzemysław Wojtaszczyk — 2011

Colloquium Mathematicae

Let X be a Banach space with a basis. We prove that X is reflexive if and only if every power-bounded linear operator T satisfies Browder’s equality x X : s u p n | | k = 1 n T k x | | < = (I-T)X . We then deduce that X (with a basis) is reflexive if and only if every strongly continuous bounded semigroup T t : t 0 with generator A satisfies A X = x X : s u p s > 0 | | 0 s T t x d t | | < . The range (I-T)X (respectively, AX for continuous time) is the space of x ∈ X for which Poisson’s equation (I-T)y = x (Ay = x in continuous time) has a solution y ∈ X; the above equalities for the ranges...

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