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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.

The uniform zero-two law for positive operators in Banach lattices

Michael Lin — 1998

Studia Mathematica

Quasi-compactness and uniform ergodicity of Markov operators

Michael Lin — 1975

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

Ergodic properties of an operator obtained from a continuous representation

Michael Lin — 1977

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

Strong ratio limit theorems for mixing Markov operators

Michael Lin — 1976

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

Weighted ergodic theorems along subsequences of density zero.

Jones, Roger L.; Lin, Michael; Olsen, James — 1998

The New York Journal of Mathematics [electronic only]

On the Range of the Generator of a Markovian Semigroup.

Ulrich Krengel; Michael Lin — 1984

Mathematische Zeitschrift

On the Fixed Point Set of Nonexpansive Order Preserving Maps.

Robert Sine; Michael Lin — 1990

Mathematische Zeitschrift

Averages of unitary representations and weak mixing of random walks

Michael Lin; Rainer 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...

Contractive projections on the fixed point set of $L_{\infty}$ contractions

Michael Lin; Robert Sine — 1991

Colloquium Mathematicae

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

Isaac Kornfeld; Michael 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 Cuny; Michael 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...

Convolution powers of spread-out probabilities

Michael Lin; Rainer Wittmann — 1996

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

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

Vladimir Fonf; Michael Lin; Alexander Rubinov — 1996

Studia Mathematica

Let T be a power-bounded linear operator in a real Banach space X. We study the equality (*) $(I - T) X = z \in X : s u p_{n} ∥ \sum_{k = 0}^{n} T^{k} z ∥ < \infty$ . 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 Markov chains...

Consistency of the LSE in Linear regression with stationary noise

Guy Cohen; Michael Lin; Arkady 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 Cohen; Christophe Cuny; Michael 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 $l i m_{n} \sum_{k = 1}^{n} (T^{k} x) / k$ . We prove that weak and strong convergence are equivalent, and in a reflexive space also $s u p_{n} | | \sum_{k = 1}^{n} (T^{k} x) / k | | < \infty$ is equivalent to the convergence. We also show that $- \sum_{k = 1}^{\infty} (T^{k}) / k$ (which converges on (I-T)X) is precisely the infinitesimal generator of the semigroup ${(I - T)}_{| \bar{(I - T) X}}^{r}$ .

Almost everywhere convergence of convolution powers on compact abelian groups

Jean-Pierre Conze; Michael 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 $∥ μ^{n} {* f - \int f d m ∥}_{p} \to 0$ for every $f \in L_{p}$ , every (some) $p \in [1, \infty)$ , if and only if $| \hat{μ} (n) | l t; 1$ for every non-zero $n \in ℤ$ ( $μ$ is strictly aperiodic). In this paper we study the a.e. convergence of $μ^{n} * f$ for every $f \in L_{p}$ 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...

On subinvariant measures for positive operators in L1

Antoine Brunel; Shlomo Horowitz; Michael Lin — 1993

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

Poisson's equation and characterizations of reflexivity of Banach spaces

Vladimir P. Fonf; Michael Lin; Przemysław 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 $x \in X : s u p_{n} | | \sum_{k = 1}^{n} T^{k} x | | < \infty$ = (I-T)X $.$ We then deduce that X (with a basis) is reflexive if and only if every strongly continuous bounded semigroup $T_{t} : t \geq 0$ with generator A satisfies $A X = x \in X : s u p_{s > 0} | | \int_{0}^{s} T_{t} x d t | | < \infty$ . 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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Currently displaying 1 – 19 of 19

Support overlapping $L_{1}$ contractions and exact non-singular transformations

The uniform zero-two law for positive operators in Banach lattices

Quasi-compactness and uniform ergodicity of Markov operators

Ergodic properties of an operator obtained from a continuous representation

Strong ratio limit theorems for mixing Markov operators

Weighted ergodic theorems along subsequences of density zero.

On the Range of the Generator of a Markovian Semigroup.

On the Fixed Point Set of Nonexpansive Order Preserving Maps.

Averages of unitary representations and weak mixing of random walks

Contractive projections on the fixed point set of $L_{\infty}$ contractions

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

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

Convolution powers of spread-out probabilities

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

Consistency of the LSE in Linear regression with stationary noise

The one-sided ergodic Hilbert transform in Banach spaces

Almost everywhere convergence of convolution powers on compact abelian groups

On subinvariant measures for positive operators in L1

Poisson's equation and characterizations of reflexivity of Banach spaces