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.

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...

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...

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...

Let T be a power-bounded linear operator in a real Banach space X. We study the equality (*) $(I-T)X=z\in X:su{p}_{n}\parallel {\sum}_{k=0}^{n}{T}^{k}z\parallel <\infty $. For X separable, we show that if T satisfies and is not uniformly ergodic, then $\overline{(I-T)X}$ 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...

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...

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 $li{m}_{n}{\sum}_{k=1}^{n}\left({T}^{k}x\right)/k$. We prove that weak and strong convergence are equivalent, and in a reflexive space also $su{p}_{n}\left|\right|{\sum}_{k=1}^{n}\left({T}^{k}x\right)/k\left|\right|<\infty $ is equivalent to the convergence. We also show that $-{\sum}_{k=1}^{\infty}\left({T}^{k}\right)/k$ (which converges on (I-T)X) is precisely the infinitesimal generator of the semigroup ${(I-T)}_{|\overline{(I-T)X}}^{r}$.

It is well-known that a probability measure $\mu $ on the circle $\mathbb{T}$ satisfies $\parallel {\mu}^{n}{*f-\int f\phantom{\rule{0.166667em}{0ex}}\mathrm{d}m\parallel}_{p}\to 0$ for every $f\in {L}_{p}$, every (some) $p\in [1,\infty )$, if and only if $|\widehat{\mu}\left(n\right)|lt;1$ for every non-zero $n\in \mathbb{Z}$ ($\mu $ is strictly aperiodic). In this paper we study the a.e. convergence of ${\mu}^{n}*f$ for every $f\in {L}_{p}$ whenever $pgt;1$. We prove a necessary and sufficient condition, in terms of the Fourier–Stieltjes coefficients of $\mu $, for the strong sweeping out property (existence of a Borel set $B$ with $lim\; sup{\mu}^{n}*{1}_{B}=1$ a.e. and $lim\; inf{\mu}^{n}*{1}_{B}=0$ a.e.). The results are extended to general compact Abelian groups $G$ with Haar...

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:su{p}_{n}\left|\right|{\sum}_{k=1}^{n}{T}^{k}x\left|\right|<\infty $ = (I-T)X$.$We then deduce that X (with a basis) is reflexive if and only if every strongly continuous bounded semigroup ${T}_{t}:t\ge 0$ with generator A satisfies
$AX=x\in X:su{p}_{s>0}\left|\right|{\int}_{0}^{s}{T}_{t}xdt\left|\right|<\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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