A Borel subset of the unit square whose vertical and horizontal sections are two-point sets admits a natural group action. We exploit this to discuss some questions about Borel subsets of the unit square on which every function is a sum of functions of the coordinates. Connection with probability measures with prescribed marginals and some function algebra questions is discussed.

We consider some descriptive properties of supports of shift invariant measures on ${ℂ}^{\mathbb{Z}}$ under the assumption that the closed linear span (in $L^2$) of the co-ordinate functions on ${ℂ}^{\mathbb{Z}}$ is all of $L^2$.

Soit $T$ un opérateur linéaire positif sur $𝒞\left(K\right)$ (où $K$ est un compact). On montre que si inf. $\{T_1^n;n\>0\}\<1$, la suite des $(T^n$) converge uniformément vers 0, et que si sup. $\{T_1^n;n\>0\}\>1$ la suite des $\left(T^n\right)$ converge uniformément vers $+\infty$.Puis on applique ces deux énoncés à l’étude des suites : $\left[{\sum }_0^{n-1}{T}^{i}f\right]/n$ et $(T^{n}f{)}^{1/n}$ ; on donne en particulier plusieurs critères de convergence uniforme de ces suites.

The notion of local mean ergodicity is introduced. Some general locally mean ergodic theorems for linear and affine operators are presented. Locally mean ergodic theorems for affine operators whose linear parts are compact or similar to subnormal operators on a Hilbert space are given.

Let U be a trigonometrically well-bounded operator on a Banach space , and denote by ${ₙ\left(U\right)}_{n=1}^{\infty }$ the sequence of (C,2) weighted discrete ergodic averages of U, that is, $ₙ\left(U\right)=1/n{\sum }_{0<\left|k\right|\le n}(1-|k|/(n+1))U^k$. We show that this sequence ${ₙ\left(U\right)}_{n=1}^{\infty }$ of weighted ergodic averages converges in the strong operator topology to an idempotent operator whose range is x ∈ : Ux = x, and whose null space is the closure of (I - U). This result expands the scope of the traditional Ergodic Theorem, and thereby serves as a link between Banach space spectral theory and...

For a bounded linear operator T in a Banach space the Ritt resolvent condition $\parallel R_{\lambda }\left(T\right)\parallel \le C/|\lambda -1|$ (|λ| > 1) can be extended (changing the constant C) to any sector |arg(λ - 1)| ≤ π - δ, $arccos\left(C^{-1}\right)<\delta <\pi /2$. This implies the power boundedness of the operator T. A key result is that the spectrum σ(T) is contained in a special convex closed domain. A generalized Ritt condition leads to a similar localization result and then to a theorem on invariant subspaces.

On reflexive spaces trigonometrically well-bounded operators have an operator-ergodic-theory characterization as the invertible operators U such that $su{p}_{n\in \mathbb{N},z\in }\left|\right|{\sum }_{0<\left|k\right|\le n}(1-|k|/(n+1))k^{-1}{z}^k{U}^k\left|\right|<\infty$. (*) Trigonometrically well-bounded operators permeate many settings of modern analysis, and this note highlights the advances in both their spectral theory and operator ergodic theory made possible by a recent rekindling of interest in the R. C. James inequalities for super-reflexive spaces. When the James inequalities are combined with Young-Stieltjes...

The classical Banach principle is an essential tool for the investigation of ergodic properties of Cesàro subsequences. The aim of this work is to extend the Banach principle to the case of stochastic convergence in operator algebras. We start by establishing a sufficient condition for stochastic convergence (stochastic Banach principle). Then we prove stochastic convergence for bounded Besicovitch sequences, and as a consequence for uniform subsequences.

We show that the set of those Markov semigroups on the Schatten class ₁ such that in the strong operator topology $li{m}_{t\to \infty }T\left(t\right)=Q$, where Q is a one-dimensional projection, form a meager subset of all Markov semigroups.

Let $T:L^p\left(\Omega \right)\to L^p\left(\Omega \right)$ be a positive contraction, with $1\&lt;p\&lt;\infty$. Assume that $T$ is analytic, that is, there exists a constant $K\ge 0$ such that $\parallel T^n-T^{n-1}\parallel \le K/n$ for any integer $n\ge 1$. Let $2\&lt;q\&lt;\infty$ and let $v^q$ be the space of all complex sequences with a finite strong $q$-variation. We show that for any $x\in L^p\left(\Omega \right)$, the sequence ${\left(\left[T^n\left(x\right)\right]\left(\lambda \right)\right)}_{n\ge 0}$ belongs to $v^q$ for almost every $\lambda \in \Omega$, with an estimate $\parallel {\left(T^n\left(x\right)\right)}_{n\ge 0}{\parallel }_{L^p\left(v^q\right)}\le C{\parallel x\parallel }_p$. If we remove the analyticity assumption, we obtain an estimate $\parallel {\left(M_n\left(T\right)x\right)}_{n\ge 0}{\parallel }_{L^p\left(v^q\right)}\le C{\parallel x\parallel }_p$, where $M_n\left(T\right)={(n+1)}^{-1}{\sum }_{k=0}^n{T}^k$ denotes the ergodic average of $T$. We also obtain similar results for strongly continuous semigroups ${\left(T_t\right)}_{t\ge 0}$ of positive...