Two problems posed by Choquet and Foias are solved:(i) Let $T$ be a positive linear operator on the space $C\left(X\right)$ of continuous real-valued functions on a compact Hausdorff space $X$. It is shown that if $n^{-1}{\sum }_{r=0}^{n-1}{T}^{r}1$ converges pointwise to a continuous limit, then the convergence is uniform on $X$.(ii) An example is given of a Choquet simplex $K$ and a positive linear operator $T$ on the space $A\left(K\right)$ of continuous affine real-valued functions on $K$, such that$$\inf \left\{\right(T^{n}1\left)\right(x):n\ge \}\<1$$for each $x$ in ${\partial }_{\ell }K$, but $\parallel T^{n}1\parallel$ does not converge to 0.

We extend the notion of Dobrushin coefficient of ergodicity to positive contractions defined on the L¹-space associated with a finite von Neumann algebra, and in terms of this coefficient we prove stability results for L¹-contractions.

For a given linear operator T in a complex Banach space X and α ∈ ℂ with ℜ (α) > 0, we define the nth Cesàro mean of order α of the powers of T by $M{ₙ}^{\alpha }={\left(A{ₙ}^{\alpha }\right)}^{-1}{\sum }_{k=0}^n{A}_{n-k}^{\alpha -1}{T}^k$. For α = 1, we find $Mₙ¹={(n+1)}^{-1}{\sum }_{k=0}^n{T}^k$, the usual Cesàro mean. We give necessary and sufficient conditions for a (C,α) bounded operator to be (C,α) strongly (weakly) ergodic.

We improve a recent result of T. Yoshimoto about the uniform ergodic theorem with Cesàro means of order α. We give a necessary and sufficient condition for the (C,α) uniform ergodicity with α > 0.

Let A denote a complex unital Banach algebra. We characterize properties such as boundedness, relative compactness, and convergence of the sequence ${x^n(x-1)}_{n\in \mathbb{N}}$ for an arbitrary x ∈ A, using σ(x) and resolvent conditions. Under these circumstances, we investigate elements in the peripheral spectrum, and give further conclusions, also involving the behaviour of ${x^n}_{n\in \mathbb{N}}$ and ${1/n{\sum }_{k=0}^{n-1}{x}^k}_{n\in \mathbb{N}}$.

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 theory, are...

For an arbitrary set $A\subseteq {\mathbb{N}}^{+}$ containing 1, an ergodic automorphism T whose set of essential values of the multiplicity function is equal to A is constructed. If A is additionally finite, T can be chosen to be an analytic diffeomorphism on a finite-dimensional torus.