In analogy to the analyticity condition $\parallel A{e}^{tA}\parallel \le C{t}^{-1}$, t > 0, for a continuous time semigroup ${\left(e^{tA}\right)}_{t\ge 0}$, a bounded operator T is called analytic if the discrete time semigroup ${\left(T^n\right)}_{n\in \mathbb{N}}$ satisfies $\parallel (T-I)T^n\parallel \le C{n}^{-1}$, n ∈ ℕ. We generalize O. Nevanlinna’s characterization of powerbounded and analytic operators T to the following perturbation result: if S is a perturbation of T such that $\parallel R({\lambda }_0,T)-R({\lambda }_0,S)\parallel$ is small enough for some ${\lambda }_0\in \varrho \left(T\right)\cap \varrho \left(S\right)$, then the type $\omega$ of the semigroup $\left(e^{t(S-I)}\right)$ also controls the analyticity of S in the sense that $\parallel (S-I)S^n\parallel \le C(\omega +n^{-1})e^{\omega n}$, n ∈ ℕ. As an application we generalize...

The aim of this note is to describe the Poisson boundary of the group of invertible triangular matrices with coefficients in a number field. It generalizes to any dimension and to any number field a result of Brofferio concerning the Poisson boundary of random rational affinities.

Nous montrons que toute probabilité de transition sur un espace mesurable correspondant à une chaîne de Markov vérifiant la condition de récurrence de Harris, admet au moins un opérateur potentiel positif ; à partir de là, nous développons une théorie du “potentiel logarithmique” pour ces probabilités de transition, en étudiant notamment de manière approfondie un cône de fonctions dites spéciales.