We discuss several convergence and ergodic properties of iterates of some bounded linear operators on Banach or Hilbert spaces, with emphasis on the unconditional Ritt property.

We study the parabolic linear fractional maps of the unit ball of C^d. We suggest a classification of these maps and apply this classification to compute their characteristic domain. This helps us to understand the composition operators associated to parabolic linear fractional maps, at least from the point of view of their spectrum and of their dynamics.

Let H be a Hilbert space and T_i be a bounded linear operator on H for 1 £ i £ n. The n-tuple T = (T₁,...,T_n) is a multicontraction if å_i=1ⁿ T_i T_i^* £ \1_H.

The Poisson kernel and the characteristic function are two important objects associated to a multicontraction T in Popescu's theory. We give their behaviour with respect to the action of the group of unitarily implemented automorphisms of the algebra generated by creation operators on the Fock space.

Some approximation properties of the weighted Hardy space for the unit disc with the weight function satisfying Muckenhoupt's (A^p) condition will be presented.

Spectral theory is a very efficient tool for estimating functions of a self-adjoint operator, but the situation is much more difficult if we consider non-normal operators. We present some remarks which have been of use to us and illustrate them by the following result:

25 years ago, Rieffel introduced an algebraic invariant for Banach algebras called topological stable rank which generalized the notion of dimension to the non-commutative setting. For example, if X is a compact Hausdorff space of dimension n, then tsr ( C_R(X) ) = n and with complex scalars tsr ( C(X) ) = ën/2 û+ 1. The topological stable rank has a left and right version, which coincide for C^*-algebras and commutative algebras. Moreover, tsr is a Banach algebra variant of the purely algebraic invariant of Bass stable rank for rings-and the left and right versions of Bass stable rank are always equal. So Rieffel asked whether they are always equal?

We have calculated the left and right topological stable ranks for the class of nest algebras, and can answer Rieffel's question negatively.

Soit T un opérateur linéaire borné sur un espace de Hilbert H. On note Lat(T) le treillis des sous espaces fermés de H invariants par T, AlgLat(T) l'algèbre des opérateurs S bornés sur H tel que Lat(T) Ì Lat(S) et W(T) la fermeture (pour la topologie faible des opérateurs) des polynômes en T. On dit que T est reflexif si AlgLat(T) = W(T). Dans cet exposé nous discuterons la réflexivité des extensions d'opérateurs réflexifs par des opérateurs algèbriques. Nous donnerons des exemples d'opérateurs dont toute extension par un opérateur algèbrique réflexif reste réflexive.

Nous discuterons d'une question récente de Brézis et Korevaar concernant certaines suites de nombres complexes de carré sommable ainsi que de certaines contributions récentes de Kahane concernant cette question.

Il s'agit d'un travail conjoint avec Luis Salinas (Chili).

Dans cet exposé, nous étudions la propriété de base de Riesz pour les familles de noyaux reproduisants dans les espaces de de Branges-Rovnyak à valeurs vectorielles. Nous donnerons également des résultats pour la complétude d'une certaine famille de fonctions.

If T is a bounded operator on a complex Banach space X and T_n denotes the direct sum TżÅT of n copies of T acting on XżÅX, we study the sequence ( m(T_n) )_{n ³ 1} of the multiplicities of the operators T_n. Answering a question of Atzmon, we show that this sequence is either eventually constant or grows to infinity at least as fast as n. Then we construct examples of operators on Hilbert spaces such that m(T_n)=d for every n ³ 1, where d is an arbitrary positive integer. This answers a question of Herrero and Wogen and characterizes convex sequences which can be realized as a sequence ( m(T_n) )_{n ³ 0} for some operator T on a Hilbert space.

This is joint work with Maria Roginskaya.

Soit T une contraction essentiellement unitaire de classe C₀ et soit f une fonction borné dans le disque unité. Nous montrons que f(T) est compact si et seulement si Tⁿ f(T)® 0. De plus si T est cyclique, inversible et f est une fonction de l'algèbre du disque, nous montrons que f(T) est compact si et seulement si f s'annule sur le spectre de T. Ceci donne une géneralisation du Théorème de Atzmon-Isaeev lorsque T est cyclique, inversible et le spectre de T est réduit à un point.

Ce travail est en collaboration avec Mohammed Zarrabi.

We will give an overview of recent results on spectral behavior of discrete Schrödinger operators (Jacobi matrices) in one dimension and on regular infinite trees (Bethe lattices, also known as Cayley trees).

The notions of left- and of right-topological stable rank for a Banach algebra were introduced by M. Rieffel in 1983 as a generalisation of the notion of the covering dimension of a topological space. For many classes of Banach algebras, including C^*-algebras and abelian Banach algebras, the two notions coincide. This led Rieffel to pose the question of whether or not the two stable ranks always agree. In this talk, we shall answer this question by studying the topological stable rank of nest algebras acting on infinite-dimensional, complex, separable Hilbert spaces. These are algebras which serve as an infinite-dimensional generalisation of the algebra of upper-triangular n ×n complex matrices.

This is joint work with K. R. Davidson, R. Levene, and H. Radjavi.