International audienceWe prove that l(2) contains vectors which are hypercyclic simultaneously for all multiples of the backward shift operator by constants of absolute value greater than 1. The set of such vectors is dense G(delta). (C) 2002 Elsevier Science (USA). All rights reserved

Abstract. We study the localization of zeros for Cauchy transforms of discrete measures on the real line. This question is motivated by the theory of canonical systems of differential equations. In particular, we prove that the spaces of Cauchy transforms having the localization property are in one-to-one correspondence with the canonical systems of special type, namely, those whose Hamiltonians consist only of indivisible intervals accumulating on the left. Various aspects of the localization phenomena are studied in details. Connections with the density of polynomials and other topics in analysis are discussed.

Let H (B d ) denote the space of holomorphic functions on the unit ball B d of C d . Given a radial doubling weight w, we construct functions f, g ∈ H (B 1 ) such that | f | + |g| is comparable to w. Also, we obtain similar results for B d , d ≥ 2, and for circular, strictly convex domains with smooth boundary. As an application, we study weighted composition operators and related integral operators on growth spaces of holomorphic functions.

We consider an infinite locally finite tree $T$ \ud equipped with nearest neighbor transition coefficients, giving rise to a space of harmonic functions. We show that, except for trivial cases, the generic harmonic function on $T$ has dense range in the complex numbers. By looking at forward-only transition coefficients, we show that the generic harmonic function induces a boundary martingale that approximates in probability all measurable functions on the boundary of $T$. We also study algebraic genericity,\ud spaceability and frequent universality of these phenomena

We extend some results of M.G. Krein to the class of entire functions which can be represented as ratios of discrete Cauchy transforms in the plane. As an application we obtain new versions of de Branges' Ordering Theorem for nearly invariant subspaces in a class of Hilbert spaces of entire functions. Examples illustrating sharpness of the obtained results are given.

Let Hol(D) denote the space of holomorphic functions on the unit disk D. We characterize those radial weights w on D for which there exist functions f, g ∈ Hol(D) such that the sum |f | + |g| is equivalent to w. Also, we obtain similar results in several complex variables for circular, strictly convex domains with smooth boundary.

We give estimates on the logarithmic Sobolev constant of some finite lamplighter graphs in terms of the spectral gap of the underlying base. Also, we give examples of application.

We construct right shift invariant subspaces of index n, 1 [ n [ ., in a p spaces, 2 < p < ., and in weighted a p spaces. © 2002 Elsevier Science (USA)