Etienne Matheron

HYPONORMAL OPERATORS, WEIGHTED SHIFTS AND WEAK FORMS OF SUPERCYCLICITY

An operator T on a Banach space X is said to be weakly supercyclic (respectively N -supercyclic) if there exists a one-dimensional (respectively N -dimensional) subspace of X whose orbit under T is weakly dense (respectively norm dense) in X. We show that a weakly supercyclic hyponormal operator is necessarily a multiple of a unitary operator, and we give an example of a weakly supercyclic unitary operator. On the other hand, we show that hyponormal operators are never N -supercyclic. Finally, we characterize N -supercyclic weighted shifts.

Descriptive set Theory of Families of Small Sets

This is a survey paper on the descriptive set theory of hereditary families of closed sets in Polish spaces. Most of the paper is devoted to ideals and σ-ideals of closed or compact sets.

Pyramidal vectors and smooth functions on Banach spaces

We prove that if X, Y are Banach spaces such that Y has nontrivial cotype and X has trivial cotype, then smooth functions from X into Y have a kind of “harmonic” behaviour. More precisely, we show that if Ω is a bounded open subset of X and f : Ω → Y is C1smooth with uniformly continuous Fréchet derivative, then f(∂Ω) is dense in f(Ω). We also give a short proof of a recent result of P. Hájek. This note is motivated by recent results of P. Hájek ([H1], [H2]) concerning (Fréchet) smooth nonlinear operators on the space c0. In [H1], Hájek proved (among other things) that if f : c0 → R is a Csmooth map with uniformly continuous derivative on Bc0 , then f (Bc0) is a relatively compact subset of l1. From this, he deduced that if Y is a Banach space with non trivial type and f : c0 → Y is Csmooth with locally uniformly continuous derivative, then f is locally compact, which means that each point x ∈ c0 has a neighbourhood V such that f(V ) is relatively compact in Y . In [H2], he also proved that the same is true if Y has an unconditional basis and does not contain c0. These striking results are to be compared with another recent theorem, due to S. M. Bates ([B]), according to which for any separable Banach space Y there exists a Csmooth surjection from c0 onto Y ; clearly, such a map cannot be locally compact unless Y is finite-dimensional, by the Baire category theorem. This note is a by-product of several vain attempts to generalize Hájek’s local compactness results to all Banach spaces Y not containing c0. We show that if X , Y are Banach spaces such that Y has finite cotype and X does not have finite cotype, then smooth functions from X into Y have a kind of “harmonic” behaviour (Theorem 1). We also prove that if Y has finite cotype, then smooth functions from c0 into Y essentially turn weakly convergent sequences into (norm) Cesaro-convergent sequences (Theorem 2). Both results rest on an elementary finite-dimensional lemma (Lemma 1) involving what we have called pyramidal vectors of c0 (Definition 1). Finally, we give a very short proof of Hájek’s basic result for scalar-valued functions, which looks rather different (at least in its form) from the original one. This proof is based on the notion of strong sequential continuity (Definition 3), which might be of independent interest. Let us now fix the notation that will be used throughout this note. The letters X, Y will always designate (real) Banach spaces. If Z is a normed space, we denote Received by the editors February 19, 1999. 2000 Mathematics Subject Classification. Primary 46B20. c ©2000 American Mathematical Society

Strongly Sequentially Continuous Functions

Abstract Given a topological abelian group G, we study the class of strongly sequentially continuous functions on G. Strong sequential continuity is a property intermediate between sequential continuity and uniform sequential continuity, which appeared naturally in the study of smooth functions on Banach spaces. In this paper, we shall mainly concentrate on the gap between strong sequential continuity and uniform sequential continuity. It turns out that if G has some completeness property—for example, if it is completely metrizable—then all strongly sequentially continuous functions on G are uniformly sequentially continuous. On the other hand, we exhibit a large and natural class of groups for which the two notions differ. This class is defined by a property reminiscent of the classical Dirichlet theorem; it includes all dense sugroups of R generated by an increasing sequence of Dirichlet sets, and groups of the form (X, w), where X is a separable Banach space failing the Schur property. Finally, we show that the family of bounded, real-valued strongly sequentially continuous functions on G is a closed subalgebra of l∞(G).

A useful lemma concerning subseries convergence

We give simple and almost identical proofs of several classical results in Functional Analysis by means of a single lemma concerning subseries convergence.