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Topological Dichotomy and Unconditional Convergence

Lefevre, Pascal — 1999

Serdica Mathematical Journal

In this paper, we give a criterion for unconditional convergence with respect to some summability methods, dealing with the topological size of the set of choices of sign providing convergence. We obtain similar results for boundedness. In particular, quasi-sure unconditional convergence implies unconditional convergence.

Essential Norms of Weighted Composition Operators on L^1 Müntz Spaces

Al Alam, IhabLefèvre, Pascal — 2014

Serdica Mathematical Journal

This paper discusses the problem of boundedness and compactness for weighted composition operators defined on a Müntz subspace of L^1([0,1]). We compute the essential norm of such operators when the symbol ϕ of the composition operator satisfies a special condition (condition (β)). As a corollary, we obtain the exact values of essential norms of composition and multiplication operators. This completes the corresponding results of the first named author in the framework of Müntz subspaces of C([0,...

On some properties of the class of stationary sets

Pascal Lefevre — 1998

Colloquium Mathematicae

Some new properties of the stationary sets (defined by G. Pisier in [12]) are studied. Some arithmetical conditions are given, leading to the non-stationarity of the prime numbers. It is shown that any stationary set is a set of continuity. Some examples of "large" stationary sets are given, which are not sets of uniform convergence.

Measures and lacunary sets

Pascal Lefèvre — 1999

Studia Mathematica

We establish new connections between some classes of lacunary sets. The main tool is the use of (p,q)-summing or weakly compact operators (for Riesz sets). This point of view provides new properties of stationary sets and allows us to generalize to more general abelian groups than the torus some properties of p-Sidon sets. We also construct some new classes of Riesz sets.

Some remarks on quasi-Cohen sets

Pascal LefèvreDaniel Li — 2001

Colloquium Mathematicae

We are interested in Banach space geometry characterizations of quasi-Cohen sets. For example, it turns out that they are exactly the subsets E of the dual of an abelian compact group G such that the canonical injection C ( G ) / C E c ( G ) L ² E ( G ) is a 2-summing operator. This easily yields an extension of a result due to S. Kwapień and A. Pełczyński. We also investigate some properties of translation invariant quotients of L¹ which are isomorphic to subspaces of L¹.

Some Banach spaces of Dirichlet series

Maxime BailleulPascal Lefèvre — 2015

Studia Mathematica

The Hardy spaces of Dirichlet series, denoted by p (p ≥ 1), have been studied by Hedenmalm et al. (1997) when p = 2 and by Bayart (2002) in the general case. In this paper we study some L p -generalizations of spaces of Dirichlet series, particularly two families of Bergman spaces, denoted p and p . Each could appear as a “natural” way to generalize the classical case of the unit disk. We recover classical properties of spaces of analytic functions: boundedness of point evaluation, embeddings between...

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