Weakly mixing operators and hypercyclicity sets

On étudie les "fréquences d'hypercyclicité" possibles pour un opérateur non faiblement mélangean

Mixing operators and small subsets of the circle

We provide complete characterizations, on Banach spaces with cotype 2, of those linear operators which happen to be weakly mixing or strongly mixing transformations with respect to some nondegenerate Gaussian measure. These characterizations involve two families of small subsets of the circle: the countable sets, and the so-called sets of uniqueness for Fourier-Stieltjes series. The most interesting part, i.e. the sufficient conditions for weak and strong mixing, is valid on an arbitrary (complex, separable) Fr\'echet space

Smoothness, asymptotic smoothness and the Blum-Hanson property

We isolate various su cient conditions for a Banach space X to have the so-called Blum-Hanson property. In particular, we show that X has the Blum-Hanson property if either the modulus of asymptotic smoothness of X has an extremal behaviour at in nity, or if X is uniformly G^ateaux smooth and embeds isometrically into a Banach space with a 1-unconditional nite-dimensional decomposition

Generic properties of l_p-contractions and similar operator topologies

If $X$ is a separable reflexive Banach space, there are several natural Polish topologies on $\mathcal{B}(X)$, the set of contraction operators on $X$ (none of which being clearly ``more natural'' than the others), and hence several a priori different notions of genericity -- in the Baire category sense -- for properties of contraction operators. So it makes sense to investigate to which extent the generic properties, i.e. the comeager sets, really depend on the chosen topology on $\mathcal{B}(X)$. In this paper, we focus on $\ell_p\,$-$\,$spaces, $1<p\neq 2<\infty$. We show that for some pairs of natural Polish topologies on $\mathcal B_1(\ell_p)$, the comeager sets are in fact the same; and our main result asserts that for $p=3$ or $3/2$ and in the real case, all topologies on $\mathcal B_1(\ell_p)$ lying between the Weak Operator Topology and the Strong$^*$ Operator Topology share the same comeager sets. Our study relies on the consideration of continuity points of the identity map for two different topologies on $\mathcal{B}_1 (\ell_p)$. The other essential ingredient in the proof of our main result is a careful examination of norming vectors for finite-dimensional contractions of a special type.Comment: 39

Rigidity sequences, Kazhdan sets and group topologies on the integers

International audienceWe study the relationships between three different classes of sequences (or sets) of integers, namely rigidity sequences, Kazhdan sequences (or sets) and nullpotent sequences. We prove that rigidity sequences are non-Kazhdan and nullpotent, and that all other implications are false. In particular, we show by probabilistic means that there exist sequences of integers which are both nullpotent and Kazhdan. Moreover, using Baire category methods, we provide general criteria for a sequence of integers to be a rigidity sequence. Finally, we give a new proof of the existence of rigidity sequences which are dense in Z for the Bohr topology, a result originally due to Griesmer

THE BLUM-HANSON PROPERTY FOR C(K) SPACES

8 pagesWe show that if K is a compact metrizable space, then the Banach space C(K) has the so-called Blum-Hanson property exactly when K has finitely many accumulation points. We also show that the space $\ell^\infty$ does not have the Blum-Hanson property

How to recognize a true Σ30Σ_3^0 set

Let X be a Polish space, and let $(A_p)_{p∈ω}$ be a sequence of $G_δ$ hereditary subsets of K(X) (the space of compact subsets of X). We give a general criterion which allows one to decide whether $∪_{p∈ω}A _p$ is a true $∑_3^0$ subset of K(X). We apply this criterion to show that several natural families of thin sets from harmonic analysis are true $∑_3^0$

Complexité de la famille des ensembles de synthèse d'un groupe abélien localement compact

On montre que si G est un groupe abélien localment compact non diskret à base dénombrable d'ouverts, alors la famille des fermés de synthèse pour l'algèbre de Fourier A(G) est une partie coanalytique non borélienne de ℱ(G), l'ensemble des fermés de G muni de la structure borélienne d'Effros. On généralise ainsi un résultat connu dans le cas du groupe