Compact composition operators on the Dirichlet space and capacity of sets of contact points

In this paper, we prove that for every compact set of the unit disk of logarithmic capacity 0, there exists a Schur function both in the disk algebra and in the Dirichlet space such that the associated composition operator is in all Schatten classes (of the Dirichlet space), and for which the set of points whose image touches the unit circle is equal to this compact set. We show that for every bounded composition operator on the Dirichlet space and for every point of the unit circle, the logarithmic capacity of the set of point having this point as image is 0. We show that every compact composition operator on the Dirichlet space is compact on the gaussian Hardy-Orlicz space; in particular, it is in every Schatten class on the usual Hilbertian Hardy space. On the other hand, there exists a Schur function such that the associated composition operator is compact on the gaussian Hardy-Orlicz space, but which is not even bounded on the Dirichlet space. We prove that the Schatten classes on the Dirichlet space can be separated by composition operators. Also, there exists a Schur function such that the associated composition operator is compact on the Dirichlet space, but in no Schatten class

Compact composition operators on Bergman-Orlicz spaces

We construct an analytic self-map $\phi$ of the unit disk and an Orlicz function $\Psi$ for which the composition operator of symbol $\phi$ is compact on the Hardy-Orlicz space $H^\Psi$, but not compact on the Bergman-Orlicz space ${\mathfrak B}^\Psi$. For that, we first prove a Carleson embedding theorem, and then characterize the compactness of composition operators on Bergman-Orlicz spaces, in terms of Carleson function (of order 2). We show that this Carleson function is equivalent to the Nevanlinna counting function of order 2.Comment: 32 page

Thin sets of integers in Harmonic analysis and p-stable random Fourier series

We investigate the behavior of some thin sets of integers defined through random trigonometric polynomial when one replaces Gaussian or Rademacher variables by p-stable ones, with 1 < p < 2. We show that in one case this behavior is essentially the same as in the Gaussian case, whereas in another case, this behavior is entirely different

The canonical injection of the Hardy-Orlicz space HΨH^\Psi into the Bergman-Orlicz space BΨ{\mathfrak B}^\Psi

We study the canonical injection from the Hardy-Orlicz space $H^\Psi$ into the Bergman-Orlicz space ${\mathfrak B}^\Psi$.Comment: 21 page

Weak compactness and Orlicz spaces

We give new proofs that some Banach spaces have Pe{\l}czy\'nski's property $(V)$

Some new properties of composition operators associated with lens maps

We give examples of results on composition operators connected with lens maps. The first two concern the approximation numbers of those operators acting on the usual Hardy space $H^2$. The last ones are connected with Hardy-Orlicz and Bergman-Orlicz spaces $H^\psi$ and $B^\psi$, and provide a negative answer to the question of knowing if all composition operators which are weakly compact on a non-reflexive space are norm-compact.Comment: 21 page

Some revisited results about composition operators on Hardy spaces

We generalize, on one hand, some results known for composition operators on Hardy spaces to the case of Hardy-Orlicz spaces $H^\Psi$: construction of a "slow" Blaschke product giving a non-compact composition operator on $H^\Psi$; construction of a surjective symbol whose composition operator is compact on $H^\Psi$ and, moreover, is in all the Schatten classes $S_p (H^2)$, $p > 0$. On the other hand, we revisit the classical case of composition operators on $H^2$, giving first a new, and simplier, characterization of closed range composition operators, and then showing directly the equivalence of the two characterizations of membership in the Schatten classes of Luecking and Luecking and Zhu.Comment: 21 page

La modelización y simulación computacional como metodología de investigación social

il presente trabajo se propone brindar una introducción crítica y sistemática a la metodología de modelado y simulación computacional en ciencias sociales. El uso de métodos computacionales para simular procesos sociales se ha expandido con vigor en los últimos veinte años en las distintas disciplinas de las ciencias sociales; sin embargo, continúa siendo una metodología poco conocida y empleada por los investigadores sociales en América Latina. El objetivo de este trabajo es contribuir a la reflexión epistemológica, teórica y metodológica de la simulación computacional en ciencias sociales, atendiendo a la dimensión política que atraviesa toda praxis científica

Confinement of Spin and Charge in High-Temperature Superconductors

By exploiting the internal gauge-invariance intrinsic to a spin-charge separated electron, we show that such degrees of freedom must be confined in two-dimensional superconductors experiencing strong inter-electron repulsion. We also demonstrate that incipient confinement in the normal state can prevent chiral spin-fluctuations from destroying the cross-over between strange and psuedo-gap regimes in under-doped high-temperature superconductors. Last, we suggest that the negative Hall anomaly observed in these materials is connected with this confinement effect.Comment: 12 pages, 1 postscript figure, to appear in PRB (RC), May 199