On approximation numbers of composition operators

We show that the approximation numbers of a compact composition operator on the weighted Bergman spaces $\mathfrak{B}_\alpha$ of the unit disk can tend to 0 arbitrarily slowly, but that they never tend quickly to 0: they grow at least exponentially, and this speed of convergence is only obtained for symbols which do not approach the unit circle. We also give an upper bounds and explicit an example

Estimates for approximation numbers of some classes of composition operators on the Hardy space

We give estimates for the approximation numbers of composition operators on $H^2$, in terms of some modulus of continuity. For symbols whose image is contained in a polygon, we get that these approximation numbers are dominated by \e^{- c \sqrt n}. When the symbol is continuous on the closed unit disk and has a domain touching the boundary non-tangentially at a finite number of points, with a good behavior at the boundary around those points, we can improve this upper estimate. A lower estimate is given when this symbol has a good radial behavior at some point. As an application we get that, for the cusp map, the approximation numbers are equivalent, up to constants, to \e^{- c \, n / \log n}, very near to the minimal value \e^{- c \, n}. We also see the limitations of our methods. To finish, we improve a result of O. El-Fallah, K. Kellay, M. Shabankhah and H. Youssfi, in showing that for every compact set $K$ of the unit circle \T with Lebesgue measure 0, there exists a compact composition operator $C_\phi \colon H^2 \to H^2$, which is in all Schatten classes, and such that $\phi = 1$ on $K$ and $|\phi | < 1$ outside $K$

A spectral radius type formula for approximation numbers of composition operators

For approximation numbers $a_n (C_\phi)$ of composition operators $C_\phi$ on weighted analytic Hilbert spaces, including the Hardy, Bergman and Dirichlet cases, with symbol $\phi$ of uniform norm $< 1$, we prove that \lim_{n \to \infty} [a_n (C_\phi)]^{1/n} = \e^{- 1/ \capa [\phi (\D)]}, where \capa [\phi (\D)] is the Green capacity of \phi (\D) in \D. This formula holds also for $H^p$ with $1 \leq p < \infty$.Comment: 25 page

Infinitesimal Carleson property for weighted measures induced by analytic self-maps of the unit disk

We prove that, for every $\alpha > -1$, the pull-back measure $\phi ({\cal A}_\alpha)$ of the measure $d{\cal A}_\alpha (z) = (\alpha + 1) (1 - |z|^2)^\alpha \, d{\cal A} (z)$, where ${\cal A}$ is the normalized area measure on the unit disk \D, by every analytic self-map \phi \colon \D \to \D is not only an $(\alpha + 2)$-Carleson measure, but that the measure of the Carleson windows of size \eps h is controlled by \eps^{\alpha + 2} times the measure of the corresponding window of size $h$. This means that the property of being an $(\alpha + 2)$-Carleson measure is true at all infinitesimal scales. We give an application by characterizing the compactness of composition operators on weighted Bergman-Orlicz spaces

Two remarks on composition operators on the Dirichlet space

We show that the decay of approximation numbers of compact composition operators on the Dirichlet space $\mathcal{D}$ can be as slow as we wish, which was left open in the cited work. We also prove the optimality of a result of O.~El-Fallah, K.~Kellay, M.~Shabankhah and A.~Youssfi on boundedness on $\mathcal{D}$ of self-maps of the disk all of whose powers are norm-bounded in $\mathcal{D}$.Comment: 15 page

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