42 research outputs found
On approximation numbers of composition operators
We show that the approximation numbers of a compact composition operator on
the weighted Bergman spaces 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
, 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
of the unit circle \T with Lebesgue measure 0, there exists a compact
composition operator , which is in all Schatten
classes, and such that on and outside
A spectral radius type formula for approximation numbers of composition operators
For approximation numbers of composition operators on
weighted analytic Hilbert spaces, including the Hardy, Bergman and Dirichlet
cases, with symbol of uniform norm , 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 with .Comment: 25 page
Infinitesimal Carleson property for weighted measures induced by analytic self-maps of the unit disk
We prove that, for every , the pull-back measure of the measure , where is the normalized area
measure on the unit disk \D, by every analytic self-map \phi \colon \D \to
\D is not only an -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 . This means that the
property of being an -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 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
of self-maps of the disk all of whose powers are norm-bounded in
.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 of the unit disk and an Orlicz
function for which the composition operator of symbol is compact
on the Hardy-Orlicz space , but not compact on the Bergman-Orlicz space
. 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