459 research outputs found
Old and new results on normality
We present a partial survey on normal numbers, including Keane's
contributions, and with recent developments in different directions.Comment: Published at http://dx.doi.org/10.1214/074921706000000248 in the IMS
Lecture Notes--Monograph Series
(http://www.imstat.org/publications/lecnotes.htm) by the Institute of
Mathematical Statistics (http://www.imstat.org
Approximation numbers of composition operators on the space of Dirichlet series
By a theorem of Gordon and Hedenmalm, generates a bounded
composition operator on the Hilbert space of Dirichlet series
with square-summable coefficients if and only if
, where is a nonnegative integer and a
Dirichlet series with the following mapping properties: maps the right
half-plane into the half-plane if and is
either identically zero or maps the right half-plane into itself if is
positive. It is shown that the th approximation numbers of bounded
composition operators on are bounded below by a constant times
for some when and bounded below by a constant times
for some when is positive. Both results are best possible.
The case when , is bounded and smooth up to the boundary of the
right half-plane, and , is discussed in depth;
it includes examples of non-compact operators as well as operators belonging to
all Schatten classes . For
with independent integers, it is shown that the th approximation
number behaves as , possibly up to a factor .
Estimates rely mainly on a general Hilbert space method involving finite linear
combinations of reproducing kernels. A key role is played by a recently
developed interpolation method for using estimates of solutions
of the equation. Finally, by a transference principle from
of the unit disc, explicit examples of compact composition operators with
approximation numbers decaying at essentially any sub-exponential rate can be
displayed.Comment: Final version, to appear in Journal of Functional Analysi
Approximation numbers of composition operators on spaces of Dirichlet series
By a theorem of Bayart, generates a bounded composition operator on
the Hardy space \Hpof Dirichlet series () only if
, where is a nonnegative integer and a
Dirichlet series with the following mapping properties: maps the right
half-plane into the half-plane \Real s >1/2 if and is either
identically zero or maps the right half-plane into itself if is positive.
It is shown that the th approximation numbers of bounded composition
operators on \Hp are bounded below by a constant times for some
when and bounded below by a constant times for some when
is positive. Both results are best possible. Estimates rely on a
combination of soft tools from Banach space theory (-numbers, type and
cotype of Banach spaces, Weyl inequalities, and Schauder bases) and a certain
interpolation method for \Ht, developed in an earlier paper, using estimates
of solutions of the equation. A transference principle
from of the unit disc is discussed, leading to explicit examples of
compact composition operators on \Ho with approximation numbers decaying at a
variety of sub-exponential rates. Finally, a new Littlewood--Paley formula is
established, yielding a sufficient condition for a composition operator on
\Hp to be compact.Comment: This is the final version of the paper, to appear in Annales de
l'Institut Fourie
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
Approximation numbers of composition operators on the Hardy space of the ball and of the polydisk
We give general estimates for the approximation numbers of composition
operators on the Hardy space on the ball and the polydisk
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
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