Entropy of $C(K)$-valued operators and some applications

Preprint series: 99-51, Reports on Analysis

MSC:
41A46 Approximation by arbitrary nonlinear expressions; widths and entropy
47B38 Operators on function spaces (including composition operators, kernel operators)
46B20 Geometry and structure of normed linear spaces

Abstract: We investigate how the entropy numbers $(e_n(T))$ of an arbitrary
Hölder-continuous operator $T:E\to C(K)$ are influenced by the
entropy numbers $(\varepsilon_n(K))$ of the underlying compact metric
space $K$ and the
geometry of $E$ in terms of the (weak) type of its dual.
We derive diverse universal inequalities relating
finitely many $\varepsilon_n(K)$'s with finitely many $e_n(T)$'s.
These inequalities yield asymptotic estimates for the sequence
$(e_n(T))$ which generalize similar results of Carl, Heinrich and
Kühn.
It is shown that these estimates are asymptotically optimal whenever
one can expect. We also discuss necessary conditions on $E$ for the
above inequalities.

Two applications are presented: First we proof an inverse form of
Carl's inequality which relates entropy numbers of operators with
several approximation quantities.
We then consider how the entropy numbers of a precompact subset $A$
of a Banach space $E$ can be used to estimate the entropy numbers of
the absolutely convex hull ${\rm aco} A$ of $A$.
This question was first considered by Dudley
in order to describe universal Donsker classes which play an
important role for certain statistics.
For Banach spaces of type $p$, $p>1$, we proof several inequalities
which estimates $e_n({\rm aco} A)$ by finitely many
$\varepsilon_n(A)$'s.
In particular we complement results of Carl, Kyrezi and Pajor.
It is also shown that these estimates are asymptotically optimal
for some subset $A$ of $E$ whenever $E$ is exactly of type $p$.
As a consequence of one of the inequalities described above
we proof that $(e_n(A))$ and $(e_n({\rm aco} A))$ have the same
asymptotic behaviour if one of these sequences decreases slowly'' and
$E$ is of type $p$ for some $p>1$.
This phenomenon is also discussed for Banach
spaces having no proper type.

Keywords: entropy numbers, metric entropy, Hoelder-continuous operators, convex bodies

Notes: Dissertation 1999 (Prof. B. Carl)