**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)

**Upload:** 1999-12-14

**Update:** 1999-12-20

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