# Álgebras de funciones analíticas acotadas. Interpolación

Abstract (Summary)

ABSTRACT
The lines studied in this thesis are the following:
² Interpolating Sequences for Uniform Algebras
² Composition Operators
² Topological Properties in Algebras of Analytic Functions
After the preliminaries, the second chapter is devoted to the study of
interpolating sequences on uniform algebras A. We ¯rst deal with the con-
nection between interpolating sequences and linear interpolating sequences.
Next, we deal with dual uniform algebras A = X¤. In this context, we
prove ¯rst that c0¡linear interpolating sequences are linear interpolating
and then, we show that c0¡interpolating sequences are, indeed, c0¡linear
interpolating, obtaining that c0¡interpolating sequences (xn) ½ MA X
become linear interpolating. Finally, we provide a di®erent approach to
prove that c0¡interpolating sequences are not c0¡linear interpolating via
composition operators.
We continue with the study of interpolating sequences for the algebras
of analytic functions H1(BE) and A1(BE) in the third chapter. The study
of interpolating sequences for H1 arises from the results of L. Carleson, W.
K. Hayman and D. J. Newman. When we deal with general Banach spaces,
we prove that the Hayman-Newman condition for the sequence of norms is
su±cient for a sequence (xn) ½ BE¤¤ to be interpolating for H1(BE) if E
is any ¯nite or in¯nite dimensional Banach space. This is a consequence of
a stronger result :
The Carleson condition for the sequence (kxnk) ½ D is su±cient for
(xn) to be interpolating for H1(BE).
Actually, the result holds for sequences in BE¤¤ thanks to the Davie-
Gamelin extension.
When we deal with A = A1(BE), the existence of interpolating se-
quences for A was proved by J. Globevnik for a wide class of in¯nite-
dimensional Banach spaces. We complete this study by proving the ex-
istence of interpolating sequences for A1(BE) for any in¯nite-dimensional
Banach space E, characterizing the separability of A1(BE) in terms of the
¯nite dimension of E.
Finally, we study the metrizability of bounded subsets of MA when we
deal with A = Au(BE).
In chapter 4 we deal with composition operators on H1(BE). First we
study the spectra of these operators. L. Zheng described the spectrum
of some composition operators on H1. Her results where extended to
H1(BE), E any complex Banach space, by P. Galindo, T. Gamelin and
M. LindstrÄom for the power compact case. In this work, the authors also
deal with the non power compact case for Hilbert spaces. Inspired by them
and using some interpolating results, we provide a general theorem which
describes the spectrum of H1(BE) for general Banach spaces. In partic-
ular, we prove that conditions on this theorem are satis¯ed by the n¡fold
product space Cn, completing the description of ¾(CÁ) in this case, which
was an open question.
Next, we study the class of Radon-Nikod¶ym composition operators from
H1(BE) to H1(BF ). We characterize these operators in terms of the As-
plund property.
Chapter 5 deals with properties related to Hankel-type operators. The
concept of tight algebra is related to these operators and was introduced
by B. Cole and T. Gamelin. They proved that A(Dn) is not tight on its
spectrum for n ¸ 2. We present a new approach to this result extending
it to algebras Au(BE) for Banach spaces E = C £ F endowed with the
supremum norm.
In addition, we show that H1(BE) is never tight on its spectrum re-
gardless the Banach space E.
Hankel-type operators are also closely related to the Dunford-Pettis prop-
erty through the so-called Bourgain algebras introduced by J. A. Cima and
R. M. Timoney. We prove that the Bourgain algebras of A(Dn) as a sub-
space of C( ¹D n) are themselves.
Bibliographical Information:

Advisor:Galindo Pastor, Pablo

School:Universitat de València

School Location:Spain

Source Type:Master's Thesis

Keywords:anàlisi matemàtica

ISBN:

Date of Publication:06/26/2008