Álgebras de funciones analíticas acotadas. Interpolación

by Miralles Montolío, Alejandro

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.