Álgebras de funciones analíticas acotadas. Interpolación

• Main Author: Miralles Montolío, Alejandro
• Other Authors: Galindo Pastor, Pablo
• Format: Doctoral Thesis
• Language: Spanish
• Published: Universitat de València 2008
• Subjects: Facultat de Matemàtiques; 512
• Online Access: http://hdl.handle.net/10803/9471; http://nbn-resolving.de/urn:isbn:9788437072517

Este trabajo resume, de forma parcial, la investigaci¶on realizada durantemi periodo predoctoral. Esta investigaci¶on pertenece, de forma general,a la teor¶³a de ¶algebras de Banach conmutativas y ¶algebras uniformes y,en particular, se desarrolla principalmente en el ¶ambito de las ¶algebras defunciones anal¶³ticas acotadas en dominios de espacios de Banach ¯nito ein¯nito dimensionales.Las l¶³neas centrales de este trabajo son las siguientes:² Sucesiones de Interpolaci¶on para ¶Algebras Uniformes² Operadores de Composici¶on² Propiedades Topol¶ogicas de ¶Algebras de Funciones Anal¶³ticasLa investigaci¶on realizada sobre sucesiones de interpolaci¶on para ¶algebrasuniformes se puede dividir en dos partes: una gen¶erica en la que se propor-cionan algunos resultados de car¶acter general sobre sucesiones de interpo-laci¶on para ¶algebras uniformes, y una parte m¶as espec¶³¯ca, en que se tratansucesiones de interpolaci¶on para algunas ¶algebras de funciones anal¶³ticasacotadas. Estos puntos se tratan en los Cap¶³tulos 2 y 3. El estudio de oper-adores de composici¶on, principalmente sobre H1(BE), centra el contenidodel Cap¶³tulo 4. En este cap¶³tulo estudiaremos una descripci¶on del espectrode estos operadores y los llamados operadores de composici¶on de Radon-Nikod¶ym. Para ello, se har¶a uso de algunos resultados de interpolaci¶on delcap¶³tulo anterior. Con respecto a la tercera l¶³nea que hemos citado, estu-diaremos los llamados operadores de tipo Hankel en el cap¶³tulo 5. ¶Estosnos permitir¶an tratar el concepto de ¶algebra tight y las ¶algebras de Bour-gain de un subespacio de C(K), que est¶an estrechamente relacionadas conla propiedad de Dunford-Pettis. === The lines studied in this thesis are the following:² Interpolating Sequences for Uniform Algebras² Composition Operators² Topological Properties in Algebras of Analytic FunctionsAfter the preliminaries, the second chapter is devoted to the study ofinterpolating 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, weprove ¯rst that c0¡linear interpolating sequences are linear interpolatingand then, we show that c0¡interpolating sequences are, indeed, c0¡linearinterpolating, obtaining that c0¡interpolating sequences (xn) ½ MA Xbecome linear interpolating. Finally, we provide a di®erent approach toprove that c0¡interpolating sequences are not c0¡linear interpolating viacomposition operators.We continue with the study of interpolating sequences for the algebrasof analytic functions H1(BE) and A1(BE) in the third chapter. The studyof 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 issu±cient for a sequence (xn) ½ BE¤¤ to be interpolating for H1(BE) if Eis any ¯nite or in¯nite dimensional Banach space. This is a consequence ofa 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-dimensionalBanach 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 wedeal with A = Au(BE).In chapter 4 we deal with composition operators on H1(BE). First westudy the spectra of these operators. L. Zheng described the spectrumof some composition operators on H1. Her results where extended toH1(BE), E any complex Banach space, by P. Galindo, T. Gamelin andM. LindstrÄom for the power compact case. In this work, the authors alsodeal with the non power compact case for Hilbert spaces. Inspired by themand using some interpolating results, we provide a general theorem whichdescribes 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¡foldproduct space Cn, completing the description of ¾(CÁ) in this case, whichwas an open question.Next, we study the class of Radon-Nikod¶ym composition operators fromH1(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. Theconcept of tight algebra is related to these operators and was introducedby B. Cole and T. Gamelin. They proved that A(Dn) is not tight on itsspectrum for n ¸ 2. We present a new approach to this result extendingit to algebras Au(BE) for Banach spaces E = C £ F endowed with thesupremum 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 andR. M. Timoney. We prove that the Bourgain algebras of A(Dn) as a sub-space of C( ¹D n) are themselves.