Álgebras de funciones analíticas acotadas. Interpolación
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Álgebras de funciones analíticas acotadas. Interpolación

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Álgebras de funciones analíticas acotadas. Interpolación

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dc.contributor.advisor Galindo Pastor, Pablo es_ES
dc.contributor.author Miralles Montolío, Alejandro es_ES
dc.contributor.other Universitat de València - ANÀLISI MATEMÀTICA es_ES
dc.date.accessioned 2010-07-07T15:33:08Z
dc.date.available 2010-07-07T15:33:08Z
dc.date.issued 2008 es_ES
dc.date.submitted 2008-06-26 es_ES
dc.identifier.uri http://hdl.handle.net/10550/15784
dc.description.abstract RESUMEN Este trabajo resume, de forma parcial, la investigación realizada durante mi periodo predoctoral. Esta investigación pertenece, de forma general, a la teoría de álgebras de Banach conmutativas y álgebras uniformes y, en particular, se desarrolla principalmente en el ámbito de las álgebras de funciones analíticas acotadas en dominios de espacios de Banach ¯nito e in¯nito dimensionales. Las líneas centrales de este trabajo son las siguientes: ² Sucesiones de Interpolación para Álgebras Uniformes ² Operadores de Composición ² Propiedades Topológicas de Álgebras de Funciones Analíticas La investigación realizada sobre sucesiones de interpolación para álgebras uniformes se puede dividir en dos partes: una genérica en la que se propor- cionan algunos resultados de carácter general sobre sucesiones de interpo- lación para álgebras uniformes, y una parte más específica, en que se tratan sucesiones de interpolación para algunas álgebras de funciones analíticas acotadas. Estos puntos se tratan en los Capítulos 2 y 3. El estudio de oper- adores de composición, principalmente sobre H1(BE), centra el contenido del Capítulo 4. En este cap¶³tulo estudiaremos una descripci¶on del espectro de estos operadores y los llamados operadores de composición de Radon- Nikod¶ym. Para ello, se harí uso de algunos resultados de interpolación del capítulo anterior. Con respecto a la tercera línea que hemos citado, estu- diaremos los llamados operadores de tipo Hankel en el capítulo 5. ¶Estos nos permitirán tratar el concepto de álgebra tight y las álgebras de Bour- gain de un subespacio de C(K), que están estrechamente relacionadas con la propiedad de Dunford-Pettis. __________________________________________________________________________________________________ es_ES
dc.format.mimetype application/pdf es_ES
dc.language cat-en-es es_ES
dc.rights spa es_ES
dc.rights Copyright information available at source archive es_ES
dc.subject none es_ES
dc.title Álgebras de funciones analíticas acotadas. Interpolación es_ES
dc.type info:eu-repo/semantics/doctoralThesis es_ES
dc.description.abstractenglish 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. es_ES

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