Function and Operator Theory on Large Bergman spaces

[eng] The theory of Bergman spaces has been a central subject of study in complex analysis during the past decades. The book [7] by S. Bergman contains the first systematic treat-ment of the Hilbert space of square integrable analytic functions with respect to Lebesgue area measure on a domain. His...

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Autor: Arroussi, Hicham
Tipo de recurso: tesis doctoral
Estado:Versión publicada
Fecha de publicación:2016
País:España
Institución:Universidad de Barcelona
Repositorio:Dipòsit Digital de la UB
OAI Identifier:oai:diposit.ub.edu:2445/102411
Acceso en línea:https://hdl.handle.net/2445/102411
http://hdl.handle.net/10803/395175
Access Level:acceso abierto
Palabra clave:Funcions analítiques
Funcions holomorfes
Operadors de Toeplitz
Equacions funcionals
Nuclis de Bergman
Analytic functions
Holomorphic functions
Toeplitz operators
Functional equations
Bergman kernel functions
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spelling Function and Operator Theory on Large Bergman spacesArroussi, HichamFuncions analítiquesFuncions holomorfesOperadors de ToeplitzEquacions funcionalsNuclis de BergmanAnalytic functionsHolomorphic functionsToeplitz operatorsFunctional equationsBergman kernel functions[eng] The theory of Bergman spaces has been a central subject of study in complex analysis during the past decades. The book [7] by S. Bergman contains the first systematic treat-ment of the Hilbert space of square integrable analytic functions with respect to Lebesgue area measure on a domain. His approach was based on a reproducing kernel that became known as the Bergman kernel function. When attention was later directed to the spaces AP over the unit disk, it was natural to call them Bergman spaces. As counterparts of Hardy spaces, they presented analogous problems. However, although many problems in Hardy spaces were well understood by the 1970s, their counterparts for Bergman spaces were generally viewed as intractable, and only some isolated progress was done. The 1980s saw the emerging of operator theoretic studies related to Bergman spaces with important contributions by several authors. Their achievements on Bergman spaces with standard weights are presented in Zhu's book [77]. The main breakthroughs came in the 1990s, where in a flurry of important advances, problems previously considered intractable began to be solved. First came Hedenmalm's construction of canonical divisors [26], then Seip's description [59] of sampling and interpolating sequences on Bergman spaces, and later on, the study of Aleman, Richter and Sundberg [1] on the invariant subspaces of A2, among others. This attracted other workers to the field and inspired a period of intense research on Bergman spaces and related topics. Nowadays there are rich theories on Bergman spaces that can be found on the textbooks [27] and [22]. Meanwhile, also in the nineties, some isolated problems on Bergman spaces with ex-ponential type weights began to be studied. These spaces are large in the sense that they contain all the Bergman spaces with standard weights, and their study presented new dif-ficulties, as the techniques and ideas that led to success when working on the analogous problems for standard Bergman spaces, failed to work on that context. It is the main goal of this work to do a deep study of the function theoretic properties of such spaces, as well as of some operators acting on them. It turns out that large Bergman spaces are close in spirit to Fock spaces [79], and many times mixing classical techniques from both Bergman and Fock spaces in an appropriate way, can led to some success when studying large Bergman spaces.Universitat de BarcelonaPau, JordiOrtega Cerdà, JoaquimUniversitat de Barcelona. Departament de Matemàtiques i Informàtica2016info:eu-repo/semantics/doctoralThesisinfo:eu-repo/semantics/publishedVersionapplication/pdfhttps://hdl.handle.net/2445/102411http://hdl.handle.net/10803/395175Tesis Doctorals - Departament - Matemàtiques i Informàticareponame:Dipòsit Digital de la UBinstname:Universidad de BarcelonaIngléscc-by, (c) Arroussi,, 2016http://creativecommons.org/licenses/by/3.0/info:eu-repo/semantics/openAccessoai:diposit.ub.edu:2445/1024112026-05-27T06:46:51Z
dc.title.none.fl_str_mv Function and Operator Theory on Large Bergman spaces
title Function and Operator Theory on Large Bergman spaces
spellingShingle Function and Operator Theory on Large Bergman spaces
Arroussi, Hicham
Funcions analítiques
Funcions holomorfes
Operadors de Toeplitz
Equacions funcionals
Nuclis de Bergman
Analytic functions
Holomorphic functions
Toeplitz operators
Functional equations
Bergman kernel functions
title_short Function and Operator Theory on Large Bergman spaces
title_full Function and Operator Theory on Large Bergman spaces
title_fullStr Function and Operator Theory on Large Bergman spaces
title_full_unstemmed Function and Operator Theory on Large Bergman spaces
title_sort Function and Operator Theory on Large Bergman spaces
dc.creator.none.fl_str_mv Arroussi, Hicham
author Arroussi, Hicham
author_facet Arroussi, Hicham
author_role author
dc.contributor.none.fl_str_mv Pau, Jordi
Ortega Cerdà, Joaquim
Universitat de Barcelona. Departament de Matemàtiques i Informàtica
dc.subject.none.fl_str_mv Funcions analítiques
Funcions holomorfes
Operadors de Toeplitz
Equacions funcionals
Nuclis de Bergman
Analytic functions
Holomorphic functions
Toeplitz operators
Functional equations
Bergman kernel functions
topic Funcions analítiques
Funcions holomorfes
Operadors de Toeplitz
Equacions funcionals
Nuclis de Bergman
Analytic functions
Holomorphic functions
Toeplitz operators
Functional equations
Bergman kernel functions
description [eng] The theory of Bergman spaces has been a central subject of study in complex analysis during the past decades. The book [7] by S. Bergman contains the first systematic treat-ment of the Hilbert space of square integrable analytic functions with respect to Lebesgue area measure on a domain. His approach was based on a reproducing kernel that became known as the Bergman kernel function. When attention was later directed to the spaces AP over the unit disk, it was natural to call them Bergman spaces. As counterparts of Hardy spaces, they presented analogous problems. However, although many problems in Hardy spaces were well understood by the 1970s, their counterparts for Bergman spaces were generally viewed as intractable, and only some isolated progress was done. The 1980s saw the emerging of operator theoretic studies related to Bergman spaces with important contributions by several authors. Their achievements on Bergman spaces with standard weights are presented in Zhu's book [77]. The main breakthroughs came in the 1990s, where in a flurry of important advances, problems previously considered intractable began to be solved. First came Hedenmalm's construction of canonical divisors [26], then Seip's description [59] of sampling and interpolating sequences on Bergman spaces, and later on, the study of Aleman, Richter and Sundberg [1] on the invariant subspaces of A2, among others. This attracted other workers to the field and inspired a period of intense research on Bergman spaces and related topics. Nowadays there are rich theories on Bergman spaces that can be found on the textbooks [27] and [22]. Meanwhile, also in the nineties, some isolated problems on Bergman spaces with ex-ponential type weights began to be studied. These spaces are large in the sense that they contain all the Bergman spaces with standard weights, and their study presented new dif-ficulties, as the techniques and ideas that led to success when working on the analogous problems for standard Bergman spaces, failed to work on that context. It is the main goal of this work to do a deep study of the function theoretic properties of such spaces, as well as of some operators acting on them. It turns out that large Bergman spaces are close in spirit to Fock spaces [79], and many times mixing classical techniques from both Bergman and Fock spaces in an appropriate way, can led to some success when studying large Bergman spaces.
publishDate 2016
dc.date.none.fl_str_mv 2016
dc.type.none.fl_str_mv info:eu-repo/semantics/doctoralThesis
info:eu-repo/semantics/publishedVersion
format doctoralThesis
status_str publishedVersion
dc.identifier.none.fl_str_mv https://hdl.handle.net/2445/102411
http://hdl.handle.net/10803/395175
url https://hdl.handle.net/2445/102411
http://hdl.handle.net/10803/395175
dc.language.none.fl_str_mv Inglés
language_invalid_str_mv Inglés
dc.rights.none.fl_str_mv cc-by, (c) Arroussi,, 2016
http://creativecommons.org/licenses/by/3.0/
info:eu-repo/semantics/openAccess
rights_invalid_str_mv cc-by, (c) Arroussi,, 2016
http://creativecommons.org/licenses/by/3.0/
eu_rights_str_mv openAccess
dc.format.none.fl_str_mv application/pdf
dc.publisher.none.fl_str_mv Universitat de Barcelona
publisher.none.fl_str_mv Universitat de Barcelona
dc.source.none.fl_str_mv Tesis Doctorals - Departament - Matemàtiques i Informàtica
reponame:Dipòsit Digital de la UB
instname:Universidad de Barcelona
instname_str Universidad de Barcelona
reponame_str Dipòsit Digital de la UB
collection Dipòsit Digital de la UB
repository.name.fl_str_mv
repository.mail.fl_str_mv
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