Invariant Subspaces in Spaces of Analytic Functions
(1999) In Doctoral Theses in Mathematical Sciences 1999:9. Abstract
 Let D be a finitely connected bounded domain with smooth boundary in the complex plane. We first study Banach spaces of analytic functions on D . The main result is a theorem which converts the study of hyperinvariant subspaces on multiply connected domains into the study of hyperinvariant subspaces on domains with fewer holes. The Banach spaces are defined by a natural set of axioms fulfilled by the familiar <i>Hardy, Dirichlet</i>, and <i>Bergman spaces</i>.
Let D_1 be a bounded domain obtained from D by adding some of the connectivity components of the complement of D ; hence D_1 has fewer holes. Let B and B_1 be the Banach spaces of analytic functions on the domains D and D_1 , respectively.... (More)  Let D be a finitely connected bounded domain with smooth boundary in the complex plane. We first study Banach spaces of analytic functions on D . The main result is a theorem which converts the study of hyperinvariant subspaces on multiply connected domains into the study of hyperinvariant subspaces on domains with fewer holes. The Banach spaces are defined by a natural set of axioms fulfilled by the familiar <i>Hardy, Dirichlet</i>, and <i>Bergman spaces</i>.
Let D_1 be a bounded domain obtained from D by adding some of the connectivity components of the complement of D ; hence D_1 has fewer holes. Let B and B_1 be the Banach spaces of analytic functions on the domains D and D_1 , respectively. Assume that I is a hyperinvariant subspace of B_1 , and consider the smallest hyperinvariant subspace of B containing I ; this is Lambda (I) , the closure in B of the span of I cdot M(B) , where M(B) denotes the <i>space of multipliers</i> of B . Under reasonable assumptions, we prove that I mapsto Lambda (I) gives a onetoone correspondence between a class of hyperinvariant subspaces of B_1 , and a class of hyperinvariant subspaces of B . The inverse mapping is given by J mapsto J cap B_1 . We then generalize the above result to the setting of the quasiBanach spaces of analytic functions on D .
In Chapter IV, we shall establish a Riesztype representation formula for superbiharmonic functions satisfying certain growth conditions on the unit disk. This representation formula can be regarded as an analogue of the PoissonJensen representation formula for subharmonic functions.
In Chapter V, this representation formula will be used to prove an approximation theorem in certain weighted Bergman spaces. More precisely, we consider those weighted Bergman spaces whose (nonradial) weights are superbiharmonic and fulfill a certain growth condition. We shall prove that the polynomials are dense in such weighted Bergman spaces. (Less)
Please use this url to cite or link to this publication:
http://lup.lub.lu.se/record/19200
 author
 Abkar, Ali ^{LU}
 opponent

 Professor Borichev, Alexander, Department of Mathematics, Bordeaux University, France.
 organization
 publishing date
 1999
 type
 Thesis
 publication status
 published
 subject
 keywords
 functional analysis, Series, Fourier analysis, weighted Bergman space., superbiharmonic function, quasiBanach space of analytic functions, index, multiplier module homomorphism, multiplier, Banach space of analytic functions, hyperinvariant subspace, Serier, fourieranalys, funktionsanalys
 in
 Doctoral Theses in Mathematical Sciences
 volume
 1999:9
 pages
 84 pages
 publisher
 Centre for Mathematical Sciences, Lund University
 defense location
 Mathematics Building, SÃ¶lvegatan 18, Room MH:C
 defense date
 20000121 10:15
 external identifiers

 other:LUNFMA10121999
 ISSN
 14040034
 ISBN
 9162839594
 language
 English
 LU publication?
 yes
 id
 a8041c5e0a0647e1ad4b7540018b5958 (old id 19200)
 date added to LUP
 20070525 08:42:59
 date last changed
 20180529 11:30:23
@phdthesis{a8041c5e0a0647e1ad4b7540018b5958, abstract = {Let D be a finitely connected bounded domain with smooth boundary in the complex plane. We first study Banach spaces of analytic functions on D . The main result is a theorem which converts the study of hyperinvariant subspaces on multiply connected domains into the study of hyperinvariant subspaces on domains with fewer holes. The Banach spaces are defined by a natural set of axioms fulfilled by the familiar <i>Hardy, Dirichlet</i>, and <i>Bergman spaces</i>.<br/><br> <br/><br> Let D_1 be a bounded domain obtained from D by adding some of the connectivity components of the complement of D ; hence D_1 has fewer holes. Let B and B_1 be the Banach spaces of analytic functions on the domains D and D_1 , respectively. Assume that I is a hyperinvariant subspace of B_1 , and consider the smallest hyperinvariant subspace of B containing I ; this is Lambda (I) , the closure in B of the span of I cdot M(B) , where M(B) denotes the <i>space of multipliers</i> of B . Under reasonable assumptions, we prove that I mapsto Lambda (I) gives a onetoone correspondence between a class of hyperinvariant subspaces of B_1 , and a class of hyperinvariant subspaces of B . The inverse mapping is given by J mapsto J cap B_1 . We then generalize the above result to the setting of the quasiBanach spaces of analytic functions on D .<br/><br> <br/><br> In Chapter IV, we shall establish a Riesztype representation formula for superbiharmonic functions satisfying certain growth conditions on the unit disk. This representation formula can be regarded as an analogue of the PoissonJensen representation formula for subharmonic functions.<br/><br> <br/><br> In Chapter V, this representation formula will be used to prove an approximation theorem in certain weighted Bergman spaces. More precisely, we consider those weighted Bergman spaces whose (nonradial) weights are superbiharmonic and fulfill a certain growth condition. We shall prove that the polynomials are dense in such weighted Bergman spaces.}, author = {Abkar, Ali}, isbn = {9162839594}, issn = {14040034}, keyword = {functional analysis,Series,Fourier analysis,weighted Bergman space.,superbiharmonic function,quasiBanach space of analytic functions,index,multiplier module homomorphism,multiplier,Banach space of analytic functions,hyperinvariant subspace,Serier,fourieranalys,funktionsanalys}, language = {eng}, pages = {84}, publisher = {Centre for Mathematical Sciences, Lund University}, school = {Lund University}, series = {Doctoral Theses in Mathematical Sciences}, title = {Invariant Subspaces in Spaces of Analytic Functions}, volume = {1999:9}, year = {1999}, }