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Invariant Subspaces in Spaces of Analytic Functions

Abkar, Ali LU (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 one-to-one 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 quasi-Banach spaces of analytic functions on D .



In Chapter IV, we shall establish a Riesz-type representation formula for super-biharmonic functions satisfying certain growth conditions on the unit disk. This representation formula can be regarded as an analogue of the Poisson-Jensen 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 (non-radial) weights are super-biharmonic and fulfill a certain growth condition. We shall prove that the polynomials are dense in such weighted Bergman spaces. (Less)
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author
opponent
  • Professor Borichev, Alexander, Department of Mathematics, Bordeaux University, France.
organization
publishing date
type
Thesis
publication status
published
subject
keywords
functional analysis, Series, Fourier analysis, weighted Bergman space., super-biharmonic function, quasi-Banach 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, Sölvegatan 18, Room MH:C
defense date
2000-01-21 10:15
external identifiers
  • other:LUNFMA-1012-1999
ISSN
1404-0034
ISBN
91-628-3959-4
language
English
LU publication?
yes
id
a8041c5e-0a06-47e1-ad4b-7540018b5958 (old id 19200)
date added to LUP
2007-05-25 08:42:59
date last changed
2016-09-19 08:44:56
@phdthesis{a8041c5e-0a06-47e1-ad4b-7540018b5958,
  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 &lt;i&gt;Hardy, Dirichlet&lt;/i&gt;, and &lt;i&gt;Bergman spaces&lt;/i&gt;.<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 &lt;i&gt;space of multipliers&lt;/i&gt; of B . Under reasonable assumptions, we prove that I mapsto Lambda (I) gives a one-to-one 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 quasi-Banach spaces of analytic functions on D .<br/><br>
<br/><br>
In Chapter IV, we shall establish a Riesz-type representation formula for super-biharmonic functions satisfying certain growth conditions on the unit disk. This representation formula can be regarded as an analogue of the Poisson-Jensen 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 (non-radial) weights are super-biharmonic and fulfill a certain growth condition. We shall prove that the polynomials are dense in such weighted Bergman spaces.},
  author       = {Abkar, Ali},
  isbn         = {91-628-3959-4},
  issn         = {1404-0034},
  keyword      = {functional analysis,Series,Fourier analysis,weighted Bergman space.,super-biharmonic function,quasi-Banach 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},
}