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Bergman spaces on disconnected domains

Aleman, Alexandru LU ; Richter, Stefan and Ross, William T (1996) In Canadian Journal of Mathematics 48(2). p.225-243
Abstract
For a bounded open set $U$ in the complex plane, we consider the Bergman space $L^p_a(U)$ of $p$th-power area-integrable analytic functions on $U$, with $p$ assumed to be in the range $1\le p<+\infty$, where the lower bound assures that $L^p_a(U)$ is a Banach space. Fix a bounded domain $G$ and a compact subset $K$ of $G$ with zero area measure. Let $M$ be a closed subspace of $L^p_a(G\sbs K)$. We say that $M$ is invariant provided that $zf\in M$ whenever $f\in M$. Aleman, Richter, and Ross study those invariant subspaces $M$ of $L^p_a(G\sbs K)$ that contain $L^p_a(G)$. They also add the requirement that $M$ have index one, which is taken to mean that $(z-\lambda)M$ has codimension $1$ in $M$, for all $\lambda\in G\sbs K$. Natural... (More)
For a bounded open set $U$ in the complex plane, we consider the Bergman space $L^p_a(U)$ of $p$th-power area-integrable analytic functions on $U$, with $p$ assumed to be in the range $1\le p<+\infty$, where the lower bound assures that $L^p_a(U)$ is a Banach space. Fix a bounded domain $G$ and a compact subset $K$ of $G$ with zero area measure. Let $M$ be a closed subspace of $L^p_a(G\sbs K)$. We say that $M$ is invariant provided that $zf\in M$ whenever $f\in M$. Aleman, Richter, and Ross study those invariant subspaces $M$ of $L^p_a(G\sbs K)$ that contain $L^p_a(G)$. They also add the requirement that $M$ have index one, which is taken to mean that $(z-\lambda)M$ has codimension $1$ in $M$, for all $\lambda\in G\sbs K$. Natural examples of such invariant subspaces are those of the form $M=L^p_a(G\sbs E)$, where $E$ is a closed subset of $K$. The authors show that for $p<2$, these are indeed all such invariant subspaces which can be found. For $p\ge2$, this is not so, but nevertheless a complete classification can be found in terms of quasi-closed subsets $E$ of $K$.



If the condition that the index of $M$ is one is dropped, then the structure of such invariant subspaces can be extremely complicated. If, however, $G\sbs K$ is connected, the authors show that $M$ automatically has index one (due to the assumption that $M$ should contain $L_a^p(G)$). (Less)
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type
Contribution to journal
publication status
published
subject
in
Canadian Journal of Mathematics
volume
48
issue
2
pages
225 - 243
publisher
Cambridge University Press
external identifiers
  • scopus:0030504629
ISSN
0008-414X
language
English
LU publication?
no
id
8b2528f1-46da-4cb3-b0d9-9eae2db94884 (old id 1467233)
date added to LUP
2016-04-01 16:27:56
date last changed
2025-10-14 10:05:11
@article{8b2528f1-46da-4cb3-b0d9-9eae2db94884,
  abstract     = {{For a bounded open set $U$ in the complex plane, we consider the Bergman space $L^p_a(U)$ of $p$th-power area-integrable analytic functions on $U$, with $p$ assumed to be in the range $1\le p&lt;+\infty$, where the lower bound assures that $L^p_a(U)$ is a Banach space. Fix a bounded domain $G$ and a compact subset $K$ of $G$ with zero area measure. Let $M$ be a closed subspace of $L^p_a(G\sbs K)$. We say that $M$ is invariant provided that $zf\in M$ whenever $f\in M$. Aleman, Richter, and Ross study those invariant subspaces $M$ of $L^p_a(G\sbs K)$ that contain $L^p_a(G)$. They also add the requirement that $M$ have index one, which is taken to mean that $(z-\lambda)M$ has codimension $1$ in $M$, for all $\lambda\in G\sbs K$. Natural examples of such invariant subspaces are those of the form $M=L^p_a(G\sbs E)$, where $E$ is a closed subset of $K$. The authors show that for $p&lt;2$, these are indeed all such invariant subspaces which can be found. For $p\ge2$, this is not so, but nevertheless a complete classification can be found in terms of quasi-closed subsets $E$ of $K$. <br/><br>
<br/><br>
If the condition that the index of $M$ is one is dropped, then the structure of such invariant subspaces can be extremely complicated. If, however, $G\sbs K$ is connected, the authors show that $M$ automatically has index one (due to the assumption that $M$ should contain $L_a^p(G)$).}},
  author       = {{Aleman, Alexandru and Richter, Stefan and Ross, William T}},
  issn         = {{0008-414X}},
  language     = {{eng}},
  number       = {{2}},
  pages        = {{225--243}},
  publisher    = {{Cambridge University Press}},
  series       = {{Canadian Journal of Mathematics}},
  title        = {{Bergman spaces on disconnected domains}},
  volume       = {{48}},
  year         = {{1996}},
}