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The backward shift on weighted Bergman spaces

Aleman, Alexandru LU and Ross, William T (1996) In Michigan Mathematical Journal 43(2). p.291-319
Abstract
For $0<p<+\infty$ and $-1<\alpha<+\infty$ the weighted Bergman space $A^p_\alpha$ is defined to be the space of analytic functions $f$ in the unit disk D for which $\Vert f\Vert^p\equiv\int_D|f(z)|^p(1-|z|)^\alpha dA(z)<+\infty$, where $dA$ is the area measure on D. When $1\leq p<+\infty,\ A^p_\alpha$ is a Banach space with the norm $\Vert\ \Vert$ above. The backward shift operator $L$ is defined on the space of analytic functions in the unit disk by $(Lf)(z)=(f(z)-f(0)/z),\ z\in{\bf D}$. It is easy to see that $L$ is a bounded linear operator on each of the weighted Bergman spaces $A^p_\alpha$.



In this paper the authors investigate the invariant subspaces of the operator $L\colon A^p_\alpha\to... (More)
For $0<p<+\infty$ and $-1<\alpha<+\infty$ the weighted Bergman space $A^p_\alpha$ is defined to be the space of analytic functions $f$ in the unit disk D for which $\Vert f\Vert^p\equiv\int_D|f(z)|^p(1-|z|)^\alpha dA(z)<+\infty$, where $dA$ is the area measure on D. When $1\leq p<+\infty,\ A^p_\alpha$ is a Banach space with the norm $\Vert\ \Vert$ above. The backward shift operator $L$ is defined on the space of analytic functions in the unit disk by $(Lf)(z)=(f(z)-f(0)/z),\ z\in{\bf D}$. It is easy to see that $L$ is a bounded linear operator on each of the weighted Bergman spaces $A^p_\alpha$.



In this paper the authors investigate the invariant subspaces of the operator $L\colon A^p_\alpha\to A^p_\alpha$ when $1\leq p<+\infty$. The study is based on duality and involves a notion called ``pseudo-continuation'', just as in the classical case of Hardy spaces.



The main result of the paper characterizes the class of invariant subspaces $M$ such that the annihilator of $M$ (under a suitable duality pairing) is generated by slightly ``smoother'' functions. When $\alpha$ is ``much bigger than'' $p$, the paper gives a complete characterization of $L$-invariant subspaces in $A^p_\alpha$. (Less)
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author
publishing date
type
Contribution to journal
publication status
published
subject
in
Michigan Mathematical Journal
volume
43
issue
2
pages
291 - 319
publisher
University of Michigan, Department of Mathematics
external identifiers
  • scopus:0040655912
ISSN
0026-2285
DOI
10.1307/mmj/1029005464
language
English
LU publication?
no
id
36856e0e-35c1-4bb7-ae03-cd1d89014312 (old id 1467224)
date added to LUP
2009-09-16 13:14:00
date last changed
2017-01-01 07:14:15
@article{36856e0e-35c1-4bb7-ae03-cd1d89014312,
  abstract     = {For $0&lt;p&lt;+\infty$ and $-1&lt;\alpha&lt;+\infty$ the weighted Bergman space $A^p_\alpha$ is defined to be the space of analytic functions $f$ in the unit disk D for which $\Vert f\Vert^p\equiv\int_D|f(z)|^p(1-|z|)^\alpha dA(z)&lt;+\infty$, where $dA$ is the area measure on D. When $1\leq p&lt;+\infty,\ A^p_\alpha$ is a Banach space with the norm $\Vert\ \Vert$ above. The backward shift operator $L$ is defined on the space of analytic functions in the unit disk by $(Lf)(z)=(f(z)-f(0)/z),\ z\in{\bf D}$. It is easy to see that $L$ is a bounded linear operator on each of the weighted Bergman spaces $A^p_\alpha$. <br/><br>
<br/><br>
In this paper the authors investigate the invariant subspaces of the operator $L\colon A^p_\alpha\to A^p_\alpha$ when $1\leq p&lt;+\infty$. The study is based on duality and involves a notion called ``pseudo-continuation'', just as in the classical case of Hardy spaces. <br/><br>
<br/><br>
The main result of the paper characterizes the class of invariant subspaces $M$ such that the annihilator of $M$ (under a suitable duality pairing) is generated by slightly ``smoother'' functions. When $\alpha$ is ``much bigger than'' $p$, the paper gives a complete characterization of $L$-invariant subspaces in $A^p_\alpha$.},
  author       = {Aleman, Alexandru and Ross, William T},
  issn         = {0026-2285},
  language     = {eng},
  number       = {2},
  pages        = {291--319},
  publisher    = {University of Michigan, Department of Mathematics},
  series       = {Michigan Mathematical Journal},
  title        = {The backward shift on weighted Bergman spaces},
  url          = {http://dx.doi.org/10.1307/mmj/1029005464},
  volume       = {43},
  year         = {1996},
}