The backward shift on weighted Bergman spaces
(1996) In Michigan Mathematical Journal 43(2). p.291319 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 $\Vert f\Vert^p\equiv\int_Df(z)^p(1z)^\alpha dA(z)<+\infty$, 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 $\Vert f\Vert^p\equiv\int_Df(z)^p(1z)^\alpha dA(z)<+\infty$, 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 ``pseudocontinuation'', 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)
Please use this url to cite or link to this publication:
http://lup.lub.lu.se/record/1467224
 author
 Aleman, Alexandru ^{LU} and Ross, William T
 publishing date
 1996
 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
 00262285
 DOI
 10.1307/mmj/1029005464
 language
 English
 LU publication?
 no
 id
 36856e0e35c14bb7ae03cd1d89014312 (old id 1467224)
 date added to LUP
 20090916 13:14:00
 date last changed
 20180107 09:30:58
@article{36856e0e35c14bb7ae03cd1d89014312, 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 $\Vert f\Vert^p\equiv\int_Df(z)^p(1z)^\alpha dA(z)<+\infty$, 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 $A^p_\alpha$. <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 $1\leq p<+\infty$. The study is based on duality and involves a notion called ``pseudocontinuation'', 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 = {00262285}, language = {eng}, number = {2}, pages = {291319}, 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}, }