Some sufficient conditions for the division property of invariant subspaces in weighted Bergman spaces
(1997) In Journal of Functional Analysis 144(2). p.542556 Abstract
 The authors consider weighted Bergman spaces of holomorphic $L\sp p\/$ integrable functions with respect to certain Borel measures on a bounded plane region. A closed subspace $\scr M $ of such a space is said to be invariant if $z {\scr M} \subset {\scr M}$. In the case of the standard Bergman space of the unit disk with $p = 2$, the authors proved in a recent joint paper with C. Sundberg [Acta Math. 177 (1996), no. 2, 275310; MR1440934 (98a:46034)] that every invariant subspace ${\scr M}\/$ is generated by a wandering subspace ${\scr M} \ominus z {\scr M}\/$ for the shift operator, thus obtaining an analogue of the celebrated theorem of Beurling.
An invariant space is said to have the division property if the... (More)  The authors consider weighted Bergman spaces of holomorphic $L\sp p\/$ integrable functions with respect to certain Borel measures on a bounded plane region. A closed subspace $\scr M $ of such a space is said to be invariant if $z {\scr M} \subset {\scr M}$. In the case of the standard Bergman space of the unit disk with $p = 2$, the authors proved in a recent joint paper with C. Sundberg [Acta Math. 177 (1996), no. 2, 275310; MR1440934 (98a:46034)] that every invariant subspace ${\scr M}\/$ is generated by a wandering subspace ${\scr M} \ominus z {\scr M}\/$ for the shift operator, thus obtaining an analogue of the celebrated theorem of Beurling.
An invariant space is said to have the division property if the quotient space ${\scr M} / (z  \lambda) \scr M$ is onedimensional for any $\lambda\/$ in the domain. Under certain additional hypotheses, this is equivalent to the codimensionone property studied earlier by S. Richter, H. Hedenmalm, K. Seip, and others. The main results of this paper are various sufficient conditions for an invariant subspace to have the division property in terms of the local boundary behavior of the functions in the subspace.
A function $f $ analytic in the unit disc is said to be locally Nevanlinna (near a point $\lambda$ on the unit circle) if the subharmonic function $\log f$ has a harmonic majorant in the Carleson type set obtained by intersecting the unit disk with some disc centered at $\lambda$. The authors show that an invariant subspace $\scr M\/$ of the standard Bergman space $A\sp{p}\sb{\alpha}\/$ with radial weights $(1z\sp{2})\sp{\alpha}$ $(p \ge 1 $, $\alpha > 1$) possesses the division property if some nonzero function $f\/$ in the subspace is locally Nevanlinna at a boundary point.
They observe that, given an invariant subspace $\scr M\/$ which does not have the division property, it will always contain two functions such that the space they generate does not have it either. It is, therefore, a question of interest to determine under what conditions the space generated by two functions will possess this property. The authors prove the following theorem. Let $1 \le p < \infty$, $1/r + 1/s = 1/p$, and let $f$ and $g$ be two nonzero functions in $A\sp{p}\/$ such that $f$ is locally $r$integrable and $g$ is locally $s$integrable, both in a neighborhood of a boundary point of the disc. Then the closed linear span of the cyclic invariant subspaces generated by $f\/$ and by $g\/$ has the division property.
An example is also given (based on the earlier results of Hedenmalm and Seip) which shows that the theorem is in a way sharp for $p=2\/$.
The main results also have some interesting corollaries, listed in the paper. (Less)
Please use this url to cite or link to this publication:
http://lup.lub.lu.se/record/1467206
 author
 Aleman, Alexandru ^{LU} and Richter, Stefan
 publishing date
 1997
 type
 Contribution to journal
 publication status
 published
 subject
 in
 Journal of Functional Analysis
 volume
 144
 issue
 2
 pages
 542  556
 publisher
 Elsevier
 external identifiers

 scopus:0031093842
 ISSN
 00221236
 language
 English
 LU publication?
 no
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 52faafdbb71243468433318e7302076d (old id 1467206)
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 date added to LUP
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@article{52faafdbb71243468433318e7302076d, abstract = {The authors consider weighted Bergman spaces of holomorphic $L\sp p\/$ integrable functions with respect to certain Borel measures on a bounded plane region. A closed subspace $\scr M $ of such a space is said to be invariant if $z {\scr M} \subset {\scr M}$. In the case of the standard Bergman space of the unit disk with $p = 2$, the authors proved in a recent joint paper with C. Sundberg [Acta Math. 177 (1996), no. 2, 275310; MR1440934 (98a:46034)] that every invariant subspace ${\scr M}\/$ is generated by a wandering subspace ${\scr M} \ominus z {\scr M}\/$ for the shift operator, thus obtaining an analogue of the celebrated theorem of Beurling. <br/><br> <br/><br> An invariant space is said to have the division property if the quotient space ${\scr M} / (z  \lambda) \scr M$ is onedimensional for any $\lambda\/$ in the domain. Under certain additional hypotheses, this is equivalent to the codimensionone property studied earlier by S. Richter, H. Hedenmalm, K. Seip, and others. The main results of this paper are various sufficient conditions for an invariant subspace to have the division property in terms of the local boundary behavior of the functions in the subspace. <br/><br> <br/><br> A function $f $ analytic in the unit disc is said to be locally Nevanlinna (near a point $\lambda$ on the unit circle) if the subharmonic function $\log f$ has a harmonic majorant in the Carleson type set obtained by intersecting the unit disk with some disc centered at $\lambda$. The authors show that an invariant subspace $\scr M\/$ of the standard Bergman space $A\sp{p}\sb{\alpha}\/$ with radial weights $(1z\sp{2})\sp{\alpha}$ $(p \ge 1 $, $\alpha > 1$) possesses the division property if some nonzero function $f\/$ in the subspace is locally Nevanlinna at a boundary point. <br/><br> <br/><br> They observe that, given an invariant subspace $\scr M\/$ which does not have the division property, it will always contain two functions such that the space they generate does not have it either. It is, therefore, a question of interest to determine under what conditions the space generated by two functions will possess this property. The authors prove the following theorem. Let $1 \le p < \infty$, $1/r + 1/s = 1/p$, and let $f$ and $g$ be two nonzero functions in $A\sp{p}\/$ such that $f$ is locally $r$integrable and $g$ is locally $s$integrable, both in a neighborhood of a boundary point of the disc. Then the closed linear span of the cyclic invariant subspaces generated by $f\/$ and by $g\/$ has the division property. <br/><br> <br/><br> An example is also given (based on the earlier results of Hedenmalm and Seip) which shows that the theorem is in a way sharp for $p=2\/$. <br/><br> <br/><br> The main results also have some interesting corollaries, listed in the paper.}, author = {Aleman, Alexandru and Richter, Stefan}, issn = {00221236}, language = {eng}, number = {2}, pages = {542556}, publisher = {Elsevier}, series = {Journal of Functional Analysis}, title = {Some sufficient conditions for the division property of invariant subspaces in weighted Bergman spaces}, volume = {144}, year = {1997}, }