Some sufficient conditions for the division property of invariant subspaces in weighted Bergman spaces
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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 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.
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.
Details
Authors  

External organisations 

Research areas and keywords  Subject classification (UKÄ) – MANDATORY

Original language  English 

Pages (fromto)  542556 
Journal  Journal of Functional Analysis 
Volume  144 
Issue number  2 
Publication status  Published  1997 
Publication category  Research 
Peerreviewed  Yes 
Externally published  Yes 