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

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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, 275--310; 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 one-dimensional for any $\lambda\/$ in the domain. Under certain additional hypotheses, this is equivalent to the codimension-one 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 $(1-|z|\sp{2})\sp{\alpha}$ $(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.


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Research areas and keywords

Subject classification (UKÄ) – MANDATORY

  • Mathematics
Original languageEnglish
Pages (from-to)542-556
JournalJournal of Functional Analysis
Issue number2
Publication statusPublished - 1997
Publication categoryResearch
Externally publishedYes