The backward shift on weighted Bergman spaces
Author
Summary, in English
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 $A^p_\alpha$.
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 $A^p_\alpha$.
Publishing year
1996
Language
English
Pages
291-319
Publication/Series
Michigan Mathematical Journal
Volume
43
Issue
2
Document type
Journal article
Publisher
University of Michigan, Department of Mathematics
Topic
- Mathematics
Status
Published
ISBN/ISSN/Other
- ISSN: 0026-2285