The backward shift on weighted Bergman spaces
(1996) In Michigan Mathematical Journal 43(2). p.291-319- 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 $\Vert f\Vert^p\equiv\int_D|f(z)|^p(1-|z|)^\alpha dA(z)<+\infty$, 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... (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 $\Vert f\Vert^p\equiv\int_D|f(z)|^p(1-|z|)^\alpha dA(z)<+\infty$, 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 $A^p_\alpha$. (Less)
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- 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
- 0026-2285
- DOI
- 10.1307/mmj/1029005464
- language
- English
- LU publication?
- no
- id
- 36856e0e-35c1-4bb7-ae03-cd1d89014312 (old id 1467224)
- date added to LUP
- 2016-04-01 16:45:22
- date last changed
- 2022-01-28 21:55:51
@article{36856e0e-35c1-4bb7-ae03-cd1d89014312, 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 $\Vert f\Vert^p\equiv\int_D|f(z)|^p(1-|z|)^\alpha dA(z)<+\infty$, 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 $A^p_\alpha$. <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 ``pseudo-continuation'', 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 = {{0026-2285}}, language = {{eng}}, number = {{2}}, pages = {{291--319}}, 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}}, doi = {{10.1307/mmj/1029005464}}, volume = {{43}}, year = {{1996}}, }