| Title | The backward shift on weighted Bergman spaces |
| Authors | Alexandru Aleman, William T Ross |
| Alternative Location | http://dx.doi.org/10.1307/m... |
| Publication | Michigan mathematical journal |
| Year | 1996 |
| Volume | 43 |
| Issue | 2 |
| Pages | 291 - 319 |
| Document type | Article |
| Status | Published |
| Quality controlled | Yes |
| Language | eng |
| Publisher | University of Michigan, Department of Mathematics |
| Abstract 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 $A^p_\alpha$. <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> 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 $A^p_\alpha$. |
| ISBN/ISSN/Other | ISSN: 00262285 |
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