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The backward shift on Dirichlet-type spaces

Author: Stephan Ramon Garcia
Journal: Proc. Amer. Math. Soc. 133 (2005), 3047-3056
MSC (2000): Primary 30D55, 47B38
Published electronically: March 31, 2005
MathSciNet review: 2159784
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Abstract: We study the backward shift operator on Hilbert spaces ${\mathcal{H}}_{\alpha}$ (for ${\alpha \geq 0}$) which are norm equivalent to the Dirichlet-type spaces $D_{\alpha}$. Although these operators are unitarily equivalent to the adjoints of the forward shift operator on certain weighted Bergman spaces, our approach is direct and completely independent of the standard Cauchy duality. We employ only the classical Hardy space theory and an elementary formula expressing the inner product on ${\mathcal{H}}_{\alpha}$ in terms of a weighted superposition of backward shifts.

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Stephan Ramon Garcia
Affiliation: Department of Mathematics, University of California at Santa Barbara, Santa Barbara, California, 93106-3080

Keywords: Backward shift operator, Dirichlet-type spaces, weighted Bergman spaces, cyclic function, noncyclic function, invariant subspaces, pseudocontinuation, Bergman shift operator
Received by editor(s): May 8, 2004
Received by editor(s) in revised form: May 31, 2004
Published electronically: March 31, 2005
Communicated by: Joseph A. Ball
Article copyright: © Copyright 2005 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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