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Reproducing kernels, de Branges-Rovnyak spaces, and norms of weighted composition operators

Author: Michael T. Jury
Journal: Proc. Amer. Math. Soc. 135 (2007), 3669-3675
MSC (2000): Primary 47B33; Secondary 47B32, 46E22
Published electronically: August 15, 2007
MathSciNet review: 2336583
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Abstract: We prove that the norm of a weighted composition operator on the Hardy space $ H^2$ of the disk is controlled by the norm of the weight function in the de Branges-Rovnyak space associated to the symbol of the composition operator. As a corollary we obtain a new proof of the boundedness of composition operators on $ H^2$ and recover the standard upper bound for the norm. Similar arguments apply to weighted Bergman spaces. We also show that the positivity of a generalized de Branges-Rovnyak kernel is sufficient for the boundedness of a given composition operator on the standard function spaces on the unit ball.

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Additional Information

Michael T. Jury
Affiliation: Department of Mathematics, University of Florida, Gainesville, Florida 32603

Received by editor(s): July 27, 2006
Received by editor(s) in revised form: September 19, 2006
Published electronically: August 15, 2007
Communicated by: Joseph A. Ball
Article copyright: © Copyright 2007 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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