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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

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
Email: mjury@math.ufl.edu

DOI: http://dx.doi.org/10.1090/S0002-9939-07-08931-9
PII: S 0002-9939(07)08931-9
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.