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

2007/07/23 by Michael T. Jury, Jury, Michael T. · 2 citations
Mathematics · #Holomorphic and Operator Theory #Advanced Topics in Algebra #Advanced Operator Algebra Research

paper · pdf · doi:10.48550/arxiv.0707.3426

Abstract

We prove that the norm of a weighted composition operator on the Hardy space H² 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 H², 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 functions spaces on the unit ball.

Cited by

• A Kernel Approach to the Stinespring and Kraus Representations
• Weighted Composition Operator and the Mittag-Leffler space

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• Weighted composition operators on de Branges-Rovnyak spaces
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