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THE ESSENTIAL NORMS OF COMPOSITION OPERATORS ON WEIGHTED DIRICHLET SPACES

Published online by Cambridge University Press:  31 January 2018

YUFEI LI*
Affiliation:
Department of Mathematical Sciences, Dalian University of Technology, Liaoning, Dalian, 116024, PR China email [email protected]
YUFENG LU
Affiliation:
Department of Mathematical Sciences, Dalian University of Technology, Liaoning, Dalian, 116024, PR China email [email protected]
TAO YU
Affiliation:
Department of Mathematical Sciences, Dalian University of Technology, Liaoning, Dalian, 116024, PR China email [email protected]
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Abstract

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Let $\unicode[STIX]{x1D711}$ be an analytic self-map of the unit disc. If $\unicode[STIX]{x1D711}$ is analytic in a neighbourhood of the closed unit disc, we give a precise formula for the essential norm of the composition operator $C_{\unicode[STIX]{x1D711}}$ on the weighted Dirichlet spaces ${\mathcal{D}}_{\unicode[STIX]{x1D6FC}}$ for $\unicode[STIX]{x1D6FC}>0$. We also show that, for a univalent analytic self-map $\unicode[STIX]{x1D711}$ of $\mathbb{D}$, if $\unicode[STIX]{x1D711}$ has an angular derivative at some point of $\unicode[STIX]{x2202}\mathbb{D}$, then the essential norm of $C_{\unicode[STIX]{x1D711}}$ on the Dirichlet space is equal to one.

Type
Research Article
Copyright
© 2018 Australian Mathematical Publishing Association Inc. 

Footnotes

This research is supported by NSFC grant no. 11671065. The third author is supported by the NSFC grant nos. 11271332 and 11431011.

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