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Composition operators with maximal norm on weighted Bergman spaces

Authors: Brent J. Carswell and Christopher Hammond
Journal: Proc. Amer. Math. Soc. 134 (2006), 2599-2605
MSC (2000): Primary 47B33
Published electronically: February 17, 2006
MathSciNet review: 2213738
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Abstract: We prove that any composition operator with maximal norm on one of the weighted Bergman spaces $ A^{2}_{\alpha}$ (in particular, on the space $ A^{2}=A^{2}_{0}$) is induced by a disk automorphism or a map that fixes the origin. This result demonstrates a major difference between the weighted Bergman spaces and the Hardy space $ H^{2}$, where every inner function induces a composition operator with maximal norm.

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Additional Information

Brent J. Carswell
Affiliation: Department of Mathematics, Allegheny College, Meadville, Pennsylvania 16335

Christopher Hammond
Affiliation: Department of Mathematics and Computer Science, Connecticut College, New London, Connecticut 06320

Keywords: Composition operator, norm, essential norm
Received by editor(s): February 2, 2005
Received by editor(s) in revised form: March 21, 2005
Published electronically: February 17, 2006
Communicated by: Joseph A. Ball
Article copyright: © Copyright 2006 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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