Composition operators with maximal norm on weighted Bergman spaces

Carswell, Brent; Hammond, Christopher

doi:10.1090/S0002-9939-06-08271-2

Composition operators with maximal norm on weighted Bergman spaces
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by Brent J. Carswell and Christopher Hammond PDF

Proc. Amer. Math. Soc. 134 (2006), 2599-2605 Request permission

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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