We prove that any composition operator with maximal norm on one of the weighted Bergman spaces 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 H2, where every inner function induces a composition operator with maximal norm.
Carswell, B. J., and C. Hammond, 2006: Composition operators with maximal norm on weighted Bergman spaces. Proceedings of the American Mathematical Society, 134, 2599-2605.
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