Composition operators with maximal norm on weighted Bergman spaces

Abstract

We prove that any composition operator with maximal norm on one of the weighted Bergman spaces A α 2 A^{2}_{\alpha } (in particular, on the space A 2 = A 0 2 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 H^{2} , where every inner function induces a composition operator with maximal norm.