### Composition Operators Induced by Disk Automorphisms

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5.1 Fundamental Properties of Composition Operators 169

Composition operators are characterized as those operators whose adjoints map the set of reproducing kernels into itself.

Theorem . An operator A on H 2 is a composition operator if and only if A maps the set of reproducing kernels into itself.

Proof. We showed above that A k k() when A C . Conversely, sup- pose that for each D, A k k for some D. Dene : D D by () . Notice that, for f H 2 ,

``                 (Af, k )  (f, A k )  (f, k() )  f (()).``

If we take f (z) z, then g Af is in H 2 , and is thus analytic. But then, by the above equation, we have

``                 g()  (g, k )  (Af,...``

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