Composition Operators Induced by Disk Automorphismsin Other (Other) by Dgoodz19
Your Price: $19.50 (30% discount)
You Save: $8.36
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,...