Composition operators on a functional Hilbert space

Abstract

Let T be a mapping from a set X into itself and let H ( X ) be a functional Hilbert space on the set X . Then the composition operator C T on H ( X ) induced by T is a bounded linear transformation from H ( X ) into itself defined by C T f = f ∘ T . In this paper composition operators are characterized in the case when H ( X ) = H 2 (π + ) in terms of the behaviour of the inducing functions in the vicinity of the point at infinity. An estimate for the lower bound of ∥ C T ∥ is given. Also the invertibility of C T is characterized in terms of the invertibility of T .