A Short Proof of a Characterization of Inner Functions in Terms of the Composition Operators they Induce

The paper contains a new prof for the sufficiency in Joel H. Shapiro’s recent characterization of inner functions saying that an analytic selfmap φ of the open unit disk is an inner function if and only if the essential norm of the composition operator of symbol φ is equal to √ 1+|φ(0)| 1−|φ(0)| . The main ingredient in the proof is a formula for the essential norm of a composition operator in terms of Aleksandrov measures obtained by J. Cima and A. Matheson. The necessity was originally proved by Joel Shapiro in 1987. A short proof of the necessity, by Aleksandrov measures techniques, was obtained by Jonathan E. Shapiro in 1998. For each holomorphic selfmap φ : U → U of the open unit disk U, the composition operator Cφ of symbol φ is defined as follows Cφf = f ◦φ, f ∈ H. In this definition H is the Hilbert Hardy space on U i.e. the set of all analytic functions on U with square summable Taylor coefficients. It is wellknown that each such φ induces a bounded composition operator Cφ on H. Recently Joel Shapiro obtained the following characterization of inner functions [7], (that is of selmaps φ whose radial limit function is unimodular a.e. on the unit circle T).