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

Abstract

The paper contains a new proof for the sufficiency in Joel H. Shapiro’s recent characterization of inner functions saying that an analytic self-map 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 Cima and Matheson. The necessity was originally proved by Joel Shapiro in 1987. A short proof of the necessity, by Aleksandrov measure techniques, was obtained by Jonathan E. Shapiro in 1998. For each holomorphic self-map 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 well known 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 self-maps φ whose radial limit function is unimodular almost everywhere on the unit circle T). Theorem 1. The function φ : U → U is inner if and only if (1) ‖Cφ‖e = √ 1 + |φ(0)| 1− |φ(0)| , where ‖Cφ‖e denotes the essential norm of Cφ. The necessity in the previous equivalence was originally proved in [6]. Jonathan Shapiro [5] observed that Aleksandrov measures can be used Received by the editors on May 7, 2003, and in revised form on May 30, 2003. Copyright c ©2005 Rocky Mountain Mathematics Consortium 1723