A short proof of a characterization of inner functions in terms of the composition operators they induce

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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.

Original languageEnglish (US)
Pages (from-to)1723-1726
Number of pages4
JournalRocky Mountain Journal of Mathematics
Volume35
Issue number5
DOIs
Publication statusPublished - Dec 1 2005

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ASJC Scopus subject areas

  • Mathematics(all)

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