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 language | English (US) |
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Pages (from-to) | 1723-1726 |
Number of pages | 4 |
Journal | Rocky Mountain Journal of Mathematics |
Volume | 35 |
Issue number | 5 |
DOIs | |
State | Published - 2005 |
ASJC Scopus subject areas
- Mathematics(all)