Vol. 9, No. 5, 2016

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Spectrum of a composition operator with automorphic symbol

Robert F. Allen, Thong M. Le and Matthew A. Pons

Vol. 9 (2016), No. 5, 813–829
Abstract

We give a complete characterization of the spectrum of composition operators, induced by an automorphism of the open unit disk, acting on a family of Banach spaces of analytic functions that includes the Bloch space and $BMOA$. We show that for parabolic and hyperbolic automorphisms the spectrum is the unit circle. For the case of elliptic automorphisms, the spectrum is either the unit circle or a finite cyclic subgroup of the unit circle.

Keywords
composition operator, spectrum, automorphism
Mathematical Subject Classification 2010
Primary: 47A10, 47B33
Secondary: 30H05