Spectra of Composition Operators with Symbols in S(2)

TL;DR

This paper characterizes the spectrum and essential spectrum of composition operators with symbols in S(2) on the Hardy space, revealing new spectral shapes and settling a longstanding conjecture.

Contribution

It provides the first detailed spectral descriptions for composition operators with symbols in S(2), including new spectral shape possibilities and a resolution of Cowen's conjecture.

Findings

Describes spectrum and essential spectrum for S(2) symbols

Identifies new spectral shape possibilities

Settles Cowen's spectral conjecture

Abstract

Let H^2(D) denote the classical Hardy space of the open unit disk D in the complex plane. We obtain descriptions of both the spectrum and essential spectrum of composition operators on H^2(D) whose symbols belong to the class S(2) introduced by Kriete and Moorhouse [Trans. Amer. Math. Soc., 359, 2007]. Our work reveals new possibilities for the shapes of composition-operator spectra, settling a conjecture of Cowen's [J. Operator Th. 9, 1983]. Our results depend on a number of lemmas, perhaps of independent interest, that provide spectral characterizations of sums of elements of a unital algebra over a field when certain pairwise products of the summands are zero.