# Journal of Operator Theory

Volume 75, Issue 1, Winter 2016  pp. 21-48.

Spectra of composition operators with symbols in $\mathcal{S}(2)$

Authors:  Paul S. Bourdon
Author institution: Department of Mathematics, University of Virginia, Charlottesville, 22904, U.S.A.

Summary: Let $H^2(\mathbb{D})$ denote the classical Hardy space of the open unit disk $\mathbb{D}$ in the complex plane. We obtain descriptions of both the spectrum and essential spectrum of composition operators on $H^2(\mathbb{D})$ whose symbols belong to the class $\mathcal{S}(2)$ introduced by Kriete and Moorhouse (Trans. Amer. Math. Soc, 359(2007), 2915-2944). Our work reveals new possibilities for the shapes of composition-operator spectra, settling a conjecture of Cowen (J. Operator Theory 9(1983), 77-106). 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.

DOI: http://dx.doi.org/10.7900/jot.2014oct11.2044
Keywords: composition operator, Hardy space, spectrum, essential spectrum