# 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

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