Spectra of Composition Operators with Symbols in S(2)

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.