Conjectures on spectra of composition operators and related issues

A conjecture posed by Cowen and MacCluer in 1995 states that the spectrum of composition operators acting on the Hardy space H2 induced by analytic selfmaps of the open unit disc having a fixed point in that disc, other than the identity or elliptic automorphisms, is always representable as the union of a connected set containing the origin, the so called, Schröder eigenvalues of the inducing selfmap, and 1. Our first result is proving that the aforementioned conjecture holds. We also consider two related conjectures regarding the spectra of composition operators, proving that one of them holds, and showing that the other one (which is still open) is satisfied in particular cases.