TY - JOUR
T1 - Conjectures on spectra of composition operators and related issues
AU - Gallardo-Gutiérrez, Eva A.
AU - Matache, Valentin
N1 - Publisher Copyright:
© 2017 Elsevier Inc.
PY - 2017/12/15
Y1 - 2017/12/15
N2 - 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.
AB - 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.
KW - Composition operators
KW - Spectra
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U2 - 10.1016/j.aim.2017.10.022
DO - 10.1016/j.aim.2017.10.022
M3 - Article
AN - SCOPUS:85033669245
SN - 0001-8708
VL - 322
SP - 1085
EP - 1098
JO - Advances in Mathematics
JF - Advances in Mathematics
ER -