The Spectrum of a Composition Operator and Calderón’s Complex Interpolation

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“…Independently, Donaway [1999, Corollary 3.11] and Martín and Vukotić [2005, Theorem 7] showed that composition operators on D induced by univalent selfmaps of ‫,ބ‬ and thus the automorphisms, have spectral radius 1. However, by direct calculation, one can see that the functions in F are not contained in the Dirichlet space; for the case of f s this is shown in [Pons 2010] (see p. 455). Despite D not satisfying all the properties of X , the spectrum of automorphism induced composition operators on D are precisely the same as those on X .…”

Compact differences of composition operators on weighted Dirichlet spaces

“…In this note we will further assume that the space X 0 ∩ X 1 is dense in both X 0 and X 1 and define the interpolation algebra I[X 0 X 1 ] to be the set of all linear operators T : X 0 ∩ X 1 → X 0 ∩ X 1 that are both 0-continuous and 1-continuous. The interpolation algebra defined above first appeared in the L -space setting in [2]; for properties and applications to the study of spectra, see [1,10,16,17].…”

“…One can see this by considering these spaces as weighted 2 -spaces or by considering the techniques developed in [13]; a direct proof of this nature can be found in [16]. [12] that boundedness of C on D α implies boundedness on D β when −1 < α < β.…”

Spectra of linear fractional composition operators on the Hardy and weighted Bergman spaces of the half-plane

“…In fact, in the unit disc setting the spectral picture of composition operators has been completely determined when the inducing maps are linear fractional transformations and the operators act on weighted Dirichlet spaces, including H 2 (D) and A 2 α (D) for all α > −1 (see, for instance, [17,2,10,8,19,7]). The spectra in the corresponding setting on the half-plane are largely unknown; in the (unweighted) Dirichlet space of Π + the spectra are known but besides that only the spectra (and the essential spectra) for invertible or self-adjoint parabolic and invertible hyperbolic composition operators acting on the Hardy space H 2 (Π + ) have been computed (see [15]).…”