Wednesday, 05 March, 2014
SPEAKER: Prof. Marie Neophytou, Belmont University
TITLE: Eigenvalues of the Adjoints of Some Composition Operators and Weighted Composition Operators
ABSTRACT: Let H 2 be the Hardy-Hilbert space of analytic functions on the unit disk. If ϕ is an analytic map of the unit disk into itself and ψ is analytic on the disk, the composition operator Cϕ with symbol ϕ is defined by Cϕ f = f ◦ ϕ, and the weighted composition operator Wψ,ϕ by Wψ,ϕ f = ψ(f ◦ ϕ), for f in H 2 .
We look at adjoints of composition operators with symbols ϕ that have a fixed point inside the disk and a fixed point on the boundary with finite angular derivative there. By imposing a few extra assumptions on ϕ, we show that the point spectrum of the adjoint contains a disk centered at the origin, and that the corresponding eigenspaces are infinite-dimensional. We also identify a subspace of H 2 which is invariant for the adjoint and on which the adjoint acts like a weighted shift. Finally, we generalize these results to weighted composition operators.