The Numerical Ranges of Automorphic Composition Operators

Abstract

We investigate the shape of the numerical range for composition operators induced on the Hardy space H 2 by conformal automorphisms of the unit disc. We show that usually, but not always, such operators have numerical ranges whose closures are discs centered at the origin. Surprising open questions arise from our investigation.

When is zero in the numerical range of a composition operator?

“…Each such map fixes the point p = α −1 (0) ∈ U. Now as we have previously noted, α = ρ ω • α p for some ω ∈ ∂U, where α p is the special (self-inverse) automorphism defined by (4), and ρ ω is the rotation defined by ρ ω (z) ≡ ωz. From this it follows easily that ϕ = α p • δ r • α p .…”

“…Zero-Inclusion for Automorphic Composition Operators. Numerical ranges of composition operators induced by conformal automorphisms are the focus of the authors' paper [4], where it is shown such automorphic composition operators usually, but not always, have numerical ranges whose closures are discs centered at the origin. Conformal automorphisms of U, which as we have already noted are always linear-fractional maps, are classified as follows.…”

“…Elliptic: Conformally conjugate to a rotation about the origin; two fixed points in the sphere, one in U, one outside its closure. By Theorem 3.1 of [4], if ϕ is an automorphism of U that is either of hyperbolic or parabolic type, then the closure of W (C ϕ ) is a disc centered at the origin, from which it follows that 0 ∈ int W (C ϕ ). The situation for elliptic automorphisms is more interesting.…”

“…Suppose that ϕ is given by (5). If ω is not a root of unity, then by Theorem 4.1 of [4], the closure of W (C ϕ ) is a disk centered at the origin so that, just as for the other automorphic composition operators, W (C ϕ ) contains 0 in its interior. To answer the Zero-Inclusion Question when ω is a root of unity, we observe that for every nonnegative integer n, the map α n p will be an eigenvector for C ϕ with eigenvalue ω n .…”

“…If ω = 1, then, of course, ϕ is the identity mapping and W (C ϕ ) = {1}. If ω = −1, then by Corollary 4.4 of [4], W (C ϕ ) is a ellipse with foci −1 and 1 that is degenerate if and only if p = 0 (in which case ϕ(z) = −z). Summarizing, we see that if ϕ is an elliptic automorphism, then 0 ∈ W (C ϕ ) unless ϕ(z) = z, and 0…”