Note on Hilbert-Schmidt composition operators on weighted Hardy spaces

Abstract

We show that if Cφ is a Hilbert-Schmidt composition operator on an appropriately weighted Hardy space, then there exists a capacity, associated to the weight sequence of the space, so that the set on which the radial limit of φ is unimodular has capacity zero. This extends recent results by Gallardo-Gutierrez and Gonzalez. Let D be the open unit disk in the complex plane and suppose that (X, ‖ · ‖) is a Hilbert space of analytic functions on D. We say that X is a weighted Hardy space if the set {z : j = 0, 1, 2, . . . } of monomials is a complete orthogonal system. We put βj = ‖z‖. Then β := {βj} is called the weight sequence and X is denoted by H(β). Many classical function spaces are weighted Hardy spaces. For example, the standard Hardy spaceH, the α-Dirichlet space Dα, 0 ≤ α < 1, of all analytic functions whose first derivative is square integrable with respect to the measure (1 − |z|)dA(z), and the Bergman space A of all square integrable analytic functions are particular instances of H(β) with βj ≡ 1, βj ∼ (1 + j)(1−α)/2 and βj = (1 + j)−1/2 respectively. Now suppose that φ : D → D is an analytic self-map of the unit disk and consider the corresponding composition operator Cφ acting on H(β), i.e. Cφ(f) = f ◦ φ, f ∈ H(β). We are interested in the behavior of those φ which induce HilbertSchmidt composition operators. 2000 Mathematics Subject Classification. 47B33, 30C85, 31A20.