Composition operator on Bloch type spaces

Abstract

Let B and B 0 denote the α−Bloch spaces and little α−Bloch spaces. An analytic map φ of the unit disk into itself induces an operator Cφ on analytic functions by composition. We study the boundedness and compactness of composition operators Cφ from B(B 0 ) to B β(B 0 ) for 0 < α, β < ∞. 2000 Mathematics Subject Classification: 47B38, 47B07, 30D55. 1. NOTATIONS AND RESULTS Let D denote the unit disk in the complex plane, H(D) be the space of all analytic functions on D. A function f ∈ H(D) is said to belong to the α−Bloch space B if ‖f‖α = |f(0)|+ sup z∈D (1− |z|2)α|f ′(z)| < ∞, and to the little α−Bloch space B 0 if lim |z|→1 (1− |z|2)αf ′(z) = 0. It is well-known that B is a Banach space under the above norm and B 0 is a closed subspace of B (see, for example, [13], [7] for more information on α−Bloch and little α−Bloch spaces). If α = 1, B and B 0 become the well-known Bloch space and little Boch space [2]. Let φ : D → D be an analytic self-map of the unit disk. The composition operator Cφ induced by such φ is the linear map on the spaces of all analytic functions on the unit disk defined by Cφ(f) = f ◦ φ. (1.1)