Necessary and sufficient conditions are established for a composition operator C(phi)f = f o phi to be bounded or compact on the Bers-type space H-alpha(infinity) and the little Bers-type space H-alpha(infinity). The boundedness and compactness of the composition operator C-phi on A(infinity)(phi) are characterized, which generalize the case of C-phi on H-alpha(infinity).
For α ∈ (0, ∞), let Hα∞ (or Hα,0∞) denote the collection of all functions f which are analytic on the unit disc D and satisfy |f(z)|(1-|z|2)α = O(1) (or |f(z)|(1 - |z|2)α = o(1) as |z| → 1). Hα∞(or Hα,0∞) is called a Bers-type space (or a little Bers-type space).In this paper, we give some basic properties of Hα∞. C, the composition operator associated with a symbol function which is an analytic self map of D, is difined by Cf = f o . We characterize the boundedness and compactness of C which sends one Bers-type space to another function space.
