Rhaly Operators Acting on Hardy, Bergman, and Dirichlet Spaces

Abstract

In this article we address the question of characterizing the sequences of complex numbers (η) = {ηn}∞ n=0 whose associated Rhaly operator R(η) is bounded or compact on the Hardy spaces H p (1 ≤ p < ∞), on the Bergman spaces Ap α, and on the Dirichlet spaces Dp α (1 ≤ p < ∞, α > −1). We give a number of conditions which are either necessary or sufficient for the boundedness (compactness) of R(η) on these spaces. These conditions have to do with the membership in certain mean Lipschitz spaces of analytic functions of the function F(η) defined by F(η)(z) = ∞ n=0 ηn zn (z ∈ D). We prove that if 2 ≤ p < ∞ and ηn = O 1 n

, then R(η) is bounded on H p. However, there exists a sequence (η) with ηn = O 1 n such that the operator R(η) is not bounded on H p for 1 ≤ p < 2. We deal also with the derivative-Hardy spaces. For p > 0 the derivative-Hardy space S p consists of those functions f , analytic in the unit disc D, such that f ∈ H p. We prove that if 1 ≤ p < ∞ and 1 < q < ∞ then R(η) is a bounded operator from S p into Sq if and only if it is compact and this happens if and only if F(η) ∈ Sq .