Rhaly Operators Acting on Hardy, Bergman, and Dirichlet Spaces

Abstract In this article we address the question of characterizing the sequences of complex numbers () =\ ₙ\₍=₀^ (η) = η n n = 0 ∞ whose associated Rhaly operator {R} () R (η) is bounded or compact on the Hardy spaces Hᵖ H p (1 p 1 ≤ p ∞), on the Bergman spaces Aᵖ_ A α p, and on the Dirichlet spaces {D}ᵖ_ D α p (1 p 1 ≤ p ∞, >-1 α > - 1). We give a number of conditions which are either necessary or sufficient for the boundedness (compactness) of {R} () 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 () F (η) defined by F () (z) = ₍=₀^ ₙzⁿ F (η) (z) = ∑ n = 0 ∞ η n z n