Product-type operators acting between Dirichlet and Zygmund-type spaces

Abstract

LetD = {z ? C : |z| < 1} be the open unit disk in the complex plane C. By H(D), denote the space of all holomorphic functions on D. For an analytic self map ? on D and u, v ? H(D), we have a product type operator Tu,v,? defined by Tu,v,? f (z) = u(z) f (?(z)) + v(z) f ?(?(z)), f ? H(D), z ? D, This operator is basically a combination of three other operators namely composition operator, multiplication operator and differentiation operator. We study the boundedness and compactness of this operator from Dirichlet-type spaces to Zygmund-type spaces.