Operators Induced by Radial Measures Acting on the Dirichlet Space

Abstract

Let {mathbb {D}} be the unit disc in the complex plane. Given a positive finite Borel measure mu on the radius [0, 1), we let mu _n denote the n-th moment of mu and we deal with the action on spaces of analytic functions in {mathbb {D}} of the operator of Hibert-type {mathcal {H}}_mu and the operator of Cesàro-type {mathcal {C}}_mu which are defined as follows: If f is holomorphic in {mathbb {D}}, f(z)=sum _{n=0}^infty a_nz^n (zin {mathbb {D}}), then {mathcal {H}}_mu (f) is formally defined by {mathcal {H}}_mu (f)(z) = sum _{n=0}^infty left( sum _{k=0}^infty mu _{n+k}a_kright) z^n (zin {mathbb {D}}) and {mathcal {C}}_mu (f) is defined by mathcal C_mu (f)(z) = sum _{n=0}^infty mu _nleft( sum _{k=0}^na_kright) z^n (zin {mathbb {D}}). These are natural generalizations of the classical Hilbert and Cesàro operators. A good amount of work has been devoted recently to study the action of these operators on distinct spaces of analytic functions in {mathbb {D}}. In this paper we study the action of the operators {mathcal {H}}_mu and {mathcal {C}}_mu on the Dirichlet space {mathcal {D}} and, more generally, on the analytic Besov spaces B^p (1le p<infty ).