Operators of Hilbert and Cesàro type acting on Dirichlet spaces

Abstract

If α > −1 the space of Dirichlet type D2 α consists of those functions f which are analytic in the unit disc D such that f belongs to the weighted Bergman space A2 α. The space D2 0 is the classical Dirichlet space D. If g is an analytic function in D, we study the generalized Hilbert operator Hg defined by Hg( f )(z) = 1 0 f (t)g (t z) dt acting on the spaces D2 α (0 ≤ α ≤ 1). We obtain a characterization of those g for which Hg is bounded, compact, or Hilbert-Schmidt on the Dirichlet space D. In addition to this, we use our results concerning the operators Hg to study certain Cesàro-type operators C(η) acting on the spaces D2 α (0 ≤ α ≤ 1). We give also a characterization of the positive finite Borel measures μ in [0, 1) for which a certain Cesàro type operator Cμ associated to μ is bounded on the Bergman space A1 α (α > −1). This is an extension of the previously known results for the spaces Ap α with p > 1 and α > −1