Operators of Hilbert and Cesàro type acting on Dirichlet spaces

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If α>-1-1$$\\end{document}]]> the space of Dirichlet type Dα2 consists of those functions f which are analytic in the unit disc D such that f′ belongs to the weighted Bergman space Aα2. The space D02 is the classical Dirichlet space D. If g is an analytic function in D, we study the generalized Hilbert operator Hg defined by Hg(f)(z)=∫01f(t)g′(tz)dtacting on the spaces Dα2 (0≤α≤1). We obtain a characterization of those g for which Hg is bounded, compact, or Hilbert-Schmidt on the Dirichlet space D. In addition to this, we use our results concerning the operators Hg to study certain Cesàro-type operators C(η) acting on the spaces Dα2 (0≤α≤1). We give also a characterization of the positive finite Borel measures μ in [0, 1) for which a certain Cesàro type operator Cμ associated to μ is bounded on the Bergman space Aα1 (α>-1-1$$\\end{document}]]>). This is an extension of the previously known results for the spaces Aαp with p>11$$\\end{document}]]> and α>-1-1$$\\end{document}]]>.