Hilbert-Type Operators Acting on Bergman Spaces

If mu is a positive Borel measure on the interval [0, 1) we let H-mu be the Hankel matrix H-mu=(mu(n,k))(n,k >= 0) with entries mu(n, k)=mu(n+k), where, for n=0,1,2,& mldr;, mu(n) denotes the moment of order n of mu. This matrix formally induces an operator, called also H-mu, on the space of all analytic functions in the unit disc D as follows: If f is an analytic function in D, f(z)=& sum;(infinity)(k=0)a(k)z(k), z is an element of D, H-mu(f) is formally defined by H-mu(f)(z)=& sum;(infinity )(n=0)(& sum;(infinity)(k=0)mu(n+k)a(k)) z(n), z is an element of D. This is a natural generalization of the classical Hilbert operator. This paper is devoted to studying the operators H-mu acting on the Bergman spaces A(p), 1 <= p < infinity. Among other results, we give a complete characterization of those mu for which H-mu is bounded or compact on the space A(p) when p is either 1 or greater than 2. We also give a number of results concerning the boundedness and compactness of H-mu on A(p) for the other values of p, as well as on its membership in the Schatten classes S-p(A(2)).