A Hankel matrix acting on Hardy and Bergman spaces

Let be mu finite positive Borel measure on [0,1). Let H(mu) = (mu(n,k))(n,k >= 0) be the Hankel matrix with entries mu(n,k) = integral([0,1)) t(n+k) d mu(t). The matrix H(mu) induces formally an operator on the space of all analytic functions in the unit disc by the formally H(mu)(f)(z) = Sigma(infinity)(n=0)(Sigma(infinity)(k=0)mu(n,k)ak)z(n), z is an element of D, where f (z) = Sigma(infinity)(n=0)a(n)z(n) is an analytic function in D. We characterize those positive Borel measures on [0,1) such that H(mu)(f)(z) = integral([0,1)) f(t)/1-tz d mu(t) for all f in the Hardy space H(1), and among them we describe those for which H(mu) is bounded and compact on H(1). We also study the analogous problem for the Bergman space A(2).