A GENERALIZED HILBERT OPERATOR ACTING ON CONFORMALLY INVARIANT SPACES

If p is a positive Borel measure on the interval (0,1), we let TLM 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,...,mu denotes the moment of order n of mu. This matrix formally induces the operator [GRAPHICS] on the space of all analytic functions f(z) = sigma(infinity)(k=o )a(k)z(k), in the unit disk D. This is a natural generalization of the classical Hilbert operator. The action of the operators H-mu on Hardy spaces has been recently studied. This article is devoted to a study of the operators H-mu acting on certain conformally invariant spaces of analytic functions on the disk such as the Bloch space, the space BMOA, the analytic Besov spaces, and the Q(s)-spaces.