If is a positive Borel measure on the interval , we let be the Hankel matrix with entries , where, for , denotes the moment of order of . This matrix formally induces the operator
on the space of all analytic functions , in the unit disk . This is a natural generalization of the classical Hilbert operator. The action of the operators on Hardy spaces has been recently studied. This article is devoted to a study of the operators 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 -spaces.