A generalized Hilbert operator acting on conformally invariant spaces

Abstract

If μ is a positive Borel measure on the interval [0,1), we let H_μ be the Hankel matrix with entries μ_{n,k}=μ_{n+k}, where μ_n denotes the moment of order n of the measure μ. This matrix formally induces an operator on the space of all analytic functions in the unit disk D. This is a natural generalization of the classical Hilbert operator. The action of the operators H_μ on Hardy spaces has been recently studied. This article is devoted to a study of the operators H_μ 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.