A generalized Hilbert operator acting on conformally invariant spaces

Abstract

If μ is a positive Borel measure on the interval [0,1), we let Hμ be the Hankel matrix Hμ=(μn,k)n,k≥0 with entries μn,k=μn+k, where, for n=0,1,2,…, μn denotes the moment of order n of μ. This matrix formally induces the operator Hμ(f)(z)=∑n=0∞(∑k=0∞μn,kak)zn on the space of all analytic functions f(z)=∑k=0∞akzk, in the unit disk D. This is a natural generalization of the classical Hilbert operator. The action of the operators Hμ on Hardy spaces has been recently studied. This article is devoted to a study of the operators Hμ 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 Qs-spaces.