If $\mu $ is a positive Borel measure on the interval $[0, 1)$ we let $\mathcal H_\mu $ be the Hankel matrix $\mathcal H_\mu =(\mu_{ n, k})_{ n,k\ge 0}$ with entries $\mu_{ n, k}=\mu_{ n+k}$, where, for $n\,=\,0, 1, 2, \dots $, $\mu_n$ denotes the moment of orden $n$ of $\mu $. This matrix induces formally the operator $$\mathcal{ H}_\mu (f)(z)= \sum_{ n=0}^{ \infty}\left(\sum_{ k=0}^{ \infty} \mu_{ n,k}{ a_k}\right)z^n$$ on the space of all analytic functions $f(z)=\sum_{ k=0}^\infty a_kz^k$, in the unit disc $\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 paper is devoted to study the operators $H_\mu $ acting on certain conformally invariant spaces of analytic functions on the disc such as the Bloch space, $BMOA$, the analytic Besov spaces, and the $Q_s$ spaces.
Categories: math.CV