Mean Lipschitz spaces and a generalized Hilbert operator

If $\mu$ is a positive Borel measure on the interval $[0,1)$ we let ${\mathcal{H}}_{\mu }$ be the Hankel matrix $\mathcal H_\mu =(\mu {n, k}){n,k\ge 0}$withentries$\mu _{n, k}=\mu _{n+k}$,where,for$n,=,0, 1, 2, \dots $,$\mu_n$denotesthemomentoforder$n$of$\mu $. This matrix induces formally the operator $${\mathcal{H}}_{\mu }(f)(z)=\sum _{n=0}^{\mathrm{\infty }}\left(\sum _{k=0}^{\mathrm{\infty }}{\mu }_{n,k}{a}_k\right)z^n$$ on the space of all analytic functions $f(z)={\sum }_{k=0}^{\mathrm{\infty }}{a}_k{z}^k$, in the unit disc $\mathbb{D}$. This is a natural generalization of the classical Hilbert operator. In this paper we study the action of the operators ${\mathcal{H}}_{\mu }$ on mean Lipschitz spaces of analytic functions.