Hilbert matrix operator acting between conformally invariant spaces

In this article we study the action of the the Hilbert matrix operator $\mathcal{H}$ from the space of bounded analytic functions into conformally invariant Banach spaces. In particular, we describe the norm of $\mathcal{H}$ from $H^{\mathrm{\infty }}$ into $\text{BMOA}$ and we characterize the positive Borel measures $\mu$ such that $\mathcal{H}$ is bounded from $H^{\mathrm{\infty }}$ into the conformally invariant Dirichlet space $M({\mathcal{D}}_{\mu })$. For particular measures $\mu$, we also provide the norm of $\mathcal{H}$ from $H^{\mathrm{\infty }}$ into $M({\mathcal{D}}_{\mu })$.