A Derivative-Hilbert operator acting on BMOA space

Abstract: Let $\mu$ be a positive Borel measure on the interval $[0,1)$. The Hankel matrix $\mathcal{H}{\mu}=(\mu{n,k}){n,k\geq 0}$ with entries $\mu{n,k}=\mu_{n+k}$, where $\mu_{n}=\int_{[0,1)}t^nd\mu(t)$, induces, formally, the Derivative-Hilbert operator $$\mathcal{DH}\mu(f)(z)=\sum{n=0}^\infty\left(\sum_{k=0}^\infty \mu_{n,k}a_k\right)(n+1)z^n , ~z\in \mathbb{D},$$ where $f(z)=\sum_{n=0}^\infty a_nz^n$ is an analytic function in $\mathbb{D}$. We characterize the measures $\mu$ for which $\mathcal{DH}\mu$ is a bounded operator on $BMOA$ space. We also study the analogous problem from the $\alpha$-Bloch space $\mathcal{B}\alpha(\alpha>0)$ into the $BMOA$ space.