The range of Hilbert operator and Derivative-Hilbert operator acting on $H^1$

Abstract: Let $\mu$ be a positive Borel measure on the interval $[0,1)$. The Hankel matrix $\mathcal{H}{\mu}=(\mu{n,k}){n,k\geq0}$ with entries $\mu{n,k}=\mu_{n+k}$, where $\mu_n=\int_{[0,1)}t^{n}d\mu(t)$. For $f(z)=\sum_{n=0}^{\infty}a_nz^n$ is an analytic function in $\mathbb{D}$, the Hilbert operator is defined by $$\mathcal{H}{\mu}(f)(z)=\sum{n=0}^{\infty}\Bigg(\sum_{k=0}^{\infty}\mu_{n,k}a_k\Bigg)z^n, \quad z\in \mathbb{D}.$$ The Derivative-Hilbert operator is defined as $$\mathcal{DH}{\mu}(f)(z)=\sum{n=0}^{\infty}\Bigg(\sum_{k=0}^{\infty}\mu_{n,k}a_k\Bigg)(n+1)z^n, \quad z\in \mathbb{D}.$$ In this paper, we determine the range of the Hilbert operator and Derivative-Hilbert operator acting on $H^{\infty}$.