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

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

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