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 ${\mathcal{H}}_{\mu }=({\mu }_{n,k}{)}_{n,k\ge 0}$ with entries ${\mu }_{n,k}={\mu }_{n+k}$, where ${\mu }_n={\int }_{[0,1)}{t}^{n}d\mu (t)$. For $f(z)={\sum }_{n=0}^{\infty }{a}_n{z}^n$ is an analytic function in $\mathbb{D}$, the Hilbert operator is defined by $${\mathcal{H}}_{\mu }(f)(z)=\sum _{n=0}^{\infty }\left(\sum _{k=0}^{\infty }{\mu }_{n,k}{a}_k\right)z^n,z\in \mathbb{D}.$$ The Derivative-Hilbert operator is defined as $${\mathcal{D}\mathcal{H}}_{\mu }(f)(z)=\sum _{n=0}^{\infty }\left(\sum _{k=0}^{\infty }{\mu }_{n,k}{a}_k\right)(n+1)z^n,z\in \mathbb{D}.$$ In this paper, we determine the range of the Hilbert operator and Derivative-Hilbert operator acting on $H^{\infty }$.

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