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

TL;DR

This paper characterizes the range of the Hilbert and Derivative-Hilbert operators when acting on the space of bounded analytic functions, $H^{ ext{infty}}$, providing insights into their behavior and boundedness.

Contribution

It determines the range of both the Hilbert and Derivative-Hilbert operators on $H^{ ext{infty}}$, a problem not previously addressed.

Findings

The range of the Hilbert operator on $H^{ extinfty}$ is explicitly characterized.

The range of the Derivative-Hilbert operator on $H^{ extinfty}$ is explicitly characterized.

Results contribute to understanding the boundedness and mapping properties of these operators.

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 $H^{\infty}$.