A Derivative-Hilbert operator acting on Hardy spaces

TL;DR

This paper introduces a derivative-Hilbert operator on Hardy spaces, characterizes the measures for which it acts as an integral operator, and studies its boundedness and compactness properties across various Hardy space settings.

Contribution

It provides a complete characterization of measures inducing the derivative-Hilbert operator and analyzes its boundedness and compactness on Hardy spaces.

Findings

Characterization of measures for the operator representation

Conditions for boundedness on Hardy spaces

Conditions for compactness on Hardy spaces

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 $\mu_n=\int_{ [0,1)}t^nd\mu(t)$, induces formally the operator $$\mathcal{DH}_\mu(f)(z)=\sum_{n=0}^\infty (\sum_{k=0}^\infty \mu_{n,k}a_k)(n+1)z^n$$ on the space of all analytic function $f(z)=\sum_{k=0}^ \infty a_k z^n$ in the unit disc $\mathbb{D}$. We characterize those positive Borel measures on $[0,1)$ such that $\mathcal{DH}_\mu(f)(z)= \int_{[0,1)} \frac{f(t)}{{(1-tz)^2}} d\mu(t)$ for all in Hardy spaces $H^p(0<p<\infty)$, and among them we describe those for which $\mathcal{DH}_\mu$ is a bounded(resp.,compact) operator from $H^p(0<p <\infty)$ into $H^q(q > p$ and $q\geq 1$). We also study the analogous problem in Hardy spaces $H^p(1\leq p\leq 2)$.