A Derivative-Hilbert operator acting on Hardy spaces

Abstract

Let $\mu$ be a positive Borel measure on the interval [0,1). The Hankel matrix ${\mathcal{H}}_{\mu }=({\mu }_{n,k}{)}_{n,k\ge 0}$ with entries ${\mu }_{n,k}={\mu }_{n+k}$, where ${\mu }_n={\int }_{[0,1)}{t}^{n}d\mu (t)$, induces formally the operator $${\mathcal{D}\mathcal{H}}_{\mu }(f)(z)=\sum _{n=0}^{\infty }(\sum _{k=0}^{\infty }{\mu }_{n,k}{a}_k)(n+1)z^n$$ 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 ${\mathcal{D}\mathcal{H}}_{\mu }(f)(z)={\int }_{[0,1)}\frac{f(t)}{(1-tz{)}^2}d\mu (t)$ for all in Hardy spaces $H^p(0<p<\infty )$, and among them we describe those for which ${\mathcal{D}\mathcal{H}}_{\mu }$ is a bounded(resp.,compact) operator from $H^p(0<p<\infty )$ into $H^q(q>p$ and $q\ge 1$). We also study the analogous problem in Hardy spaces $H^p(1\le p\le 2)$.