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 $H^p(1\leq p\leq 2)$.
Categories: math.CV math.FA