A generalized Hilbert matrix acting on Hardy spaces

Abstract

If $\mu$ is a positive Borel measure on the interval $[0,1)$, the Hankel matrix ${\mathcal{H}}_{\mu }=({\mu }_{n,k}{)}_{n,k\ge 0}$ with entries ${\mu }_{n,k}={\int }_{[0,1)}{t}^{n+k}d\mu (t)$ induces formally the operator $${\mathcal{H}}_{\mu }(f)(z)=\sum _{n=0}^{\infty }\left(\sum _{k=0}^{\infty }{\mu }_{n,k}{a}_k\right)z^n$$ on the space of all analytic functions $f(z)={\sum }_{k=0}^{\infty }{a}_k{z}^k$, in the unit disc $\mathbb{D}$. In this paper we describe those measures $\mu$ for which ${\mathcal{H}}_{\mu }$ is a bounded (compact) operator from $H^p$ into $H^q$, $0<p,q<\infty$. We also characterize the measures $\mu$ for which ${\mathcal{H}}_{\mu }$ lies in the Schatten class $S_p(H^2)$, $1<p<\infty$.