Generalized Hilbert Operator Acting on Hardy Spaces

Abstract

Let $\hat{I}\pm >0$ and $\hat{I}¼$ be a positive Borel measure on the interval $[0,1)$. The Hankel matrix ${\mathcal{H}}_{\hat{I}¼,\hat{I}\pm }=(\hat{I}{¼}_{n,k,\hat{I}\pm }{)}_{n,k\ge 0}$ with entries $\hat{I}{¼}_{n,k,\hat{I}\pm }={\int }_{[0,1)}^{}\frac{\hat{I}“(n+\hat{I}\pm )}{\hat{I}“(n+1)\hat{I}“(\hat{I}\pm )}{t}^{n+k}d\hat{I}¼(t)$, induces, formally, the generalized-Hilbert operator as $${\mathcal{H}}_{\hat{I}¼,\hat{I}\pm }\left(f\right)\left(z\right)=\sum _{n=0}^{\infty }\left(\sum _{k=0}^{\infty }\hat{I}{¼}_{n,k,\hat{I}\pm }{a}_k\right)z^n,z\in \mathbb{D}$$ where $f(z)={\sum }_{k=0}^{\infty }{a}_k{z}^k$ is an analytic function in $\mathbb{D}$. This article is devoted study the measures $\hat{I}¼$ for which ${\mathcal{H}}_{\hat{I}¼,\hat{I}\pm }$ is a bounded(resp., compact) operator from $H^p(0<p\le 1)$ into $H^p(1\le q<\infty )$. Then, we also study the analogous problem in the Hardy spaces $H^p(1\le p\le 2)$. Finally, we obtain the essential norm of ${\mathcal{H}}_{\hat{I}¼,\hat{I}\pm }$ from $H^p(0<p\le 1)$ into $H^p(1\le q<\infty )$.