Generalized Hilbert operators arising from Hausdorff matrices

Abstract

For a finite, positive, Borel measure $\mu$ on $(0,1)$ we consider an infinite matrix ${\Gamma }_{\mu }$, related to the classical Hausdorff matrix defined by the same measure $\mu$, in the same algebraic way that the Hilbert matrix is related to the Cesáro matrix. When $\mu$ is the Lebesgue measure, ${\Gamma }_{\mu }$ reduces to the classical Hilbert matrix. We prove that the matrices ${\Gamma }_{\mu }$ are not Hankel, unless $\mu$ is a constant multiple of the Lebesgue measure, we give necessary and sufficient conditions for their boundedness on the scale of Hardy spaces $H^p,1\le p<\infty$, and we study their compactness and complete continuity properties. In the case $2\le p<\infty$, we are able to compute the exact value of the norm of the operator.

16 pages

Cited by in corpus (2)

• Hilbert matrix operator acting between conformally invariant spaces
• Generalized Hilbert matrix operators acting on Bergman spaces