Generalized Hilbert operators arising from Hausdorff matrices

TL;DR

This paper introduces a class of generalized Hilbert operators derived from Hausdorff matrices, analyzes their boundedness, compactness, and norm on Hardy spaces, and characterizes when they are Hankel matrices.

Contribution

It defines and studies a new family of operators related to Hausdorff matrices, extending classical Hilbert matrix theory to a broader measure-based context.

Findings

Matrices are not Hankel unless measure is Lebesgue.

Provides necessary and sufficient conditions for boundedness on Hardy spaces.

Calculates exact operator norm for p ≥ 2.

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 Ces\'aro 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 $H^p, \, 1 \leq p < \infty$, and we study their compactness and complete continuity properties. In the case $2\leq p<\infty$, we are able to compute the exact value of the norm of the operator.