On the boundedness of certain generalized Hilbert operators in $\ell^p$

TL;DR

This paper investigates the boundedness of generalized Hilbert operators in ^p, providing necessary and sufficient conditions and computing their norms, extending classical results about the Hilbert matrix.

Contribution

It introduces a comprehensive criterion for the boundedness of generalized Hilbert operators derived from measures on [0,1], generalizing the classical Hilbert matrix case.

Findings

Derived a necessary and sufficient condition for boundedness in ^p

Calculated the operator norms explicitly

Extended classical Hilbert matrix results to a broader class of measures

Abstract

The Hilbert matrix $\mathcal{H}_{n,m} = (n+m+ 1)^{-1}$ has been extensively studied in previous literature. In this paper we look at generalized Hilbert operators arising from measures on the interval $[0, 1]$, such that the Hilbert matrix is obtained by the Lebesgue measure. We provide a necessary and sufficient condition for these operators to be bounded in $\ell^p$ and calculate their norm.