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

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.

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