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

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 ${\mathrm{\ell }}^p$ and calculate their norm.