A Hilbert-type linear operator with the norm and its applications (Q1035509)

The author defines a Hilbert-type linear operator on the space \(l^p_\varphi:=\{\{a_n\}: \left(\sum_{n=0}^\infty\varphi(n)|a_n|^p\right)^{1/p}<\infty\},\) where \(\varphi\) is a weight function. He gives a more precise operator inequality with the norm and its equivalent forms and their equivalent reverses. He also proves that the presented constant factors in these inequalities are the best possible. See also \textit{B.-C.\thinspace Yang} [J.~Math.\ Anal.\ Appl.\ 325, No.\,1, 529--541 (2007; Zbl 1114.47010)] and \textit{Y.-J.\thinspace Li}, \textit{Z.-P.\thinspace Wang} and \textit{B.\,He} [J.~Inequal.\ Appl.\ 2007, Article ID 82138 (2007; Zbl 1137.47011)].