Generalized Hilbert series operators

Abstract

In this note we study the generalized Hilbert series operators $H_{\mu}$, induced by a positive Bore measure $\mu$ on $[0, 1)$, between the weighted sequence spaces. We characterize the measures $\mu$ for which  $H_{\mu}$ is bounded between different weighted sequence spaces. Finally, for certain special measures, we obtain the sharp norm estimates of operators and establish some new generalized Hilbert series inequalities with the best constant factors.