Generalized Hilbert series operators

TL;DR

This paper investigates the properties of generalized Hilbert series operators induced by positive measures, characterizing their boundedness between weighted sequence spaces and deriving sharp norm estimates and inequalities.

Contribution

It provides a characterization of measures for boundedness, sharp norm estimates, and new inequalities for generalized Hilbert series operators.

Findings

Characterized measures for boundedness of $H_{mu}$

Derived sharp norm estimates for specific measures

Established new generalized Hilbert series inequalities with optimal constants

Abstract

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