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

In this paper, we study the linear dynamical properties of shift operators on some classes Segal algebras. Moreover, characterize hypercyclic generalized bilateral standard Hilbert module.

We introduce a new concept of general $G$-$eta$-monotone operator generalizing the general $(H,eta)$-monotone operator cite{arvar2, arvar1}, general $H-$ monotone operator cite{xiahuang} in Banach spaces, and also generalizing $G$-$eta$-monotone operator cite{zhang}, $(A, eta)$-monotone operator cite{verma2}, $A$-monotone operator cite{verma0}, $(H, eta)$-monotone operator cite{fanghuang}...