A difference operator of negative order and its associated sequence space

Abstract

Let $\Delta^{-n}$ be the backward difference operator of order $-n$. In this paper, we firstly study some properties of the matrix domain associated with this matrix and after computing $\alpha$-, $\beta$-, $\gamma$-duals, we characterize some matrix classes and compact operators on this space. Moreover, we introduce two factorizations for the infinite Ces\`{a}ro and Hilbert operators based on the backward difference operator of order $-n$ which lead us to a generalization of the Hilbert's inequality. We also investigate the problem of finding the norm of Ces\`{a}ro and Hilbert operators from the sequence space $\ell_p (\Delta^{-n})$ into the sequence space $\ell_p$.