Cesàro sequence spaces via (p, q)-calculus and compact matrix operators

Abstract

In this article, we construct (p, q)-analogue C(p, q) of Cesàro matrix $C_1$ of order 1 and study its properties. We introduce (p, q)-Cesàro sequence spaces ${\mathcal {X}}_s^{p,q}$ and ${\mathcal {X}}_{\infty }^{p,q}$ generated by the domain of matrix C(p, q) in the spaces $\ell _s$ and $\ell _{\infty },$ respectively. We study some topological properties and inclusion relations, obtain Schauder basis of ${\mathcal {X}}_s^{p,q}$ and $\alpha -,$ $\beta -$ and $\gamma -$duals of the spaces ${\mathcal {X}}_s^{p,q}$ and ${\mathcal {X}}_{\infty }^{p,q}.$ We characterize matrix mappings from the spaces ${\mathcal {X}}_s^{p,q}$ and ${\mathcal {X}}_{\infty }^{p,q}$ to space $\mu \in \{\ell _{\infty }, c, c_0\}.$ Finally, we characterize certain classes of compact operators on the newly defined spaces using Hausdorff measure of non compactness.