On some Lambda–Pascal sequence spaces and compact operators

Abstract

We introduce Lambda–Pascal sequence spaces ${\ell }_q(G)$, $c_0(G)$, $c(G)$ and ${\ell }_{\infty }(G)$ generated by the matrix $G$ which is obtained by the product of Pascal matrix and $\mathrm{\Lambda }$-matrix. It is proved that the Lambda–Pascal sequence spaces ${\ell }_q(G)$, $c_0(G)$, $c(G)$ and ${\ell }_{\infty }(G)$ are $\mathrm{BK}$-spaces and linearly isomorphic to ${\ell }_q$, $c_0$, $c$ and ${\ell }_{\infty }$, respectively. We construct Schauder bases and obtain $\alpha$-, $\beta$- and $\gamma$-duals of the new spaces. We state and prove characterization theorems related to matrix transformation from the space ${\ell }_q(G)$ to the spaces ${\ell }_{\infty }$, $c$ and $c_0$. Finally, we determine necessary and sufficient conditions for a matrix operator to be compact from the space $c_0(G)$ to any one of the spaces ${\ell }_{\infty }$, $c$, $c_0$ or ${\ell }_1$.