On quantum generalization of Catalan sequence spaces

Abstract

In this study, we construct $q$-analog $\mc(q)$ of Catalan matrix and study the sequence spaces $c_0 (\mc(q))$ and $c(\mc(q))$ defined as the domain of $q$-Catalan matrix $\mc(q)$ in the spaces $c_0$ and $c,$ respectively. We exhibit some topological properties, obtain Schauder bases and determine $\alpha$-, $\beta$-, and $\gamma$-duals of the spaces $c_0 (\mc(q))$ and $c(\mc(q)).$ Finally, we characterize certain class of matrix mappings from the spaces $c_0 (\mc(q))$ and $c(\mc(q))$ to the space $\mu=\{\ell_{\infty},c_0,c,\ell_1\}$ and give the necessary and sufficient conditions for a matrix operator to be compact.