On the domain of $q$-Euler matrix in $c$ and $c_0$ with its point spectra

Abstract

We introduce new Banach spaces $\mse^{\alpha,\beta}_0(q)$ and $\mse^{\alpha,\beta}_{c}(q)$ defined as the domain of generalized $q$-Euler matrix $\msE^{\alpha,\beta}(q)$ in the spaces $c_0$ and $c,$ respectively. Some topological properties and inclusion relations related to the newly defined spaces are exhibited. We determine the bases and obtain K\"othe duals of the spaces $\mse^{\alpha,\beta}_0(q)$ and $\mse^{\alpha,\beta}_{c}(q).$ We characterize certain matrix mappings from the spaces $\mse^{\alpha,\beta}_0(q)$ and $\mse^{\alpha,\beta}_{c}(q)$ to the space $\msS\in\{\ell_{\infty},c,c_0,\ell_{1},bs,cs,cs_0\}.$ We compute necessary and sufficient conditions for a matrix operator to be compact from the space $\mse^{\alpha,\beta}_0(q)$ to the space $\msS\in\{\ell_{\infty},c,c_0,\ell_{1},bs,cs,cs_0\}$ using Hausdorff measure of non-compactness. Finally, we give point spectrum of the matrix $\msE^{\alpha,\beta}(q)$ in the space $c.$