Banach spaces of sequences arising from infinite matrices

Given an infinite matrix $M=(m_{nk})$ we study a family of sequence spaces ${\mathrm{\ell }}_M^p$ associated with it. When equipped with a suitable norm $\mathrm{\parallel }\cdot {\mathrm{\parallel }}_{M,p}$ we prove some basic properties of the Banach spaces of sequences $({\mathrm{\ell }}_M^p,\mathrm{\parallel }\cdot {\mathrm{\parallel }}_{M,p})$. In particular we show that such spaces are separable and strictly/uniformly convex for a considerably large class of infinite matrices $M$ for all $p>1$. A special attention is given to the identification of the dual space $({\mathrm{\ell }}_M^p{)}^{\ast }$. Building on the earlier works of Bennett and Jägers, we extend and apply some classical factorization results to the sequence spaces ${\mathrm{\ell }}_M^p$.