Banach spaces of sequences arising from infinite matrices

Abstract

Given an infinite matrix $M=(m_{nk})$ we study a family of sequence spaces ${\ell }_M^p$ associated with it. When equipped with a suitable norm $\Vert \cdot {\Vert }_{M,p}$ we prove some basic properties of the Banach spaces of sequences $({\ell }_M^p,\Vert \cdot {\Vert }_{M,p})$. 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 $({\ell }_M^p{)}^{\ast }$. Building on the earlier works of Bennett and Jägers, we extend and apply some classical factorization results to the sequence spaces ${\ell }_M^p$.