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 $(\ell _M^p )^*$. Building on the earlier works of Bennett and Jägers, we extend and apply some classical factorization results to the sequence spaces $\ell _M^p$.