Banach spaces of sequences arising from infinite matrices

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