Banach spaces of sequences arising from infinite matrices

TL;DR

The paper develops a general framework for Banach spaces generated by infinite matrices via the family $\ell^p_M$ with norm $\|x\|_{M,p}$, generalizing Ces\'aro-type spaces. It proves that $\ell^p_M$ is a separable Banach space (when $M$ has no vanishing columns) and establishes strict and uniform convexity under natural structural assumptions on $M$, leading to reflexivity in the uniform-convexity regime. A central contribution is the Bennett–Jägers-inspired factorization, introducing $d_M(p)$ and $g_M(q)$ and showing $\ell^p_M=\ell^p\cdot g_M(q)$ under suitable conditions, which yields a precise identification $(\ell^p_M)^*\cong d_M(q)$. The results unify duality theory for a broad class of matrix-induced sequence spaces, with Ces\'aro and related summability matrices as concrete examples and applications. This framework links infinite-matrix methods with classical sequence-space theory, providing tools to analyze duals, reflexivity, and convexity across a wide family of matrix-based spaces.

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 $\|\cdot\|_{M,p}$ we prove some basic properties of the Banach spaces of sequences $(\ell_M^p,\|\cdot\|_{M,p})$. 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 $\ell_M^p$.

Theorems & Definitions (35)

• Lemma 2.1
• Proposition 3.1
• proof
• Theorem 3.1
• proof
• Proposition 3.2
• proof
• Proposition 3.3
• proof
• Theorem 3.2