On lower and upper bounds of matrices

TL;DR

This paper provides alternative proofs for bounds on non-negative matrices acting on non-increasing sequences in various $l^p$ spaces, and explicitly determines these bounds for specific matrix families.

Contribution

It offers an alternate proof approach for known bounds and explicitly calculates bounds for certain matrix classes, extending previous results.

Findings

Established lower bounds for $p \,\geq\, 1$

Derived upper bounds for $0 < p \leq 1$

Explicit bounds for specific matrix families

Abstract

Using an approach of Bergh, we give an alternate proof of Bennett's result on lower bounds for non-negative matrices acting on non-increasing non-negative sequences in $l^p$ when $p \geq 1$ and its dual version, the upper bounds when $0<p \leq 1$. We also determine such bounds explicitly for some families of matrices.