Homotonic Algebras

TL;DR

This paper introduces the concept of homotonic algebras, proves the equivalence of two definitions, and provides a simple inequality to characterize sub-multiplicativity and stability of weighted sup norms within these algebras.

Contribution

It establishes the equivalence of different definitions of homotonicity and offers a straightforward inequality for analyzing norm properties in homotonic algebras.

Findings

Equivalence of two definitions of homotonicity.

A simple inequality characterizing sub-multiplicativity.

Criteria for strong stability of weighted sup norms.

Abstract

An algebra $\mathcal{A}$ of real or complex valued functions defined on a set $\mathbf{T}$ shall be called \textit{homotonic} if $\mathcal{A}$ is closed under forming of absolute values, and for all $f$ and $g$ in $\mathcal{A}$, the product $f\times g$ satisfies $|f\times g|\le|f|\times|g|$. Our main purpose in this paper is two-fold: To show that the above definition is equivalent to an earlier definition of homotonicity, and to provide a simple inequality which characterizes sub-multiplicativity and strong stability for weighted sup norms on homotonic algebras.