Galois connection between Lipschitz and linear operator ideals and minimal Lipschitz operator ideals

TL;DR

This paper explores the relationship between Lipschitz and linear operator ideals using Galois connections, introduces minimal Lipschitz operator ideals, and characterizes their properties and examples.

Contribution

It establishes a Galois connection framework between Lipschitz and linear operator ideals and introduces the concept of minimal Lipschitz operator ideals.

Findings

Criteria for recognizing composition type Lipschitz operator ideals

Introduction of minimal Banach Lipschitz operator ideals

Examples of minimal Lipschitz operator ideals not of composition type

Abstract

We establish a relation between Lipschitz operator ideals and linear operator ideals, which fits in the framework of Galois connection between lattices. We use this relationship to give a criterion which allow us to recognize when a Banach Lipschitz operator ideal is of composition type or not. Also, we introduce the concept of minimal Banach Lipschitz operator ideal, which have analogous properties to minimal Banach operator ideals. Also we characterize minimal Banach Lipschitz operator ideals which are of composition type and present examples which are not of this class.