Non-linear traces on the algebra of compact operators and majorization

TL;DR

This paper explores non-linear traces of Choquet and Sugeno types on compact operators, establishing their properties, relations to majorization, and their role in non-commutative integration theory.

Contribution

It characterizes non-linear traces of Choquet and Sugeno types via partial additivities and links them to majorization and Banach space structures.

Findings

Non-linear traces of Choquet type relate to majorization theory.

Conditions when Schatten-von Neumann classes form Banach spaces.

Triangle inequality properties for Sugeno type traces.

Abstract

We study non-linear traces of Choquet type and Sugeno type on the algebra of compact operators. They have certain partial additivities. We show that these partial additivities characterize non-linear traces of both Choquet type and Sugeno type respectively. There exists a close relation between non-linear traces of Choquet type and majorization theory. We study trace class operators for non-linear traces of Choquet type. More generally we discuss Schatten-von Neumann $p$-class operators for non-linear traces of Choquet type. We determine when they form Banach spaces. This is an attempt of non-commutative integration theory for non-linear traces of Choquet type on the algebra of compact operators. We also consider the triangle inequality for non-linear traces of Sugeno type.