Logarithmic submajorization, uniform majorization and H\"older type inequalities for $\tau$-measurable operators

TL;DR

This paper extends the determinant function to a broader class of $ au$-measurable operators in semifinite von Neumann algebras, establishing submultiplicativity and deriving H"older type inequalities and majorization results.

Contribution

It introduces a generalized determinant function for $ au$-measurable operators and proves its submultiplicativity, enabling new inequalities in operator theory.

Findings

The determinant function is shown to be submultiplicative on the algebra of $ au$-measurable operators.

H"older type inequalities are derived using the generalized determinant and Araki-Lieb-Thirring inequalities.

A Weyl-type theorem for uniform majorization is established.

Abstract

We extend the notion of the determinant function $\Lambda$, originally introduced by T.Fack for $\tau$-compact operators, to a natural algebra of $\tau$-measurable operators affiliated with a semifinite von Neumann algebra which coincides with that defined by Haagerup and Schultz in the finite case and on which the determinant function is shown to be submultiplicative. Application is given to H\"older type inequalities via general Araki-Lieb-Thirring inequalities due to Kosaki and Han and to a Weyl-type theorem for uniform majorization.