On a conjecture of A. Bikchentaev

TL;DR

This paper proves Bikchentaev's conjecture that for positive operators in a semifinite von Neumann algebra, a certain submajorization relation holds, with applications to operator ideals, inequalities, and traces.

Contribution

It establishes the conjecture in full generality and explores its implications for symmetric operator ideals and related inequalities.

Findings

Proof of Bikchentaev's conjecture in general setting

Applications to symmetric operator ideals and Golden-Thompson inequality

Insights into singular traces and operator submajorization

Abstract

In \cite{bik1}, A. M. Bikchentaev conjectured that for positive $\tau-$measurable operators $a$ and $b$ affiliated with an arbitrary semifinite von Neumann algebra $\mathcal M$, the operator $b^{1/2}ab^{1/2}$ is submajorized by the operator $ab$ in the sense of Hardy-Littlewood. We prove this conjecture in full generality and present a number of applications to fully symmetric operator ideals, Golden-Thompson inequality and (singular) traces.