Document Type : Research Paper

Author

• Seyed Mahmoud Manjegani

Department of Mathematical Sciences, Isfahan University of Technology, Isfahan, Iran, 84156-83111

Abstract

‎Let ${\lambda }_1,\dots ,{\lambda }_n$  be positive real numbers such that $\sum _{k=1}^n{\lambda }_k=1$. In this paper, we prove that for any positive operators $a_1,a_2,\dots ,a_n$ in semifinite von Neumann algebra $M$ with faithful normal trace that $\text{t}(1)<\mathrm{\infty }$, $$\prod _{k=1}^n(det{a}_k{)}^{{\lambda }_k}\le det(\sum _{k=1}^n{\lambda }_k{a}_k),$$
where $deta=exp({\int }_0^{\text{t}(1)}{\mu }_a(t)dt)$. If furthermore $\text{t}(a_i)<\mathrm{\infty }$ for every $1\le i\le n$ and $\prod _{k=1}^n(det{a}_k{)}^{{\lambda }_k}\ne 0$,
 then equality holds if and only if $a_1=a_2=\cdots =a_n$. A log-majorisation version of Young inequality are given as well.

Keywords

• ‎Singular values‎
• ‎Semifinite trace‎
• ‎Majorisation‎
• ‎log-majorisation