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 $a_1,a_2,\ldots, a_n$ in semifinite von Neumann algebra $M$ with faithful normal trace that $\t(1)<\infty$, $$\prod_{k=1}^n(\det a_k)^{\lambda_k}\,\le\,\det (\sum_{k=1}^n \lambda_k a_k),$$ where $\det a=exp(\int_0^{\t(1)} \mu_a(t)\,dt)$. If furthermore $\t(a_i)<\infty$ for every $1\le i\le n$ and $ \prod_{k=1}^n(\det a_k)^{\lambda_k}\neq 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.
Manjegani,S M . (2016). A determinant inequality and log-majorisation for operators. International Journal of Nonlinear Analysis and Applications, 7(1), 131-140. doi: 10.22075/ijnaa.2015.301
MLA
Manjegani,S M . "A determinant inequality and log-majorisation for operators", International Journal of Nonlinear Analysis and Applications, 7, 1, 2016, 131-140. doi: 10.22075/ijnaa.2015.301
HARVARD
Manjegani S M. (2016). 'A determinant inequality and log-majorisation for operators', International Journal of Nonlinear Analysis and Applications, 7(1), pp. 131-140. doi: 10.22075/ijnaa.2015.301
CHICAGO
S M Manjegani, "A determinant inequality and log-majorisation for operators," International Journal of Nonlinear Analysis and Applications, 7 1 (2016): 131-140, doi: 10.22075/ijnaa.2015.301
VANCOUVER
Manjegani S M. A determinant inequality and log-majorisation for operators. IJNAA. 2016;7(1):131-140. doi: 10.22075/ijnaa.2015.301
International Journal of Nonlinear Analysis and Applications