A determinant inequality and log-majorisation for operators

‎Let \lambda_۱,\dots,\lambda_n  be positive real numbers such that \sum_{k=۱}^n \lambda_k=۱. In this paper, we prove that for any positive operators a_۱,a_۲,\ldots, a_n in semifinite von Neumann algebra M with faithful normal trace that \t(۱)<\infty, \prod_{k=۱}^n(\det a_k)^{\lambda_k}\,\le\,\det (\sum_{k=۱}^n \lambda_k a_k),where \det a=exp(\int_۰^{\t(۱)} \mu_a(t)\,dt). If furthermore \t(a_i)<\infty for every ۱\le i\le n and \prod_{k=۱}^n(\det a_k)^{\lambda_k}\neq ۰, then equality holds if and only if a_۱=a_۲=\cdots =a_n. A log-majorisation version of Young inequality are given as well.

Keywords:

‎Singular values‎ , ‎Semifinite trace‎ , ‎Majorisation‎ , ‎log-majorisation