On submajorization inequalities for matrices of measurable operators

Abstract

Let $${\mathcal {M}}$$ be a semifinite von Neumann algebra. We proved that if $$\left( \begin{array}{cc} x &{} z\\ z^* &{} y \end{array}\right)$$ and $$\left( \begin{array}{cc} x &{} z^*\\ z &{} y \end{array}\right)$$ are positive matrices with entries in $${\mathcal M}$$ , then z is logarithmically submajorized by $$x^\frac{1}{2}y^\frac{1}{2}$$ . Using this, we proved some related submajorization inequalities.