Submajorization inequalities for matrices of τ-measurable operators.

Let (M , τ) be a semi-finite von Neumann algebra, L 0 (M) be the set of all τ-measurable operators, μ t (x) be the generalized singular number of x ∈ L 0 (M) and f : [ 0 , ∞) → [ 0 , ∞) be a concave function. We proved that if x 1 , x 2 , ... , x n are normal operators in L 0 (M) , then μ (f (| ∑ k = 1 n x k |)) is submajorized by μ (f (∑ k = 1 n | x k |)). As an application, we obtained that if x is a matrix of normal operators x i j in L 0 (M) , then μ (f (| x |)) is submajorized by μ (∑ i , j = 1 n f (| x i j |)). [ABSTRACT FROM AUTHOR]