On submajorisation of the Rotfeld’s inequality

Abstract

Let (M, ?) be a semi-finite von Neumann algebra, L0(M) be the set of all ?-measurable operators, ?t(x) be the generalized singular number of x ? L0(M). We proved that if 1 : [0,?) ? [0,?) is an increasing continuous function, then for any x, y in L0(M), ?t(1(|x + y|)) ? ?t(1( 1/2 |x| + |y| x* + y* x + y |x*| + |y*| ! )), 0 < t < ?(1). We also obtained that if f : [0,?) ? [0,?) is a concave function, then ?(f(1/2 |x| + |y| x* + y* x + y |x*| + |y*|!)) is submajorized by ?(f(|x|)) + ?(f(|y|)).