On submajorisation of the Rotfeld's inequality

Abstract

Let $(\M,\tau)$ be a semi-finite von Neumann algebra, $L_0(\M)$ be the set of all $\tau$-measurable operators, $\mu_t(x)$ be the generalized singular
number of $x\in L_0(\M)$. We proved that if $g:[0,\8)\rightarrow[0,\8)$ is an increasing continuous function, then for any $x,y$ in $L_0(\mathcal{M})$,
$$
\mu_t(g(|x+y|))\le \mu_t(g(\frac{1}{2}\left(\begin{array}{cc}
|x|+|y| & x^*+y^*\\
x+y& |x^*|+|y^*|
\end{array}\right)))\qquad 0<t<\t(1).
$$ We also obtained that if $f:[0,\8)\rightarrow[0,\8)$ is a concave function, then $ \mu(f(\frac{1}{2}\left(\begin{array}{cc}
|x|+|y| & x^*+y^*\\
x+y& |x^*|+|y^*|
\end{array}\right)))$ is submajorized by $\mu(f(|x|))+\mu(f(|y|))$.