Some Singular Value Inequalities for Convex Functions of Matrices

Abstract

In this paper, we obtain some upper bounds for singular value inequalities for convex functions of matrices. Some applications involving the spectral norm and numerical radii of matrices were given. Among other inequalities, we prove that for \(A,B\in \mathcal{\mathbb{M}}_{n}\mathcal{\mathbb{(C)}}\) and for any nonnegative increasing convex function \(f\) on \([0,\infty )\) with \(f(0)=0,\) we have\[
\begin{aligned}
&s_j\!\left(f(|aA^*B+bB^*A|)\right)
\le \tfrac12 \Big[
s_j\!\left(f(a|A|^2+b|B|^2)
\oplus f(b|A|^2+a|B|^2)\right) \\
&\quad +\, s_{j-i+1}\!\left(
f(|bA^*B+aB^*A|)
\oplus f(|aA^*B+bB^*A|)
\right)
\Big],
\end{aligned}
\]where \(a,b\geq 0\) and \(j=1,...,n.\) Also, an upper bound for \(\left\Vert {Re}A\right\Vert \) were given.