Norm and anti-norm inequalities for positive semi-definite matrices

Abstract

Some subadditivity results involving symmetric (unitarily invariant) norms are obtained. For instance, if $g(t)=\sum_{k=0}^m a_kt^k$ is a polynomial of degree $m$ with non-negative coefficients, then, for all positive operators $A,\,B$ and all symmetric norms, $\|g(A+B)\|^{1/m} \le \|g(A)\|^{1/m} + \|g(B)\|^{1/m}$. To give parallel superadditivity results, we investigate anti-norms, a class of functionals containing the Schatten $q$-norms for $q\in(0,1]$ and $q<0$. The results are extensions of the Minkowski determinantal inequality. A few estimates for block-matrices are derived. For instance, let $f:[0,\infty) \to [0,\infty)$ be concave and $p\in(1,\infty)$. If $f^p(t)$ is superadditive, then $Tr f(A) \ge (\sum_{i=1}^m f^p(a_{ii}))^{1/p}$ for all positive $m\times m$ matrix $A=[a_{ij}]$. Furthermore, for the normalized trace $\tau$, we consider functions $\phi(t)$ and $f(t)$ for which the functional $A\mapsto\phi\circ\tau\circ f(A)$ is convex or concave, and obtain a simple analytic criterion.