A concavity inequality for symmetric norms

Abstract

We review some recent convexity results for Hermitian matrices and we add a new one to the list: Let A be semidefinite positive, let Z be expansive, $Z^{\ast }Z⩾I$, and let $f\text{:}[0\text{,}\infty )\to [0\text{,}\infty )$ be a concave function. Then, for all symmetric norms

$$\Vert f(Z^{\ast }\mathit{AZ})\Vert ⩽\Vert Z^{\ast }f(A)Z\Vert \text{.}$$

This inequality complements a classical trace inequality of Brown–Kosaki.