A matrix subadditivity inequality for symmetric norms

Abstract:

Let $f(t)$ be a non-negative concave function on $[0,\infty )$. We prove that \[ \Vert f(|A+B|) \Vert \le \Vert f(|A|)+f(|B|) \Vert \] for all normal $n$-by-$n$ matrices $A$, $B$ and all symmetric norms. This result has several applications. For instance, for a Hermitian ${\mathbb {A}}=[A_{i, j}]$ partitioned in blocks of the same size, \[ \left \| f(|{\mathbb {A}}|) \right \| \le \left \| \sum f(|A_{i, j}|) \right \|. \] We also prove, in a similar way, that given $Z$ expansive and $A$ normal of the same size, \[ \Vert f(|Z^*AZ|) \Vert \le \Vert Z^*f(|A|)Z \Vert .\]