A matrix subadditivity inequality for symmetric norms

Abstract

Let f(t) be a non-negative concave function on [0,∞). We prove that ∥f(|A+B|) ∥ ≤ ∥ f(|A|) + f(|B|) ∥ for all normal n-by-n matrices A, B and all symmetric norms. This result has several applications. For instance, for a Hermitian A = [A i,j ] partitioned in blocks of the same size, ∥ f(|A|) ∥≤∥∑f(|A i,j |)∥. We also prove, in a similar way, that given Z expansive and A normal of the same size, ∥f(|Z*AZ|)∥ ≤ ∥Z*f(|A|)Z∥.