Extension of Rotfel’d Theorem

Let f(t)f(t) be a non-negative concave function on the positive half-line. Given an arbitrary partitioned positive semi-definite matrix, we show thatfAXX∗B≤f(A)+f(B)for all symmetric (i.e. unitarily invariant) norms. This characterization of concave functions extends a famous trace inequality of Rotfel’d,Trf(A+B)≤Trf(A)+Trf(B)Trf(A+B)≤Trf(A)+Trf(B)and contains several classical matrix inequalities.