Some characterizations of the trace norm triangle equality

In this paper, we investigate some equalities for the trace-class operators on a Hilbert space. For two trace-class operators A and B, we get some equivalent conditions for parallel to A parallel to(1) + parallel to B parallel to(1) = parallel to A + B parallel to(1), where parallel to A parallel to(1) is the trace norm of the trace-class operator A. Particularly, we show that parallel to A parallel to(1) + parallel to B parallel to(1) = parallel to A + B parallel to(1) if and only if vertical bar A vertical bar + vertical bar B vertical bar = vertical bar A + B vertical bar for two trace-class operators A and B. This condition is related to a result of Ando and Hayashi. Moreover, some characterizations of the equality parallel to A parallel to(1) + parallel to A*parallel to(1) = parallel to A+ A*parallel to(1) and other relevant results are given. (C) 2015 Elsevier Inc. All rights reserved.