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 ‖A‖1+‖B‖1=‖A+B‖1‖A‖1+‖B‖1=‖A+B‖1, where ‖A‖1‖A‖1 is the trace norm of the trace-class operator A  . Particularly, we show that ‖A‖1+‖B‖1=‖A+B‖1‖A‖1+‖B‖1=‖A+B‖1 if and only if |A|+|B|=|A+B||A|+|B|=|A+B| 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 ‖A‖1+‖A⁎‖1=‖A+A⁎‖1‖A‖1+‖A⁎‖1=‖A+A⁎‖1 and other relevant results are given.