  We will show that for any two bounded linear operators $X,Y$ on a Hilbert
space ${\frak H}$, if they satisfy the triangle equality $|X+Y|=|X|+|Y|$, there
exists a partial isometry $U$ on ${\frak H}$ such that $X=U|X|$ and $Y=U|Y|$.
This is a generalization of Thompson's theorem to the matrix case proved by
using a trace.
