A characterization of The operator-valued triangle equality

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