2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/94868We 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.6 pagesOperator AlgebrasFunctional Analysis47A05, 47A10, 47A12text