Journal of Operator Theory

Volume 58, Issue 2, Fall 2007 pp. 463-468.

A characterization of the operator-valued triangle equality

Authors: Tsuyoshi Ando (1) and Tomohiro Hayashi (2)
Author institution: (1) Hokkaido University (Emeritus), Japan
(2) Graduate School of Mathematics, Kyushu University, 33, Fukuoka, 812-8581, Japan

Summary: 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.

Keywords: operator theory, Hilbert space, triangle inequality

Contents Full-Text PDF