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.

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