A characterization of The operator-valued triangle equality

Tsuyoshi Ando; Tomohiro Hayashi

A characterization of The operator-valued triangle equality

Operator Algebras 2007-05-23 v1 Functional Analysis

Authors: Tsuyoshi Ando , Tomohiro Hayashi

Abstract

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 on hilbert spaces operator theory numerical range

Cite

@article{arxiv.math/0509539,
  title  = {A characterization of The operator-valued triangle equality},
  author = {Tsuyoshi Ando and Tomohiro Hayashi},
  journal= {arXiv preprint arXiv:math/0509539},
  year   = {2007}
}

Comments

6 pages

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