Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: A characterization of The operator-valued triangle equality   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.

1
The categories of this paper are: cs
7
The categories of this paper are: hep-th
25
The categories of this paper are: patt-sol
8
The categories of this paper are: hep-ex
23
The categories of this paper are: solv-int
35
The categories of this paper are: mtrl-th
18
The categories of this paper are: econ
32
The categories of this paper are: atom-ph
27
The categories of this paper are: comp-gas
36
The categories of this paper are: cmp-lg
5
The categories of this paper are: gr-qc
13
The categories of this paper are: quant-ph
28
The categories of this paper are: alg-geom
30
The categories of this paper are: q-alg
24
The categories of this paper are: dg-ga
6
The categories of this paper are: astro-ph