The main goal of this paper is to clarify the antisymmetric nature of a binary relation ≪ which is defined for normal operators A and B by: A ≪ B if there exists an operator T such that for all Borel subset Δ of the complex plane ℂ, where and are spectral measures of A and B, respectively (the operators A and B are allowed to act in different complex Hilbert spaces). It is proved that if A ≪ B and B ≪ A, then A and B are unitarily equivalent, which shows that the relation ≪ is a partial order modulo unitary equivalence.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-sm204-3-4,
author = {Takateru Okayasu and Jan Stochel and Yasunori Ueda},
title = {On a binary relation between normal operators},
journal = {Studia Mathematica},
volume = {204},
year = {2011},
pages = {247-264},
zbl = {1236.47015},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-sm204-3-4}
}
Takateru Okayasu; Jan Stochel; Yasunori Ueda. On a binary relation between normal operators. Studia Mathematica, Tome 204 (2011) pp. 247-264. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-sm204-3-4/