1Yamagata University Yamagata 990-8560, Japan 2Instytut Matematyki Uniwersytet Jagielloński Łojasiewicza 6 30-348 Kraków, Poland 3Nihon University Yamagata Senior High School Yamagata 990-2433, Japan
Studia Mathematica, Tome 204 (2011) no. 3, pp. 247-264
The main goal of this paper is to clarify the
antisymmetric nature of a binary relation $\ll$ which
is defined for normal operators $A$ and $B$ by: $A \ll
B$ if there exists an operator $T$ such that
$E_A(\varDelta) \le T^* E_B(\varDelta) T$ for all
Borel subset $\varDelta$ of the complex plane $\mathbb C$,
where $E_A$ and $E_B$ 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 $A \ll B$ and $B \ll A$, then $A$
and $B$ are unitarily equivalent, which shows that the
relation $\ll$ is a partial order modulo unitary equivalence.
Keywords:
main paper clarify antisymmetric nature binary relation which defined normal operators there exists operator vardelta * vardelta borel subset vardelta complex plane mathbb where spectral measures respectively operators allowed act different complex hilbert spaces proved unitarily equivalent which shows relation partial order modulo unitary equivalence
1
Yamagata University Yamagata 990-8560, Japan
2
Instytut Matematyki Uniwersytet Jagielloński Łojasiewicza 6 30-348 Kraków, Poland
3
Nihon University Yamagata Senior High School Yamagata 990-2433, Japan
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Takateru Okayasu; Jan Stochel; Yasunori Ueda. On a binary relation between normal operators. Studia Mathematica, Tome 204 (2011) no. 3, pp. 247-264. doi: 10.4064/sm204-3-4
