Geodesic
On a binary relation between normal operators
Takateru Okayasu 1   ; Jan Stochel 2   ; Yasunori Ueda 3
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
Studia Mathematica, Tome 204 (2011) no. 3, pp. 247-264 Cet article a éte moissonné depuis la source Institute of Mathematics Polish Academy of Sciences

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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.
DOI : 10.4064/sm204-3-4
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

Takateru Okayasu  1   ; Jan Stochel  2   ; Yasunori Ueda  3

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

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