On a binary relation between normal operators

Volume 204 / 2011

Takateru Okayasu, Jan Stochel, Yasunori Ueda Studia Mathematica 204 (2011), 247-264 MSC: Primary 47A63; Secondary 47B15. DOI: 10.4064/sm204-3-4


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.


  • Takateru OkayasuYamagata University
    Yamagata 990-8560, Japan
  • Jan StochelInstytut Matematyki
    Uniwersytet Jagielloński
    Łojasiewicza 6
    30-348 Kraków, Poland
  • Yasunori UedaNihon University
    Yamagata Senior High School
    Yamagata 990-2433, Japan

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