On a binary relation between normal operators
Tom 204 / 2011
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.