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Commutativity up to a factor of bounded operators in complex Hilbert space. (English) Zbl 1037.81009
Motivated by the canonical commutation relations and quantum measurement theory the authors study commuting up to a scalar factor of bounded operators on a (complex) Hilbert space. Let $A$ and $B$ be bounded operators acting on a Hilbert space such that $AB=\lambda BA$ for a scalar $\lambda$. The main result shows that (i) if $A$ or $B$ is self-adjoint, then $\lambda$ is real; (ii) if both $A$ and $B$ is self-adjoint, then $\lambda\in \{1,-1\};$ (iii) if $A$ and $B$ are self-adjoint and one of them is positive, then $\lambda=1$. Moreover, it is proved that $AB=UBA$ for bounded self-adjoint operators $A$ and $B$ and a unitary operator $U$ if, and only if, $AB\sp 2=B\sp 2A$ and $BA\sp 2=A\sp 2B$. In case of $A$ and $B$ being positive it means that $A$ and $B$ must commute. Many examples realizing the relation $AB=\lambda BA$ are given, including the quantum enveloping algebra. As a consequence, it is deduced that the transformations describing local measurement $X\to AXA$, $X\to BXA$ on the space of bounded operators given by positive operators $A$ and $B$ commute if, and only if, $A$ and $B$ commute.

MSC:
81P15Quantum measurement theory
47N50Applications of operator theory in quantum physics
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