zbMATH — the first resource for mathematics

Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
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.

81P15Quantum measurement theory
47N50Applications of operator theory in quantum physics
Full Text: DOI arXiv