About: Stone's theorem on one-parameter unitary groups

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In mathematics, Stone's theorem on one-parameter unitary groups is a basic theorem of functional analysis that establishes a one-to-one correspondence between self-adjoint operators on a Hilbert space and one-parameter families of unitary operators that are strongly continuous, i.e., and are homomorphisms, i.e., Such one-parameter families are ordinarily referred to as strongly continuous one-parameter unitary groups.

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rdf:type	yago:WikicatTheoremsInFunctionalAnalysis yago:Abstraction100002137 yago:Communication100033020 yago:Message106598915 yago:Proposition106750804 yago:Statement106722453 yago:Theorem106752293
rdfs:label	Stone's theorem on one-parameter unitary groups (en) Теорема Стоуна о группах унитарных операторов в гильбертовом пространстве (ru)
rdfs:comment	In mathematics, Stone's theorem on one-parameter unitary groups is a basic theorem of functional analysis that establishes a one-to-one correspondence between self-adjoint operators on a Hilbert space and one-parameter families of unitary operators that are strongly continuous, i.e., and are homomorphisms, i.e., Such one-parameter families are ordinarily referred to as strongly continuous one-parameter unitary groups. (en) Теорема Стоуна о группах унитарных операторов в гильбертовом пространстве — важный результат функционального анализа, утверждающий, что всякая сильно непрерывная однопараметрическая группа унитарных операторов представляется в виде: , где — некоторый самосопряженный оператор, а — параметр. Верно и обратное: всякому самосопряженному оператору с помощью представления Стоуна можно поставить в соответствие сильно непрерывную однопараметрическую группу унитарных операторов. Сильно непрерывная однопараметрическая группа унитарных операторов обладает следующими свойствами: . (ru)
dcterms:subject	Theorems in functional analysis
Wikipage page ID	893698 (xsd:integer)
Wikipage revision ID	1098658485 (xsd:integer)
Link from a Wikipage to another Wikipage	Canonical commutation relation Quantum mechanics One-parameter group Derivative Annals of Mathematics Riesz–Markov–Kakutani representation theorem Unitary operator Lie group Compact support Continuous function Convolution Theorems in functional analysis Mathematics Norm (mathematics) Separable space Contraction (operator theory) Proceedings of the National Academy of Sciences Lie algebra Stone–von Neumann theorem Compact operator Functional analysis Projection-valued measure Banach space Time evolution Momentum operator Position operator Extensions of symmetric operators Fourier transform Hille–Yosida theorem Hamiltonian (quantum theory) Hilbert space Weakly measurable function Spectral theorem Group algebra of a locally compact group Self-adjoint operator Map (mathematics) Strong operator topology Unitary group Riemann–Lebesgue lemma
sameAs	Stone's theorem on one-parameter unitary groups Stone's theorem on one-parameter unitary groups Stone's theorem on one-parameter unitary groups Stone's theorem on one-parameter unitary groups Stone's theorem on one-parameter unitary groups
dbp:wikiPageUsesTemplate	dbt:Citation dbt:Further dbt:Mvar dbt:Reflist dbt:Short_description dbt:Harvs dbt:Functional_analysis
authorlink	John von Neumann (en) Marshall Stone (en)
first	John (en) Marshall (en)
last	Stone (en) von Neumann (en)
year	1930 (xsd:integer) 1932 (xsd:integer)
has abstract	In mathematics, Stone's theorem on one-parameter unitary groups is a basic theorem of functional analysis that establishes a one-to-one correspondence between self-adjoint operators on a Hilbert space and one-parameter families of unitary operators that are strongly continuous, i.e., and are homomorphisms, i.e., Such one-parameter families are ordinarily referred to as strongly continuous one-parameter unitary groups. The theorem was proved by Marshall Stone , and John von Neumann showed that the requirement that be strongly continuous can be relaxed to say that it is merely weakly measurable, at least when the Hilbert space is separable. This is an impressive result, as it allows to define the derivative of the mapping which is only supposed to be continuous. It is also related to the theory of Lie groups and Lie algebras. (en) Теорема Стоуна о группах унитарных операторов в гильбертовом пространстве — важный результат функционального анализа, утверждающий, что всякая сильно непрерывная однопараметрическая группа унитарных операторов представляется в виде: , где — некоторый самосопряженный оператор, а — параметр. Верно и обратное: всякому самосопряженному оператору с помощью представления Стоуна можно поставить в соответствие сильно непрерывную однопараметрическую группу унитарных операторов. Теорема была доказана американским математиком Маршаллом Стоуном в 1930 году и имела большое значение для становления квантовой механики, а также послужила толчком к созданию теории Купмана — фон Неймана. Сильно непрерывная однопараметрическая группа унитарных операторов обладает следующими свойствами: . Важность результата для физики заключается в том, что он гарантирует существование и единственность решений уравнений Шрёдингера и Лиувилля, а также сохранение нормировок волновых функций. (ru)
prov:wasDerivedFrom	wikipedia-en:Stone's_theorem_on_one-parameter_unitary_groups?oldid=1098658485&ns=0
page length (characters) of wiki page	9767 (xsd:nonNegativeInteger)
foaf:isPrimaryTopicOf	wikipedia-en:Stone's_theorem_on_one-parameter_unitary_groups
is rdfs:seeAlso of	One-parameter group
is Link from a Wikipage to another Wikipage of	List of eponyms (L–Z) List of functional analysis topics One-parameter group Resolvent formalism Unbounded operator Mathematical formulation of quantum mechanics Ehrenfest theorem Stone–von Neumann theorem Hamiltonian (quantum mechanics) Stone's theorem Ergodic flow Hille–Yosida theorem Unitarity (physics) Marshall H. Stone Borel functional calculus CCR and CAR algebras Koopman–von Neumann classical mechanics Self-adjoint operator List of theorems Stone theorem on one-parameter unitary groups
is Wikipage redirect of	Stone theorem on one-parameter unitary groups

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