Projection-valued measure (original) (raw)
In der Mathematik, insbesondere in der Funktionalanalysis ist ein Spektralmaß eine Abbildung, die gewissen Teilmengen einer fest gewählten Menge orthogonale Projektionen eines Hilbertraums zuordnet. Spektralmaße werden verwendet, um Ergebnisse in der Spektraltheorie linearer Operatoren zu formulieren, wie z. B. den Spektralsatz für normale Operatoren. Daneben wird der Begriff, jedoch mit anderer Bedeutung, in der Stochastik verwendet.
| Property | Value |
|---|---|
| dbo:abstract | In der Mathematik, insbesondere in der Funktionalanalysis ist ein Spektralmaß eine Abbildung, die gewissen Teilmengen einer fest gewählten Menge orthogonale Projektionen eines Hilbertraums zuordnet. Spektralmaße werden verwendet, um Ergebnisse in der Spektraltheorie linearer Operatoren zu formulieren, wie z. B. den Spektralsatz für normale Operatoren. Daneben wird der Begriff, jedoch mit anderer Bedeutung, in der Stochastik verwendet. (de) En matemáticas, en especial en análisis funcional una medida espectral es una aplicación cuyo dominio es una σ-álgebra y cuyos valores son proyecciones autoadjuntas en un espacio de Hilbert. Medidas espectrales se utilizan en la teoría espectral de operadores autoadjuntos. (es) In mathematics, particularly in functional analysis, a projection-valued measure (PVM) is a function defined on certain subsets of a fixed set and whose values are self-adjoint projections on a fixed Hilbert space. Projection-valued measures are formally similar to real-valued measures, except that their values are self-adjoint projections rather than real numbers. As in the case of ordinary measures, it is possible to integrate complex-valued functions with respect to a PVM; the result of such an integration is a linear operator on the given Hilbert space. Projection-valued measures are used to express results in spectral theory, such as the important spectral theorem for self-adjoint operators. The Borel functional calculus for self-adjoint operators is constructed using integrals with respect to PVMs. In quantum mechanics, PVMs are the mathematical description of projective measurements. They are generalized by positive operator valued measures (POVMs) in the same sense that a mixed state or density matrix generalizes the notion of a pure state. (en) En mathématiques, plus précisément en analyse fonctionnelle, une mesure spectrale est une application définie sur une tribu à valeurs dans l'espace des projections orthogonales d'un espace hilbertien et vérifiant des axiomes semblables à ceux qui définissent les mesures positives. Les mesures spectrales sont utilisées pour exprimer des résultats en théorie spectrale, tels que le théorème spectral pour les opérateurs auto-adjoints. Les mesures spectrales ont des propriétés similaires aux mesures réelles positives. (fr) Hermitowska miara spektralna (albo hermitowski rozkład jedynki) – przeliczalnie addytywna miara wektorowa, określona na σ-ciele zbiorów borelowskich pewnej przestrzeni topologicznej o wartościach w przestrzeni operatorów liniowych i ciągłych pewnej przestrzeni Hilberta, spełniająca określone warunki. Hermitowskie miary spektralne pojawiają się w sformułowaniu twierdzenia spektralnego. (pl) In de functionaalanalyse, een deelgebied van de wiskunde, is een spectraalmaat een afbeelding, die bepaalde deelverzamelingen van een geselecteerde verzameling aan orthogonale projecties van een Hilbertruimte toewijst. Spectraalmaten worden gebruikt om de resultaten in de spectraaltheorie van lineaire operatoren, zoals bijvoorbeeld de spectraalstelling voor normale operatoren, te formuleren. Daarnaast wordt het begrip, evenwel met een andere betekenis, in de stochastiek gebruikt. (nl) In matematica, in particolare in analisi funzionale, una misura a valori di proiettore è una funzione definita su un certo sottoinsieme di un insieme fissato i cui valori restituiti sono proiettori autoaggiunti su uno spazio di Hilbert. Le misure a valori di proiettore sono usate per esprimere i risultati della teoria spettrale, come il teorema spettrale per operatori autoaggiunti. (it) Na matemática, particularmente em análise funcional a Medida espectral é uma função definida em certos subconjuntos de um conjunto fixo no qual todos os valores possíveis são operadores autoadjuntos no espaço de Hilbert. (pt) |
| dbo:wikiPageExternalLink | https://www.mat.univie.ac.at/~gerald/ftp/book-schroe/, |
| dbo:wikiPageID | 1046155 (xsd:integer) |
| dbo:wikiPageLength | 12480 (xsd:nonNegativeInteger) |
| dbo:wikiPageRevisionID | 1119695225 (xsd:integer) |
| dbo:wikiPageWikiLink | dbr:Quantum_information_theory dbr:Quantum_mechanics dbr:Barry_Simon dbr:Borel_algebra dbc:Spectral_theory dbr:Σ-algebra dbr:Mathematics dbr:Measure_(mathematics) dbr:Orthogonal dbr:Spectral_theory_of_compact_operators dbr:POVM dbr:Function_(mathematics) dbr:Gerald_Teschl dbr:Lp_space dbr:Sigma_additivity dbr:Complex_measure dbr:Density_matrix dbr:Functional_analysis dbr:Spectral_theory dbr:Step_function dbr:Measurable_function dbr:Measurable_space dbr:Michael_C._Reed dbc:Linear_algebra dbr:Direct_integral dbr:Projection_(mathematics) dbr:Hilbert_space dbc:Measures_(measure_theory) dbr:Borel_functional_calculus dbr:Spectral_theorem dbr:Spectral_theory_of_normal_C*-algebras dbr:Pure_state dbr:If_and_only_if dbr:Indicator_function dbr:Self-adjoint_operator dbr:Observable dbr:Self-adjoint dbr:Ring_homomorphism dbr:Mixed_state_(physics) dbr:Quantum_measurement dbr:Projection_operator dbr:Positive_operator-valued_measure |
| dbp:date | May 2015 (en) |
| dbp:reason | Partition of unity in the operator sense is not defined in the article "partition of unity". (en) What is a measure class? A measure up to measure-preserving equivalence? Should the measure be completed? (en) Is this a novel term? It's not defined in the linked article. (en) |
| dbp:wikiPageUsesTemplate | dbt:Anchor dbt:Citation dbt:Clarify dbt:Pi dbt:Reflist dbt:Short_description dbt:Spectral_theory dbt:Analysis_in_topological_vector_spaces dbt:Functional_analysis dbt:Measure_theory dbt:Narici_Beckenstein_Topological_Vector_Spaces dbt:Schaefer_Wolff_Topological_Vector_Spaces dbt:Trèves_François_Topological_vector_spaces,_distributions_and_kernels |
| dcterms:subject | dbc:Spectral_theory dbc:Linear_algebra dbc:Measures_(measure_theory) |
| gold:hypernym | dbr:Function |
| rdf:type | dbo:Disease |
| rdfs:comment | In der Mathematik, insbesondere in der Funktionalanalysis ist ein Spektralmaß eine Abbildung, die gewissen Teilmengen einer fest gewählten Menge orthogonale Projektionen eines Hilbertraums zuordnet. Spektralmaße werden verwendet, um Ergebnisse in der Spektraltheorie linearer Operatoren zu formulieren, wie z. B. den Spektralsatz für normale Operatoren. Daneben wird der Begriff, jedoch mit anderer Bedeutung, in der Stochastik verwendet. (de) En matemáticas, en especial en análisis funcional una medida espectral es una aplicación cuyo dominio es una σ-álgebra y cuyos valores son proyecciones autoadjuntas en un espacio de Hilbert. Medidas espectrales se utilizan en la teoría espectral de operadores autoadjuntos. (es) En mathématiques, plus précisément en analyse fonctionnelle, une mesure spectrale est une application définie sur une tribu à valeurs dans l'espace des projections orthogonales d'un espace hilbertien et vérifiant des axiomes semblables à ceux qui définissent les mesures positives. Les mesures spectrales sont utilisées pour exprimer des résultats en théorie spectrale, tels que le théorème spectral pour les opérateurs auto-adjoints. Les mesures spectrales ont des propriétés similaires aux mesures réelles positives. (fr) Hermitowska miara spektralna (albo hermitowski rozkład jedynki) – przeliczalnie addytywna miara wektorowa, określona na σ-ciele zbiorów borelowskich pewnej przestrzeni topologicznej o wartościach w przestrzeni operatorów liniowych i ciągłych pewnej przestrzeni Hilberta, spełniająca określone warunki. Hermitowskie miary spektralne pojawiają się w sformułowaniu twierdzenia spektralnego. (pl) In de functionaalanalyse, een deelgebied van de wiskunde, is een spectraalmaat een afbeelding, die bepaalde deelverzamelingen van een geselecteerde verzameling aan orthogonale projecties van een Hilbertruimte toewijst. Spectraalmaten worden gebruikt om de resultaten in de spectraaltheorie van lineaire operatoren, zoals bijvoorbeeld de spectraalstelling voor normale operatoren, te formuleren. Daarnaast wordt het begrip, evenwel met een andere betekenis, in de stochastiek gebruikt. (nl) In matematica, in particolare in analisi funzionale, una misura a valori di proiettore è una funzione definita su un certo sottoinsieme di un insieme fissato i cui valori restituiti sono proiettori autoaggiunti su uno spazio di Hilbert. Le misure a valori di proiettore sono usate per esprimere i risultati della teoria spettrale, come il teorema spettrale per operatori autoaggiunti. (it) Na matemática, particularmente em análise funcional a Medida espectral é uma função definida em certos subconjuntos de um conjunto fixo no qual todos os valores possíveis são operadores autoadjuntos no espaço de Hilbert. (pt) In mathematics, particularly in functional analysis, a projection-valued measure (PVM) is a function defined on certain subsets of a fixed set and whose values are self-adjoint projections on a fixed Hilbert space. Projection-valued measures are formally similar to real-valued measures, except that their values are self-adjoint projections rather than real numbers. As in the case of ordinary measures, it is possible to integrate complex-valued functions with respect to a PVM; the result of such an integration is a linear operator on the given Hilbert space. (en) |
| rdfs:label | Spektralmaß (de) Medida espectral (es) Mesure spectrale (fr) Misura a valori di proiettore (it) Spectraalmaat (nl) Projection-valued measure (en) Hermitowska miara spektralna (pl) Medida espectral (pt) |
| owl:sameAs | freebase:Projection-valued measure wikidata:Projection-valued measure dbpedia-de:Projection-valued measure dbpedia-es:Projection-valued measure dbpedia-fr:Projection-valued measure dbpedia-he:Projection-valued measure dbpedia-it:Projection-valued measure dbpedia-nl:Projection-valued measure dbpedia-pl:Projection-valued measure dbpedia-pt:Projection-valued measure https://global.dbpedia.org/id/38Gw5 |
| prov:wasDerivedFrom | wikipedia-en:Projection-valued_measure?oldid=1119695225&ns=0 |
| foaf:isPrimaryTopicOf | wikipedia-en:Projection-valued_measure |
| is dbo:wikiPageDisambiguates of | dbr:PVM_(disambiguation) |
| is dbo:wikiPageRedirects of | dbr:Projective_measurement |
| is dbo:wikiPageWikiLink of | dbr:Quantum_entanglement dbr:Quantum_logic dbr:System_of_imprimitivity dbr:Mathematical_formulation_of_quantum_mechanics dbr:Measure_(mathematics) dbr:Measurement_in_quantum_mechanics dbr:Normal_operator dbr:POVM dbr:Quantum_nondemolition_measurement dbr:Projective_measurement dbr:Fuglede's_theorem dbr:PVM_(disambiguation) dbr:Stone's_theorem_on_one-parameter_unitary_groups dbr:Position_operator dbr:Naimark's_dilation_theorem dbr:Quantum_operation dbr:Effect_algebra dbr:Borel_functional_calculus dbr:Born_rule dbr:Spectral_theorem dbr:Fidelity_of_quantum_states dbr:Set_function dbr:Quantum_indeterminacy dbr:Weak_value |
| is rdfs:seeAlso of | dbr:Measurement_in_quantum_mechanics dbr:Spectral_theory_of_normal_C*-algebras |
| is foaf:primaryTopic of | wikipedia-en:Projection-valued_measure |