About: Self-adjoint

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In mathematics, and more specifically in abstract algebra, an element x of a *-algebra is self-adjoint if . A self-adjoint element is also Hermitian, though the reverse doesn't necessarily hold. A collection C of elements of a star-algebra is self-adjoint if it is closed under the involution operation. For example, if then since in a star-algebra, the set {x,y} is a self-adjoint set even though x and y need not be self-adjoint elements. In a dagger category, a morphism is called self-adjoint if ; this is possible only for an endomorphism .

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rdfs:label	Autoadjoint (fr) Self-adjoint (en)
rdfs:comment	In mathematics, and more specifically in abstract algebra, an element x of a -algebra is self-adjoint if . A self-adjoint element is also Hermitian, though the reverse doesn't necessarily hold. A collection C of elements of a star-algebra is self-adjoint if it is closed under the involution operation. For example, if then since in a star-algebra, the set {x,y} is a self-adjoint set even though x and y need not be self-adjoint elements. In a dagger category, a morphism is called self-adjoint if ; this is possible only for an endomorphism . (en) En mathématiques, un élément x d'une algèbre involutive A est dit autoadjoint si x = x ; plus généralement, une partie de A est dite autoadjointe si elle est stable par l'involution * (comme la partie {y, y}, pour tout élément y de A). Sur la C-algèbre des opérateurs bornés sur un espace de Hilbert H, l'involution est l'application qui à tout opérateur borné associe son adjoint, et les éléments autoadjoints sont appelés les opérateurs autoadjoints. Dans une catégorie à involution, un endomorphisme f est dit autoadjoint si f † = f. (fr)
dcterms:subject	Linear algebra Abstract algebra
Wikipage page ID	302202 (xsd:integer)
Wikipage revision ID	1090839461 (xsd:integer)
Link from a Wikipage to another Wikipage	Dagger category *-algebra Conjugate transpose Mathematics Matrix (mathematics) Symmetric matrix Endomorphism Morphism Linear operator Functional analysis Orthonormal basis Linear algebra Finite-dimensional Hermitian matrix Hilbert space Involution (mathematics) Abstract algebra Abstract algebra Self-adjoint operator Unitary element Normal element Adjoint operator Star-algebra
Link from a Wikipage to an external page	https://www.mat.univie.ac.at/~gerald/ftp/book-schroe/
sameAs	Self-adjoint Self-adjoint Self-adjoint Self-adjoint
dbp:wikiPageUsesTemplate	dbt:Cite_book dbt:Reflist dbt:Short_description dbt:Sup dbt:Functional_analysis dbt:SpectralTheory
has abstract	En mathématiques, un élément x d'une algèbre involutive A est dit autoadjoint si x* = x ; plus généralement, une partie de A est dite autoadjointe si elle est stable par l'involution * (comme la partie {y, y}, pour tout élément y de A). Sur la C-algèbre des opérateurs bornés sur un espace de Hilbert H, l'involution est l'application qui à tout opérateur borné associe son adjoint, et les éléments autoadjoints sont appelés les opérateurs autoadjoints. Si H est de dimension finie, un endomorphisme linéaire de H est autoadjoint si et seulement si sa matrice dans une base orthonormée fixée est autoadjointe, ou hermitienne, c'est-à-dire égale à sa matrice adjointe (il en est alors de même pour sa matrice dans toute autre base orthonormée). Dans une catégorie à involution, un endomorphisme f est dit autoadjoint si f † = f. (fr) In mathematics, and more specifically in abstract algebra, an element x of a *-algebra is self-adjoint if . A self-adjoint element is also Hermitian, though the reverse doesn't necessarily hold. A collection C of elements of a star-algebra is self-adjoint if it is closed under the involution operation. For example, if then since in a star-algebra, the set {x,y} is a self-adjoint set even though x and y need not be self-adjoint elements. In functional analysis, a linear operator on a Hilbert space is called self-adjoint if it is equal to its own adjoint A∗. See self-adjoint operator for a detailed discussion. If the Hilbert space is finite-dimensional and an orthonormal basis has been chosen, then the operator A is self-adjoint if and only if the matrix describing A with respect to this basis is Hermitian, i.e. if it is equal to its own conjugate transpose. Hermitian matrices are also called self-adjoint. In a dagger category, a morphism is called self-adjoint if ; this is possible only for an endomorphism . (en)
prov:wasDerivedFrom	wikipedia-en:Self-adjoint?oldid=1090839461&ns=0
page length (characters) of wiki page	2237 (xsd:nonNegativeInteger)
foaf:isPrimaryTopicOf	wikipedia-en:Self-adjoint
is Link from a Wikipage to another Wikipage of	Bra–ket notation Jordan algebra Resolvent formalism Dagger category Dagger symmetric monoidal category Operator algebra Spectral triple -algebra Operator (physics) Quantum statistical mechanics Converse relation Approximate identity Sturm–Liouville theory Completely positive map Householder operator Projection-valued measure Prolate spheroidal wave function Noncommutative topology Stokes operator Gårding domain Expectation value (quantum mechanics) Extensions of symmetric operators Normal matrix Frame (linear algebra) Hilbert C-module Molecular Hamiltonian Real rank (C-algebras) Heat equation Baer ring Cotlar–Stein lemma AW-algebra Affiliated operator Albert algebra Biconjugate gradient method Transform theory Wigner–Weyl transform Dirac delta function Discrete Laplace operator Spectral theorem Spectral theory of normal C*-algebras Spectral theory of ordinary differential equations Octonion Quantum indeterminacy Skew-Hermitian matrix Superselection Quantum dilogarithm Pokhozhaev's identity Von Neumann's theorem Statistical physics Semidefinite programming Representation of a Lie superalgebra Riemann–Hilbert problem Self adjoint

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