Self-adjoint operator (original) (raw)
في الرياضيات، مؤثر المساعد الذاتي (بالإنجليزية: Self-adjoint operator) على فضاء متجهي عقدي لا نهائي الأبعاد V مع الجداء الداخلي (بالتساوي، مؤثر هيرميتي (بالإنجليزية: Hermitian operator) في حالة ذات أبعاد محدودة) هو تحويل خطي A (من V إلى نفسه) وهو النقطة المرافقة له.
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| dbo:abstract | في الرياضيات، مؤثر المساعد الذاتي (بالإنجليزية: Self-adjoint operator) على فضاء متجهي عقدي لا نهائي الأبعاد V مع الجداء الداخلي (بالتساوي، مؤثر هيرميتي (بالإنجليزية: Hermitian operator) في حالة ذات أبعاد محدودة) هو تحويل خطي A (من V إلى نفسه) وهو النقطة المرافقة له. (ar) Samoadjungovaný operátor je lineární operátor se zvláštními vlastnostmi. Operátory a především samoadjungované operátory studuje funkcionální analýza. Samoadjungovaný operátor je zobecněním samoadjungované matice. (cs) Στα μαθηματικά, ένας αυτoσυζυγής τελεστής σε ένα μιγαδικό διανυσματικό χώρο V με εσωτερικό γινόμενο είναι ένας τελεστής ( μία γραμμική απεικόνιση A από τον V στον εαυτό του) που είναι ο ίδιος ο συζυγής του: . Αν V είναι πεπερασμένης διάστασης με μία δοσμένη βάση,αυτό είναι ισοδύναμο με την συνθήκη ότι ο πίνακας A είναι , δηλαδή, ίσος με το συζυγή ανάστροφό του πίνακα, τον A*. Από το πεπερασμένης διάστασης, ο V έχει τέτοια ώστε ο πίνακας A σε σχέση με τη βάση αυτή να είναι ένας διαγώνιος πίνακας με στοιχεία πραγματικούς αριθμούς. Σ'αυτό το άρθρο, οι γενικεύσεις αυτής της έννοιας αντιστοιχούν σε τελεστές για χώρους Hilbert αυθαίρετης διάστασης. Οι αυτοσυζυγείς τελεστές χρησιμοποιούνται στη συναρτησιακή ανάλυση και στην κβαντική μηχανική. Η σημασία τους στην κβαντική μηχανική επεκτείνεται στον τύπο του Dirac–von Neumann της κβαντικής μηχανικής, στον οποίο φυσικές παρατηρήσιμες μεταβλητές όπως η θέση, η ορμή,η στροφορμή και η περιστροφή αντιπροσωπεύονται από τους αυτο-συζυγείς τελεστές στο χώρο Hilbert. Ιδιαίτερης σημασίας είναι η Χαμιλτονιανή η οποία ως παρατηρήσιμη μεταβλητή αντιστοιχεί στη συνολική ενέργεια του σωματιδίου μάζας m σ'ένα πραγματικό δυναμικό πεδίο V. Οι διαφορικοί τελεστές είναι μία σημαντική κατηγορία των . Η δομή των αυτοσυζυγών τελεστών σε απειροδιάστατους χώρους Hilbert μοιάζει ουσιαστικά με την περίπτωση των χώρων Hilbert πεπερασμένης διάστασης. Δηλαδή οι τελεστές είναι αυτοσυζυγείς αν και μόνο αν είναι μοναδικά ισοδύναμοι με πραγματικούς τελεστές πολλαπλασιασμού. Με τις κατάλληλες τροποποιήσεις το αποτέλεσμα αυτό μπορεί να επεκταθεί ενδεχομένως και στους μη φραγμένους τελεστές σε απειροδιάστατους χώρους. Δεδομένου ότι ένας ορισμένος παντού τελεστής είναι αναγκαστικά φραγμένος, χρειάζεται κανείς να είναι πιο προσεκτικός στον ορισμό του πεδίου ορισμού στη μη φραγμένη περίπτωση. Αυτό εξηγείται παρακάτω με περισσότερες λεπτομέρειες. (el) Ein selbstadjungierter Operator ist ein linearer Operator mit besonderen Eigenschaften. Operatoren und insbesondere selbstadjungierte Operatoren werden im mathematischen Teilgebiet der Funktionalanalysis untersucht. Der selbstadjungierte Operator ist eine Verallgemeinerung der selbstadjungierten Matrix. (de) In mathematics, a self-adjoint operator on an infinite-dimensional complex vector space V with inner product (equivalently, a Hermitian operator in the finite-dimensional case) is a linear map A (from V to itself) that is its own adjoint. If V is finite-dimensional with a given orthonormal basis, this is equivalent to the condition that the matrix of A is a Hermitian matrix, i.e., equal to its conjugate transpose A∗. By the finite-dimensional spectral theorem, V has an orthonormal basis such that the matrix of A relative to this basis is a diagonal matrix with entries in the real numbers. In this article, we consider generalizations of this concept to operators on Hilbert spaces of arbitrary dimension. Self-adjoint operators are used in functional analysis and quantum mechanics. In quantum mechanics their importance lies in the Dirac–von Neumann formulation of quantum mechanics, in which physical observables such as position, momentum, angular momentum and spin are represented by self-adjoint operators on a Hilbert space. Of particular significance is the Hamiltonian operator defined by which as an observable corresponds to the total energy of a particle of mass m in a real potential field V. Differential operators are an important class of unbounded operators. The structure of self-adjoint operators on infinite-dimensional Hilbert spaces essentially resembles the finite-dimensional case. That is to say, operators are self-adjoint if and only if they are unitarily equivalent to real-valued multiplication operators. With suitable modifications, this result can be extended to possibly unbounded operators on infinite-dimensional spaces. Since an everywhere-defined self-adjoint operator is necessarily bounded, one needs be more attentive to the domain issue in the unbounded case. This is explained below in more detail. (en) エルミート作用素（エルミートさようそ、英: Hermitian operator, Hermitian）とは、複素ヒルベルト空間上の線形作用素で、自分自身と形式共役になるようなもののことである。 物理学の特に量子力学の文脈では作用素のことを「演算子」と呼ぶ。そのため、エルミート作用素はエルミート演算子と呼ばれる。 エルミート作用素という名称は、エルミート行列などの研究で知られるフランス人数学者シャルル・エルミートに因む。 (ja) En mathématiques et plus précisément en algèbre linéaire, un endomorphisme autoadjoint ou opérateur hermitien est un endomorphisme d'espace de Hilbert qui est son propre adjoint (sur un espace de Hilbert réel on dit aussi endomorphisme symétrique). Le prototype d'espace de Hilbert est un espace euclidien, c'est-à-dire un espace vectoriel sur le corps des réels, de dimension finie, et muni d'un produit scalaire. L'analogue sur le corps des complexes s'appelle un espace hermitien. Sur ces espaces de Hilbert de dimension finie, un endomorphisme autoadjoint est diagonalisable dans une certaine base orthonormale et ses valeurs propres (même dans le cas complexe) sont réelles. Les applications des propriétés structurelles d'un endomorphisme autoadjoint (donc de sa forme quadratique associée) sont nombreuses. (fr) 작용소 이론에서 자기 수반 작용소(自己隨伴作用素, 영어: self-adjoint operator)는 스스로의 에르미트 수반이 자신과 같은 작용소이다. 유한 차원에서의 에르미트 행렬을 일반화한 개념이다. (ko) In matematica, in particolare in algebra lineare, un operatore autoaggiunto è un operatore lineare su uno spazio di Hilbert che è uguale al suo aggiunto. In letteratura si usa talvolta chiamare operatore simmetrico un operatore definito in un sottospazio di uno spazio vettoriale, il cui aggiunto non è in generale simmetrico, e operatore hermitiano un operatore densamente definito in tale spazio. Nel caso di uno spazio finito-dimensionale alcuni autori utilizzano inoltre il termine operatore simmetrico per denotare un operatore autoaggiunto nel caso reale. Per il teorema di Hellinger-Toeplitz un operatore simmetrico definito ovunque è anche limitato, e se il suo aggiunto è definito ovunque ed è limitato allora l'operatore è limitato. In particolare, se un operatore simmetrico limitato non è definito su tutto lo spazio allora può essere esteso in modo unico ad un operatore definito ovunque. La matrice che rappresenta un operatore autoaggiunto è una hermitiana, ed in dimensione finita il teorema spettrale asserisce che ogni operatore autoaggiunto di uno spazio vettoriale reale dotato di un prodotto scalare definito positivo ha una base ortonormale formata da autovettori. Equivalentemente, ogni matrice simmetrica reale è simile ad una matrice diagonale tramite una matrice ortogonale i cui coefficienti sono reali. Gli operatori autoaggiunti sono fondamentali in vari settori della matematica e della fisica, come ad esempio la geometria differenziale, l'analisi funzionale e la meccanica quantistica. (it) Um operador autoadjunto, hermitiano (português brasileiro) ou hermítico (português europeu) é um operador linear em um espaço vetorial com produto interno que é o adjunto de si mesmo. No caso de espaços de dimensão finita, a matriz que representa esse operador é igual à sua transposta conjugada. * Propriedades * Um operador é autoadjunto se e somente se * Todo autovalor de um operador autoadjunto é real: * Se e são autovalores diferentes associados a autovetores e . Então :Como e são distintos, temos , portanto . (pt) Operator samosprzężony (hermitowski) – odwzorowanie liniowe działające na skończenie wymiarowej, zespolonej przestrzeni wektorowej takie że gdzie: – iloczyn skalarny wektorów w przestrzeni – wektor powstały w wyniku działania operatora na wektor – sprzężenie hermitowskie wektora Operatory samosprzężone używane są w analizie funkcjonalnej. W mechanice kwantowej operatory samosprzężone reprezentują wielkości mierzone – nazywa się je obserwablami. Przydatność operatorów hermitowskich wynika stąd, że ich wartości własne są liczbami rzeczywistymi i z tej racji mogą określać wyniki pomiarów fizycznych. Operator samosprzężony skończenie wymiarowy można reprezentować za pomocą macierzy hermitowskiej (samosprzężonej). (pl) 在數學裏，作用於一個有限維的内积空間，一個自伴算子（self-adjoint operator）等於自己的伴隨算子；等價地說，在一组单位酉正交基下，表達自伴算子的矩陣是埃爾米特矩陣。埃爾米特矩陣等於自己的共軛轉置。根據有限維的譜定理，必定存在著一個正交歸一基，可以表達自伴算子為一個實值的對角矩陣。 (zh) |
| dbo:wikiPageExternalLink | https://www.mat.univie.ac.at/~gerald/ftp/book-schroe/ https://books.google.com/books%3Fid=UirEAgAAQBAJ&q=%22Self-adjoint+operator%22&pg=PP1 https://doi.org/10.1073/pnas.71.5.1952 |
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| dbp:date | September 2015 (en) |
| dbp:proof | Let be self-adjoint. Self-adjoint operators are symmetric. The initial steps of this proof are carried out based on the symmetry alone. Self-adjointness of is not used directly until step 1b. Let Denote Using the notations from the section on symmetric operators , it suffices to prove that # Let The goal is to prove the existence and boundedness of the inverted resolvent operator and show that We begin by showing that and # The operator has now been proven to be bijective, so the set-theoretic inverse exists and is everywhere defined. The graph of is the set Since is closed , so is By closed graph theorem, is bounded, so (en) # By assumption, is symmetric; therefore For every Let . If then Since and are not in the spectrum, the operators are bijective. Moreover, # Indeed, If one had then would not be injective, i.e. one would have As discussed in the article about Adjoint operator, and, hence, This contradicts the bijectiveness. # The equality shows that i.e. is self-adjoint. Indeed, it suffices to prove that For every and (en) |
| dbp:reason | Is this really true? We did not require the domain of A to be dense, yet S is densely defined. It appears that S has acquired a property that A did not possess. (en) |
| dbp:title | Proof: self-adjoint operator has real spectrum (en) Proof: Symmetric operator with real spectrum is self-adjoint (en) |
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| rdfs:comment | في الرياضيات، مؤثر المساعد الذاتي (بالإنجليزية: Self-adjoint operator) على فضاء متجهي عقدي لا نهائي الأبعاد V مع الجداء الداخلي (بالتساوي، مؤثر هيرميتي (بالإنجليزية: Hermitian operator) في حالة ذات أبعاد محدودة) هو تحويل خطي A (من V إلى نفسه) وهو النقطة المرافقة له. (ar) Samoadjungovaný operátor je lineární operátor se zvláštními vlastnostmi. Operátory a především samoadjungované operátory studuje funkcionální analýza. Samoadjungovaný operátor je zobecněním samoadjungované matice. (cs) Ein selbstadjungierter Operator ist ein linearer Operator mit besonderen Eigenschaften. Operatoren und insbesondere selbstadjungierte Operatoren werden im mathematischen Teilgebiet der Funktionalanalysis untersucht. Der selbstadjungierte Operator ist eine Verallgemeinerung der selbstadjungierten Matrix. (de) エルミート作用素（エルミートさようそ、英: Hermitian operator, Hermitian）とは、複素ヒルベルト空間上の線形作用素で、自分自身と形式共役になるようなもののことである。 物理学の特に量子力学の文脈では作用素のことを「演算子」と呼ぶ。そのため、エルミート作用素はエルミート演算子と呼ばれる。 エルミート作用素という名称は、エルミート行列などの研究で知られるフランス人数学者シャルル・エルミートに因む。 (ja) 작용소 이론에서 자기 수반 작용소(自己隨伴作用素, 영어: self-adjoint operator)는 스스로의 에르미트 수반이 자신과 같은 작용소이다. 유한 차원에서의 에르미트 행렬을 일반화한 개념이다. (ko) Um operador autoadjunto, hermitiano (português brasileiro) ou hermítico (português europeu) é um operador linear em um espaço vetorial com produto interno que é o adjunto de si mesmo. No caso de espaços de dimensão finita, a matriz que representa esse operador é igual à sua transposta conjugada. * Propriedades * Um operador é autoadjunto se e somente se * Todo autovalor de um operador autoadjunto é real: * Se e são autovalores diferentes associados a autovetores e . Então :Como e são distintos, temos , portanto . (pt) 在數學裏，作用於一個有限維的内积空間，一個自伴算子（self-adjoint operator）等於自己的伴隨算子；等價地說，在一组单位酉正交基下，表達自伴算子的矩陣是埃爾米特矩陣。埃爾米特矩陣等於自己的共軛轉置。根據有限維的譜定理，必定存在著一個正交歸一基，可以表達自伴算子為一個實值的對角矩陣。 (zh) Στα μαθηματικά, ένας αυτoσυζυγής τελεστής σε ένα μιγαδικό διανυσματικό χώρο V με εσωτερικό γινόμενο είναι ένας τελεστής ( μία γραμμική απεικόνιση A από τον V στον εαυτό του) που είναι ο ίδιος ο συζυγής του: . Αν V είναι πεπερασμένης διάστασης με μία δοσμένη βάση,αυτό είναι ισοδύναμο με την συνθήκη ότι ο πίνακας A είναι , δηλαδή, ίσος με το συζυγή ανάστροφό του πίνακα, τον A*. Από το πεπερασμένης διάστασης, ο V έχει τέτοια ώστε ο πίνακας A σε σχέση με τη βάση αυτή να είναι ένας διαγώνιος πίνακας με στοιχεία πραγματικούς αριθμούς. Σ'αυτό το άρθρο, οι γενικεύσεις αυτής της έννοιας αντιστοιχούν σε τελεστές για χώρους Hilbert αυθαίρετης διάστασης. (el) En mathématiques et plus précisément en algèbre linéaire, un endomorphisme autoadjoint ou opérateur hermitien est un endomorphisme d'espace de Hilbert qui est son propre adjoint (sur un espace de Hilbert réel on dit aussi endomorphisme symétrique). Le prototype d'espace de Hilbert est un espace euclidien, c'est-à-dire un espace vectoriel sur le corps des réels, de dimension finie, et muni d'un produit scalaire. L'analogue sur le corps des complexes s'appelle un espace hermitien. Sur ces espaces de Hilbert de dimension finie, un endomorphisme autoadjoint est diagonalisable dans une certaine base orthonormale et ses valeurs propres (même dans le cas complexe) sont réelles. Les applications des propriétés structurelles d'un endomorphisme autoadjoint (donc de sa forme quadratique associée) son (fr) In mathematics, a self-adjoint operator on an infinite-dimensional complex vector space V with inner product (equivalently, a Hermitian operator in the finite-dimensional case) is a linear map A (from V to itself) that is its own adjoint. If V is finite-dimensional with a given orthonormal basis, this is equivalent to the condition that the matrix of A is a Hermitian matrix, i.e., equal to its conjugate transpose A∗. By the finite-dimensional spectral theorem, V has an orthonormal basis such that the matrix of A relative to this basis is a diagonal matrix with entries in the real numbers. In this article, we consider generalizations of this concept to operators on Hilbert spaces of arbitrary dimension. (en) In matematica, in particolare in algebra lineare, un operatore autoaggiunto è un operatore lineare su uno spazio di Hilbert che è uguale al suo aggiunto. In letteratura si usa talvolta chiamare operatore simmetrico un operatore definito in un sottospazio di uno spazio vettoriale, il cui aggiunto non è in generale simmetrico, e operatore hermitiano un operatore densamente definito in tale spazio. Nel caso di uno spazio finito-dimensionale alcuni autori utilizzano inoltre il termine operatore simmetrico per denotare un operatore autoaggiunto nel caso reale. (it) Operator samosprzężony (hermitowski) – odwzorowanie liniowe działające na skończenie wymiarowej, zespolonej przestrzeni wektorowej takie że gdzie: – iloczyn skalarny wektorów w przestrzeni – wektor powstały w wyniku działania operatora na wektor – sprzężenie hermitowskie wektora Operatory samosprzężone używane są w analizie funkcjonalnej. Operator samosprzężony skończenie wymiarowy można reprezentować za pomocą macierzy hermitowskiej (samosprzężonej). (pl) |
| rdfs:label | مؤثر مساعد ذاتي (ar) Samoadjungovaný operátor (cs) Selbstadjungierter Operator (de) Αυτοσυζυγής τελεστής (el) Endomorphisme autoadjoint (fr) Operatore autoaggiunto (it) 자기 수반 작용소 (ko) エルミート作用素 (ja) Self-adjoint operator (en) Operator samosprzężony (pl) Operador autoadjunto (pt) 自伴算子 (zh) |
| rdfs:seeAlso | dbr:Extensions_of_symmetric_operators |
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