Operators in Rigged Hilbert Spaces, Gel’fand Bases and Generalized Eigenvalues
Given a self-adjoint operator <i>A</i> in a Hilbert space <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="script">H</mi></semantics></math></inline-for...
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| Format: | Article |
| Language: | English |
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MDPI AG
2022-12-01
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| Series: | Mathematics |
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| Online Access: | https://www.mdpi.com/2227-7390/11/1/195 |
| _version_ | 1797406806723526656 |
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| author | Jean-Pierre Antoine Camillo Trapani |
| author_facet | Jean-Pierre Antoine Camillo Trapani |
| author_sort | Jean-Pierre Antoine |
| collection | DOAJ |
| description | Given a self-adjoint operator <i>A</i> in a Hilbert space <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="script">H</mi></semantics></math></inline-formula>, we analyze its spectral behavior when it is expressed in terms of generalized eigenvectors. Using the formalism of Gel’fand distribution bases, we explore the conditions for the generalized eigenspaces to be one-dimensional, i.e., for <i>A</i> to have a simple spectrum. |
| first_indexed | 2024-03-09T03:30:58Z |
| format | Article |
| id | doaj.art-e43fb271e0234b358a0c0a0912a59d71 |
| institution | Directory Open Access Journal |
| issn | 2227-7390 |
| language | English |
| last_indexed | 2024-03-09T03:30:58Z |
| publishDate | 2022-12-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Mathematics |
| spelling | doaj.art-e43fb271e0234b358a0c0a0912a59d712023-12-03T14:55:36ZengMDPI AGMathematics2227-73902022-12-0111119510.3390/math11010195Operators in Rigged Hilbert Spaces, Gel’fand Bases and Generalized EigenvaluesJean-Pierre Antoine0Camillo Trapani1Institut de Recherche en Mathématique et Physique, Université Catholique de Louvain, B-1348 Louvain-la-Neuve, BelgiumDipartimento di Matematica e Informatica, Università degli Studi di Palermo, Via Archirafi n. 34, I-90123 Palermo, ItalyGiven a self-adjoint operator <i>A</i> in a Hilbert space <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="script">H</mi></semantics></math></inline-formula>, we analyze its spectral behavior when it is expressed in terms of generalized eigenvectors. Using the formalism of Gel’fand distribution bases, we explore the conditions for the generalized eigenspaces to be one-dimensional, i.e., for <i>A</i> to have a simple spectrum.https://www.mdpi.com/2227-7390/11/1/195rigged Hilbert spacegeneralized eigenvectorssimple spectrum |
| spellingShingle | Jean-Pierre Antoine Camillo Trapani Operators in Rigged Hilbert Spaces, Gel’fand Bases and Generalized Eigenvalues Mathematics rigged Hilbert space generalized eigenvectors simple spectrum |
| title | Operators in Rigged Hilbert Spaces, Gel’fand Bases and Generalized Eigenvalues |
| title_full | Operators in Rigged Hilbert Spaces, Gel’fand Bases and Generalized Eigenvalues |
| title_fullStr | Operators in Rigged Hilbert Spaces, Gel’fand Bases and Generalized Eigenvalues |
| title_full_unstemmed | Operators in Rigged Hilbert Spaces, Gel’fand Bases and Generalized Eigenvalues |
| title_short | Operators in Rigged Hilbert Spaces, Gel’fand Bases and Generalized Eigenvalues |
| title_sort | operators in rigged hilbert spaces gel fand bases and generalized eigenvalues |
| topic | rigged Hilbert space generalized eigenvectors simple spectrum |
| url | https://www.mdpi.com/2227-7390/11/1/195 |
| work_keys_str_mv | AT jeanpierreantoine operatorsinriggedhilbertspacesgelfandbasesandgeneralizedeigenvalues AT camillotrapani operatorsinriggedhilbertspacesgelfandbasesandgeneralizedeigenvalues |