On the orbit of invariant subspaces of linear operators in finite-dimensional spaces (new proof of a Halmos's result)
P.R. Halmos proved that for a linear operator A over a finite-dimensional complex vector space E, every A-invariant subspace of E is the range of a commutant of A. His proof was based on a generalization of the concept of eigenvector. In this note, we give an invariant proof of this Halmos's th...
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| Veröffentlicht in: | Linear algebra and its applications Jg. 329; H. 1; S. 171 - 174 |
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| Format: | Journal Article |
| Sprache: | Englisch |
| Veröffentlicht: |
New York, NY
Elsevier Inc
15.05.2001
Elsevier Science |
| Schlagworte: | |
| ISSN: | 0024-3795, 1873-1856 |
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| Abstract | P.R. Halmos proved that for a linear operator
A over a finite-dimensional complex vector space
E, every
A-invariant subspace of
E is the range of a commutant of
A. His proof was based on a generalization of the concept of eigenvector. In this note, we give an invariant proof of this Halmos's theorem. |
|---|---|
| AbstractList | P.R. Halmos proved that for a linear operator
A over a finite-dimensional complex vector space
E, every
A-invariant subspace of
E is the range of a commutant of
A. His proof was based on a generalization of the concept of eigenvector. In this note, we give an invariant proof of this Halmos's theorem. |
| Author | Faouzi, Abdelkhalek |
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| Keywords | Invariant subspaces Functor 47A15 Commutant Invariant Invariant subspace Orbit Eigenvector Linear operator |
| Language | English |
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| References | L. Fuchs, Infinite Abelian Groups, vols. I and II, Academic Press, New York, 1970 and 1973 Halmos (BIB5) 1975; 4 Carlson (BIB3) 1972; 5 Barraa, Charles (BIB1) 1990; 135 Barraa, Charles (BIB2) 1991; 153 I. Kaplansky, Infinite Abelian Groups, The University of Michigan, 1954 10.1016/S0024-3795(01)00239-7_BIB4 Barraa (10.1016/S0024-3795(01)00239-7_BIB1) 1990; 135 Barraa (10.1016/S0024-3795(01)00239-7_BIB2) 1991; 153 Carlson (10.1016/S0024-3795(01)00239-7_BIB3) 1972; 5 10.1016/S0024-3795(01)00239-7_BIB6 Halmos (10.1016/S0024-3795(01)00239-7_BIB5) 1975; 4 |
| References_xml | – volume: 135 start-page: 167 year: 1990 end-page: 170 ident: BIB1 article-title: Sur l'orbite d'un espace de Banach sous l'action du commutant d'un opérateur nilpotent publication-title: Linear Algebra Appl. – volume: 5 start-page: 293 year: 1972 end-page: 298 ident: BIB3 article-title: Inequalities for the degrees of elementary divisors of modules publication-title: Linear Algebra Appl. – volume: 4 start-page: 11 year: 1975 end-page: 15 ident: BIB5 article-title: Eigenvectors and adjoints publication-title: Linear Algebra Appl. – reference: L. Fuchs, Infinite Abelian Groups, vols. I and II, Academic Press, New York, 1970 and 1973 – reference: I. Kaplansky, Infinite Abelian Groups, The University of Michigan, 1954 – volume: 153 start-page: 177 year: 1991 end-page: 182 ident: BIB2 article-title: Sous-espaces invariants d'un opérateur nilpotent sur un espace de Banach publication-title: Linear Algebra Appl. – volume: 135 start-page: 167 year: 1990 ident: 10.1016/S0024-3795(01)00239-7_BIB1 article-title: Sur l'orbite d'un espace de Banach sous l'action du commutant d'un opérateur nilpotent publication-title: Linear Algebra Appl. doi: 10.1016/0024-3795(90)90119-W – ident: 10.1016/S0024-3795(01)00239-7_BIB4 – volume: 5 start-page: 293 year: 1972 ident: 10.1016/S0024-3795(01)00239-7_BIB3 article-title: Inequalities for the degrees of elementary divisors of modules publication-title: Linear Algebra Appl. doi: 10.1016/0024-3795(72)90010-9 – ident: 10.1016/S0024-3795(01)00239-7_BIB6 – volume: 153 start-page: 177 year: 1991 ident: 10.1016/S0024-3795(01)00239-7_BIB2 article-title: Sous-espaces invariants d'un opérateur nilpotent sur un espace de Banach publication-title: Linear Algebra Appl. doi: 10.1016/0024-3795(91)90217-K – volume: 4 start-page: 11 year: 1975 ident: 10.1016/S0024-3795(01)00239-7_BIB5 article-title: Eigenvectors and adjoints publication-title: Linear Algebra Appl. doi: 10.1016/0024-3795(71)90025-5 |
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| Snippet | P.R. Halmos proved that for a linear operator
A over a finite-dimensional complex vector space
E, every
A-invariant subspace of
E is the range of a commutant... |
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| SubjectTerms | Commutant Exact sciences and technology Function theory, analysis Functor Invariant subspaces Mathematical analysis Mathematical methods in physics Mathematics Operator theory Physics Sciences and techniques of general use |
| Title | On the orbit of invariant subspaces of linear operators in finite-dimensional spaces (new proof of a Halmos's result) |
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