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| Natura: | Artículo científico |
| Lingua: | en |
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Universidad Industrial de Santander
2015
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| Accesso online: | https://www.redalyc.org/articulo.oa?id=327038640002 |
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| _version_ | 1866591105409089536 |
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| author | Jeovanny de Jesus Muentes Acevedo |
| author_facet | Jeovanny de Jesus Muentes Acevedo |
| contents | On the continuity of the map square root of nonnegative isomorphisms in Hilbert spaces Jeovanny de Jesus Muentes Acevedo Física, Astronomía y Matemáticas spec tral theory Hilbert s paces Nonnegative operators functions of operators Let H be a real (or complex) Hilbert space. Every nonnegative operator L ∈ L ( H ) admits a unique nonnegative square root R ∈ L ( H ) , i.e., a nonnegative operator R ∈ L ( H ) such that R 2 = L . Let GL + S ( H ) be the set of nonnegative isomorphisms in L ( H ) . First we will show that GL + S ( H ) is a convex (real) Banach manifold. Denoting by L 1 / 2 the nonnegative square root of L . In [3], Richard Bouldin proves that L 1 / 2 depends continuously on L (this proof is non-trivial). This result has several applic ations. For example, it is used to find the polar decomposition of a bounde d operator. This polar decomposition allows us to determine the positiv e and negative spectral subespaces of any self-adjoint operator, and more over, allows us to define the Maslov index. The autor of the paper under review pr ovides an alternative proof (and a little more simplified) that L 1 / 2 depends continuously on L , and moreover, he shows that the map R : GL + S ( H ) → GL + S ( H ) L 7→ L 1 / 2 is a homeomorphism. 2015 artículo científico 0120-419X https://www.redalyc.org/articulo.oa?id=327038640002 en http://www.redalyc.org/revista.oa?id=3270 Revista Integración application/pdf Universidad Industrial de Santander Revista Integración (Colombia) Num.1 Vol.33 |
| format | Artículo científico |
| id | redalyc_327038640002 |
| language | en |
| publishDate | 2015 |
| publisher | Universidad Industrial de Santander |
| spellingShingle | On the continuity of the map square root of nonnegative isomorphisms in Hilbert spaces Jeovanny de Jesus Muentes Acevedo Física, Astronomía y Matemáticas spec tral theory Hilbert s paces Nonnegative operators functions of operators On the continuity of the map square root of nonnegative isomorphisms in Hilbert spaces Jeovanny de Jesus Muentes Acevedo Física, Astronomía y Matemáticas spec tral theory Hilbert s paces Nonnegative operators functions of operators Let H be a real (or complex) Hilbert space. Every nonnegative operator L ∈ L ( H ) admits a unique nonnegative square root R ∈ L ( H ) , i.e., a nonnegative operator R ∈ L ( H ) such that R 2 = L . Let GL + S ( H ) be the set of nonnegative isomorphisms in L ( H ) . First we will show that GL + S ( H ) is a convex (real) Banach manifold. Denoting by L 1 / 2 the nonnegative square root of L . In [3], Richard Bouldin proves that L 1 / 2 depends continuously on L (this proof is non-trivial). This result has several applic ations. For example, it is used to find the polar decomposition of a bounde d operator. This polar decomposition allows us to determine the positiv e and negative spectral subespaces of any self-adjoint operator, and more over, allows us to define the Maslov index. The autor of the paper under review pr ovides an alternative proof (and a little more simplified) that L 1 / 2 depends continuously on L , and moreover, he shows that the map R : GL + S ( H ) → GL + S ( H ) L 7→ L 1 / 2 is a homeomorphism. 2015 artículo científico 0120-419X https://www.redalyc.org/articulo.oa?id=327038640002 en http://www.redalyc.org/revista.oa?id=3270 Revista Integración application/pdf Universidad Industrial de Santander Revista Integración (Colombia) Num.1 Vol.33 |
| title | On the continuity of the map square root of nonnegative isomorphisms in Hilbert spaces |
| topic | Física, Astronomía y Matemáticas spec tral theory Hilbert s paces Nonnegative operators functions of operators |
| url | https://www.redalyc.org/articulo.oa?id=327038640002 |