Symmetric multilinear forms on Hilbert spaces: Where do they attain their norm?
We characterize the sets of norm one vectors x1,…,xk in a Hilbert space H such that there exists a k-linear symmetric form attaining its norm at (x1,…,xk). We prove that in the bilinear case, any two vectors satisfy this property. However, for k≥3 only collinear vectors satisfy this property in the...
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2019
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| Acceso en línea: | https://bibliotecadigital.exactas.uba.ar/collection/paper/document/paper_00243795_v563_n_p178_Carando http://hdl.handle.net/20.500.12110/paper_00243795_v563_n_p178_Carando |
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paper:paper_00243795_v563_n_p178_Carando2023-06-08T14:52:09Z Symmetric multilinear forms on Hilbert spaces: Where do they attain their norm? Hilbert spaces Multilinear forms Norm attaining mappings Hilbert spaces Tensors Multilinear forms Real case Symmetric tensors Unit ball Vector spaces We characterize the sets of norm one vectors x1,…,xk in a Hilbert space H such that there exists a k-linear symmetric form attaining its norm at (x1,…,xk). We prove that in the bilinear case, any two vectors satisfy this property. However, for k≥3 only collinear vectors satisfy this property in the complex case, while in the real case this is equivalent to x1,…,xk spanning a subspace of dimension at most 2. We use these results to obtain some applications to symmetric multilinear forms, symmetric tensor products and the exposed points of the unit ball of Ls(Hk). © 2018 Elsevier Inc. 2019 https://bibliotecadigital.exactas.uba.ar/collection/paper/document/paper_00243795_v563_n_p178_Carando http://hdl.handle.net/20.500.12110/paper_00243795_v563_n_p178_Carando |
| institution |
Universidad de Buenos Aires |
| institution_str |
I-28 |
| repository_str |
R-134 |
| collection |
Biblioteca Digital - Facultad de Ciencias Exactas y Naturales (UBA) |
| topic |
Hilbert spaces Multilinear forms Norm attaining mappings Hilbert spaces Tensors Multilinear forms Real case Symmetric tensors Unit ball Vector spaces |
| spellingShingle |
Hilbert spaces Multilinear forms Norm attaining mappings Hilbert spaces Tensors Multilinear forms Real case Symmetric tensors Unit ball Vector spaces Symmetric multilinear forms on Hilbert spaces: Where do they attain their norm? |
| topic_facet |
Hilbert spaces Multilinear forms Norm attaining mappings Hilbert spaces Tensors Multilinear forms Real case Symmetric tensors Unit ball Vector spaces |
| description |
We characterize the sets of norm one vectors x1,…,xk in a Hilbert space H such that there exists a k-linear symmetric form attaining its norm at (x1,…,xk). We prove that in the bilinear case, any two vectors satisfy this property. However, for k≥3 only collinear vectors satisfy this property in the complex case, while in the real case this is equivalent to x1,…,xk spanning a subspace of dimension at most 2. We use these results to obtain some applications to symmetric multilinear forms, symmetric tensor products and the exposed points of the unit ball of Ls(Hk). © 2018 Elsevier Inc. |
| title |
Symmetric multilinear forms on Hilbert spaces: Where do they attain their norm? |
| title_short |
Symmetric multilinear forms on Hilbert spaces: Where do they attain their norm? |
| title_full |
Symmetric multilinear forms on Hilbert spaces: Where do they attain their norm? |
| title_fullStr |
Symmetric multilinear forms on Hilbert spaces: Where do they attain their norm? |
| title_full_unstemmed |
Symmetric multilinear forms on Hilbert spaces: Where do they attain their norm? |
| title_sort |
symmetric multilinear forms on hilbert spaces: where do they attain their norm? |
| publishDate |
2019 |
| url |
https://bibliotecadigital.exactas.uba.ar/collection/paper/document/paper_00243795_v563_n_p178_Carando http://hdl.handle.net/20.500.12110/paper_00243795_v563_n_p178_Carando |
| _version_ |
1768544120366694400 |