Differential geometry on Thompson's components of positive operators
Consider the algebra L(H) of bounded linear operators on a Hilbert space H, and let L(H)+ be the set of positive elements of L(H). For each A ∈ L(H)+ we study differential geometry of the Thompson component of A, CA = {B ∈ L(H)+ : A ≤ rB and B ≤ sA for some s, r > 0}. The set of components is...
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todo:paper_00344877_v45_n1_p23_Corach2023-10-03T14:45:53Z Differential geometry on Thompson's components of positive operators Corach, G. Maestripieri, A.L. Consider the algebra L(H) of bounded linear operators on a Hilbert space H, and let L(H)+ be the set of positive elements of L(H). For each A ∈ L(H)+ we study differential geometry of the Thompson component of A, CA = {B ∈ L(H)+ : A ≤ rB and B ≤ sA for some s, r > 0}. The set of components is parametrized by means of all operator ranges of H. Each CA is a differential manifold modelled in an appropriate Banach space and a homogeneous space with a natural connection. Morover, given arbitrary B, C ∈ CA, there exists a unique geodesic with endpoints B and C. Finally, we introduce a Finsler metric on CA for which the geodesics are short and we show that it coincides with the so-called Thompson metric. JOUR info:eu-repo/semantics/openAccess http://creativecommons.org/licenses/by/2.5/ar http://hdl.handle.net/20.500.12110/paper_00344877_v45_n1_p23_Corach |
| institution |
Universidad de Buenos Aires |
| institution_str |
I-28 |
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R-134 |
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Biblioteca Digital - Facultad de Ciencias Exactas y Naturales (UBA) |
| description |
Consider the algebra L(H) of bounded linear operators on a Hilbert space H, and let L(H)+ be the set of positive elements of L(H). For each A ∈ L(H)+ we study differential geometry of the Thompson component of A, CA = {B ∈ L(H)+ : A ≤ rB and B ≤ sA for some s, r > 0}. The set of components is parametrized by means of all operator ranges of H. Each CA is a differential manifold modelled in an appropriate Banach space and a homogeneous space with a natural connection. Morover, given arbitrary B, C ∈ CA, there exists a unique geodesic with endpoints B and C. Finally, we introduce a Finsler metric on CA for which the geodesics are short and we show that it coincides with the so-called Thompson metric. |
| format |
JOUR |
| author |
Corach, G. Maestripieri, A.L. |
| spellingShingle |
Corach, G. Maestripieri, A.L. Differential geometry on Thompson's components of positive operators |
| author_facet |
Corach, G. Maestripieri, A.L. |
| author_sort |
Corach, G. |
| title |
Differential geometry on Thompson's components of positive operators |
| title_short |
Differential geometry on Thompson's components of positive operators |
| title_full |
Differential geometry on Thompson's components of positive operators |
| title_fullStr |
Differential geometry on Thompson's components of positive operators |
| title_full_unstemmed |
Differential geometry on Thompson's components of positive operators |
| title_sort |
differential geometry on thompson's components of positive operators |
| url |
http://hdl.handle.net/20.500.12110/paper_00344877_v45_n1_p23_Corach |
| work_keys_str_mv |
AT corachg differentialgeometryonthompsonscomponentsofpositiveoperators AT maestripierial differentialgeometryonthompsonscomponentsofpositiveoperators |
| _version_ |
1807323703523934208 |