Löwner's theorem and the differential geometry of the space of positive operators
Let A be a untel C*-algebra and G+ the space of all positive invertible elements of A. In this largely expository paper we collect several geometrical features of G+ which relate its structure with that of Riemannian manifolds with non positive curvature. The main result of the paper is the equivale...
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todo:paper_00015504_v49_n2_p70_Andruchow2023-10-03T13:51:18Z Löwner's theorem and the differential geometry of the space of positive operators Andruchow, E. Corach, G. Stojanoff, D. Norm inequalities Positive operators Let A be a untel C*-algebra and G+ the space of all positive invertible elements of A. In this largely expository paper we collect several geometrical features of G+ which relate its structure with that of Riemannian manifolds with non positive curvature. The main result of the paper is the equivalence of the so-called Löwner-Heinz-Cordes inequality ∥StTt∥ ≤ ∥ST∥t (valid for positive operators S, T on a Hilbert space and t ∈ [0, 1]) with the geometrical fact that for every pair γ, δ of geodesics of G+ the real function t → d(γ(t), δ(t)) is convex, where d denotes the geodesic distance. Fil:Andruchow, E. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales; Argentina. Fil:Corach, G. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales; Argentina. Fil:Stojanoff, D. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales; Argentina. JOUR info:eu-repo/semantics/openAccess http://creativecommons.org/licenses/by/2.5/ar http://hdl.handle.net/20.500.12110/paper_00015504_v49_n2_p70_Andruchow |
| institution |
Universidad de Buenos Aires |
| institution_str |
I-28 |
| repository_str |
R-134 |
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Biblioteca Digital - Facultad de Ciencias Exactas y Naturales (UBA) |
| topic |
Norm inequalities Positive operators |
| spellingShingle |
Norm inequalities Positive operators Andruchow, E. Corach, G. Stojanoff, D. Löwner's theorem and the differential geometry of the space of positive operators |
| topic_facet |
Norm inequalities Positive operators |
| description |
Let A be a untel C*-algebra and G+ the space of all positive invertible elements of A. In this largely expository paper we collect several geometrical features of G+ which relate its structure with that of Riemannian manifolds with non positive curvature. The main result of the paper is the equivalence of the so-called Löwner-Heinz-Cordes inequality ∥StTt∥ ≤ ∥ST∥t (valid for positive operators S, T on a Hilbert space and t ∈ [0, 1]) with the geometrical fact that for every pair γ, δ of geodesics of G+ the real function t → d(γ(t), δ(t)) is convex, where d denotes the geodesic distance. |
| format |
JOUR |
| author |
Andruchow, E. Corach, G. Stojanoff, D. |
| author_facet |
Andruchow, E. Corach, G. Stojanoff, D. |
| author_sort |
Andruchow, E. |
| title |
Löwner's theorem and the differential geometry of the space of positive operators |
| title_short |
Löwner's theorem and the differential geometry of the space of positive operators |
| title_full |
Löwner's theorem and the differential geometry of the space of positive operators |
| title_fullStr |
Löwner's theorem and the differential geometry of the space of positive operators |
| title_full_unstemmed |
Löwner's theorem and the differential geometry of the space of positive operators |
| title_sort |
löwner's theorem and the differential geometry of the space of positive operators |
| url |
http://hdl.handle.net/20.500.12110/paper_00015504_v49_n2_p70_Andruchow |
| work_keys_str_mv |
AT andruchowe lownerstheoremandthedifferentialgeometryofthespaceofpositiveoperators AT corachg lownerstheoremandthedifferentialgeometryofthespaceofpositiveoperators AT stojanoffd lownerstheoremandthedifferentialgeometryofthespaceofpositiveoperators |
| _version_ |
1782028951726587904 |