A complement of the Ando–Hiai inequality
| Autor: | Yuki Seo, Masaru Tominaga |
|---|---|
| Rok vydání: | 2008 |
| Předmět: |
Complement (group theory)
Operator inequality Numerical Analysis Algebra and Number Theory Operator mean Mathematical analysis Positive operator Araki–Cordes inequality Crystallography Generalized Kantorovich constant Ando–Hiai inequality Discrete Mathematics and Combinatorics Geometry and Topology Mathematics Normed vector space |
| Zdroj: | Linear Algebra and its Applications. 429(7):1546-1554 |
| ISSN: | 0024-3795 |
| DOI: | 10.1016/j.laa.2008.04.025 |
| Popis: | In this paper, we present a complement of a generalized Ando–Hiai inequality due to Fujii and Kamei [M. Fujii, E. Kamei, Ando–Hiai inequality and Furuta inequality, Linear Algebra Appl. 416 (2006) 541–545]. Let A and B be positive operators on a Hilbert space H such that 0 m 1 ⩽ A ⩽ M 1 and 0 m 2 ⩽ B ⩽ M 2 for some scalars m i ⩽ M i ( i = 1 , 2 ), and let α ∈ [ 0 , 1 ] . Put h i = M i m i for i = 1 , 2 . Then for each 0 r ⩽ 1 and s ⩾ 1 A r ♯ α r ( 1 - α ) s + α r B s ⩽ K h 1 s h 2 s , r s - ( 1 - α ) s ( 1 - α ) s + α r h 2 ( 1 - α ) s ( s - r ) ( 1 - α ) s + α r ‖ A ♯ α B ‖ rs ( 1 - α ) s + α r , where A ♯ α B ≔ A 1 2 ( A - 1 2 BA - 1 2 ) α A 1 2 is the α -geometric mean and a generalized Kantorovich constant K ( h , p ) is defined for h > 0 as K ( h , p ) ≔ h p - h ( p - 1 ) ( h - 1 ) p - 1 p h p - 1 h p - h p for all real numbers p ∈ R . |
| Databáze: | OpenAIRE |
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