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Some inequalities of matrix power and Karcher means for positive linear maps

2017/02/24 by Rahmatollah Lashkaripour, Lashkaripour, Rahmatollah, Monire Hajmohamadi +3
Computer Science · Mathematics · #47A64 #Advanced Topics in Algebra #FOS: Mathematics #Functional Analysis (math.FA) #Mathematical Inequalities and Applications #Matrix Theory and Algorithms #Operator Algebras (math.OA) #math.FA #math.OA #msc:47A64

paper · pdf · doi:10.48550/arxiv.1702.07488

arxiv created 2017/02/24 · openalex publication_date 2017/02/24 · arxiv updated 2017/02/27 · openalex created_date 2022/09/01 · openalex updated_date 2026/07/28

Abstract

In this paper, we generalize some matrix inequalities involving matrix power and Karcher means of positive definite matrices. Among other inequalities, it is shown that if \mathbb A=(A₁,...,A_n) is a n-tuple of positive definite matrices such that 0<m≤ A_i≤ M (i=1,⋯,n) for some scalars m< M and ω=(w₁,⋯,w_n) is a weight vector with w_i≥0 and ∑_i=1ⁿw_i=1, then Φ^p(∑_i=1ⁿw_iA_i)≤ α^pΦ^p(P_t(ω; \mathbb A)) and Φ^p(∑_i=1ⁿw_iA_i)≤ α^pΦ^p(Λ(ω; \mathbb A)), where p>0, α=max\\frac(M+m)²4Mm, \frac(M+m)²4^(2)/(p)Mm\, Φ is a positive unital linear map and t∈ [-1, 1]\backslash \0\.

Some inequalities of matrix power and Karcher means for positive linear maps

Abstract

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