On reverses of the Golden-Thompson type inequalities

2017/08/20 by Mohammad Bagher Ghaemi, Ghaemi, Mohammad Bagher, Venus Kaleibary +3
Computer Science · Mathematics · #Mathematical Inequalities and Applications #Mathematical functions and polynomials #Matrix Theory and Algorithms #math.FA

paper · pdf · doi:10.48550/arxiv.1708.05951

arxiv created 2017/08/20 · arxiv updated 2018/04/17

Abstract

In this paper we present some reverses of the Golden-Thompson type inequalities: Let H and K be Hermitian matrices such that e^s e^H \preceq_ols e^K \preceq_ols e^t e^H for some scalars s ≤ t, and α∈ [0 , 1]. Then for all p>0 and k =1,2,…, n λ_k (e^{(1-α)H + αK} ) ≤ (max \lbrace S(e^sp), S(e^tp)\rbrace)^(1)/(p) λ_k (e^pH \sharp_αe^pK)^(1)/(p), where A\sharp_αB = A^(1)/(2) ( A^-(1)/(2) B^(1)/(2) A^-(1)/(2) ) ^αA^(1)/(2) is α-geometric mean, S(t) is the so called Specht's ratio and \preceq_ols is the so called Olson order. The same inequalities are also provided with other constants. The obtained inequalities improve some known results.

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