A converse of Loewner-Heinz inequality and applications to operator means

JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS

卷:413

Article

Uchiyama, Mitsuru¹  Yamazaki, Takeaki² 

[1] Shimane Univ, Dept Math, Matsue, Shimane, Japan

[2] Toyo Univ, Dept Elect Elect & Comp Engn, Kawagoe, Saitama 3508585, Japan

关键词: Positive definite operators;    Loewner-Heinz inequality;    Operator mean;    Operator monotone function;    Operator concave function;    Ando-Hiai inequality;    Geometric mean;    Karcher mean;    Power mean;   

DOI  :  10.1016/j.jmaa.2013.11.055

来源: Elsevier

【 摘 要 】

Let f (t) be an operator monotone function. Then A <= B implies f (A) <= f (B), but the converse implication is not true. Let A # B be the geometric mean of A, B >= 0. If A <= B,then B-1 # A <= I; the converse implication is not true either. We will show that if f (lambda B + I)(-1) # f (lambda A + I) <= I for all sufficiently small lambda > 0, then f (lambda A + I) <= f (lambda B + I) and A <= B. Moreover, we extend it to multi-Variable matrices means. (C) 2013 Elsevier Inc. All rights reserved.

【 授权许可】

Free