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

• Published in: Journal of mathematical analysis and applications Vol. 413; no. 1; pp. 422 - 429
• Main Authors: Uchiyama, Mitsuru, Yamazaki, Takeaki
• Format: Journal Article
• Language: English
• Published: Elsevier Inc 01-05-2014
• Subjects: Ando–Hiai inequality; Geometric mean; Karcher mean; Loewner–Heinz inequality; Operator concave function; Operator mean; Operator monotone function; Positive definite operators; Power mean; Operator concave function; Geometric mean; Loewner–Heinz inequality; Operator monotone function; Karcher mean; Positive definite operators; Ando–Hiai inequality; Operator mean; Power mean
• ISSN: 0022-247X; 1096-0813
• DOI: 10.1016/j.jmaa.2013.11.055

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(λB+I)−1♯f(λA+I)⩽I for all sufficiently small λ>0, then f(λA+I)⩽f(λB+I) and A⩽B. Moreover, we extend it to multi-variable matrices means.