New characterizations of operator monotone functions

• Published in: Linear algebra and its applications Vol. 546; pp. 169 - 186
• Main Authors: Dinh, Trung Hoa, Dumitru, Raluca, Franco, Jose A.
• Format: Journal Article
• Language: English
• Published: Amsterdam Elsevier Inc 01.06.2018; American Elsevier Company, Inc
• Subjects: Arithmetic; Kubo-Ando means; Lattice of functions; Linear algebra; Mathematical analysis; Mathematical functions; Monotone functions; Operator monotone functions; Operators (mathematics); Self-adjoint means; Symmetric means; Symmetry; Operator monotone functions; Self-adjoint means; Kubo-Ando means; 47A56; 15B48; 47A64; Symmetric means; Lattice of functions; 47A63; 46E05
• ISSN: 0024-3795; 1873-1856
• DOI: 10.1016/j.laa.2018.02.004

If σ is a symmetric mean and f is an operator monotone function on [0,∞), thenf(2(A−1+B−1)−1)≤f(AσB)≤f((A+B)/2). Conversely, Ando and Hiai showed that if f is a function that satisfies either one of these inequalities for all positive operators A and B and a symmetric mean different than the arithmetic and the harmonic mean, then the function is operator monotone. In this paper, we show that the arithmetic and the harmonic means can be replaced by the geometric mean to obtain similar characterizations. Moreover, we give characterizations of operator monotone functions using self-adjoint means and general means subject to a constraint due to Kubo and Ando.