Furuta Inequality and Its Related Topics

This article is devoted to a brief survey of Furuta inequality and its related topics. It consists of 4 sections: 1. From Löwner-Heinz inequality to Furuta inequality, 2. Ando–Hiai inequality, 3. Grand Furuta inequality, and 4. Chaotic order. 1. From Löwner-Heinz inequality to Furuta inequality. The noncommutativity of operators appears in the fact that t is not orderpreserving. That is, there is a pair of positive operators A and B such that A ≥ B and A 6≥ B. The following is a quite familiar example; A = ( 2 1 1 1 ) , B = ( 1 0 0 0 ) . This implies that t is not order-preserving for p > 1 by assuming the following fact, see [20, 23, 24]: Theorem 1.1 (Löwner-Heinz inequality (LH)). The fuction t is order-preserving for 0 ≤ p ≤ 1, i.e., A ≥ B ≥ 0 =⇒ A ≥ B. The essense of the Löwner-Heinz inequality is the case p = 1 2 : A ≥ B ≥ 0 =⇒ A 1 2 ≥ B 1 2 . It is rephrased as follows: For A, B ≥ 0, ABA ≤ 1 =⇒ A 1 2BA 1 2 ≤ 1. Date: Received: 1 December 2010; Accepted: 21 December 2010. 2010 Mathematics Subject Classification. Primary 47A63; Secondary 47A64.