Eizaburo Kamei

Furuta's inequality and its application to Ando's theorem

Abstract As a continuation of preceding notes, we discuss Furutas inequality under the “chaotic order” defined by log A ⩾ log B for positive invertible operators A and B , which is applied to a generalization of Andos theorem. Consequently we obtain Furutas inequality under the chaotic order.

Mean theoretic approach to the grand Furuta inequality

Very recently, Furuta obtained the grand Furuta inequality which is a parameteric formula interpolating the Furuta inequality and the AndoHiai inequality as follows : If A ≥ B ≥ 0 and A is invertible, then for each t ∈ [0, 1], Fp,t(A,B, r, s) = A −r/2{Ar/2(A−t/2BpA−t/2)sAr/2} 1−t+r (p−t)s+r A−r/2 is a decreasing function of both r and s for all r ≥ t, p ≥ 1 and s ≥ 1. In this note, we employ a mean theoretic approach to the grand Furuta inequality. Consequently we propose a basic inequality, by which we present a simple proof of the grand Furuta inequality.

Characterization of chaotic order and its application to Furuta inequality

In this note, we give a simple characterization of the chaotic order log A > log B among positive invertible operators A, B on a Hilbert space. As an application, we discuss Furutas type operator inequality.

A Characterization of Chaotic Order and a Problem

MASATOSHI FUJII a,,, JIAN FEI JIANG b, EIZABURQ KAMEI c and KOTARO TANAHASHI d a Department of Mathematics, Osaka Kyoiku University, Asahigaoka, Kashiwara, Osaka 582, Japan; b Department of Basic Science and Technology, China Textile University, Shanghai, Postal code 200051, China; c Maebashi Institute of Technology, Kamisadori, Maebashi, Gunma 371, Japan; d Department of Mathematics, Tohoku College of Pharmacy, Komatsushima, Aoba-ku, Sendai 981, Japan

Variants of Ando-Hiai Inequality

We discuss two variables version of the Ando-Hiai inequality: For A, B > 0 and α ∈ [0, 1], if A Open image in new window B ≤ I, then Open image in new window Here Open image in new window is the α-geometric mean in the sense of Kubo-Ando. In this context, the Furuta inequality is understood as the one-sided version (the case of s = 1): If A Open image in new window B ≤ I, then Open image in new window As a consequence, the Furuta inequality has an alternative simple proof. In addition, we point out that the obtained inequality is understood as the case t = 1 in the grand Furuta inequality.

Characterization of chaotic order and its applications to furuta's type operator inequalities

Very recently, we obtained a simple characterization of the chaotic order log Alog B among positive invertible operators A B on a Hilbert space. In this note, we discuss Furutas type operator inequalities as applications of our characterization of the chaotic order.

An Application of Furuta’s Inequality to Ando’s Theorem

Several authors have given mean theoretic considerations to Furuta’s inequality which is an extension of Lowner-Heinz inequality. Ando discussed it on the geometric mean. In this note, Furuta’s inequality is applied to a generalization of Ando’s theorem.