Ando–Hiai inequality and Furuta inequality

Complete Form of Furuta Inequality

Let A and B be bounded linear operators on a Hilbert space satisfying A ≥ B ≥ 0. The well-known Furuta inequality is given as follows: Let r ≥ 0 and p > 0; then A r 2 Amin{1,p}A r 2 ≥ (A r 2 BpA r 2 ) min{1,p}+r p+r . In order to give a self-contained proof of it, Furuta (1989) proved that if 1 ≥ r ≥ 0, p > p0 > 0 and 2p0 + r ≥ p > p0, then (A r 2 Bp0A r 2 ) p+r p0+r ≥ (A r 2 BpA r 2 ) p+r p+r ...

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 ...

Corruption And Inequality

Education , Inequality and Income Inequality

In this paper we propose to measure inequality of educational achievements by constructing a Gini index on educational attainments. We then use the proposed measure to analyse the relationship between inequality in incomes and educational achievements (in terms of both the average attainments and its concentration). Even if theoretical considerations suggest a non-linear relationship between th...

An application of grand Furuta inequality to a type of operator equation

The existence of positive semidefinite solutions of the operator equation n ∑ j=1 AXA = Y is investigated by applying grand Furuta inequality. If there exists positive semidefinite solutions of the operator equation, one of the special types of Y is obtained, which extends the related result before. Finally, an example is given based on our result.