Furuta inequality of indefinite type

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 ...

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.

The Brunn-Minkowski-Type Inequality

a cauchy-schwarz type inequality for fuzzy integrals

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