Researches On The Furuta Inequality Under Chaotic Order And Other Operator Inequality Problems

Operator theory is an important part of the functional analysis, in which operator inequality is an important branch. Some operator inequalities have a wide range of applications in differential equations, optimization theory, statistics etc. In 1934, Lowner established the Lowner-Heinz inequality, which is a foundation of operator inequality theory. Furuta inequality, a famous operator inequality, is proposed by Japanese mathematician Furuta in 1987 and based on the Lowner-Heinz inequality. Afterwards, Furuta inequality has played an important role in operator inequality, which has attracted wide attentions.In Hilbert space, Chaotic order of self-adjoint operators is weaker than the ordinary order, so the conclusions about operator inequality under Chaotic order are weaker than under the ordinary order. The further discussion for the characteristics of Chaotic order are given in this paper, and we give a completely form of Furuta inequality under the chaotic order. The monotony of operator functions is also an important problem on operator inequalities, and the monotony of some operator functions under strict Chaotic order are discussed in this paper. This article also gives a promotion type of Furuta inequality under the order by (?)which is weaker then the order by C≥A≥B≥0 Matrix inequalities play an important role in the practical applications, in which Wielandt inequality becomes widespread concern due to the statistical theory of linear models in research applications. In this paper we will extend Wielandt inequality on matrix into bounded linear operators on separatable infinite dimensional Hilbert space and give Promotion type of Wielandt inequality under some weaker conditions.