Discussion Of Some Problem Via The Furuta Inequality

In this thesis we study generalization and application of the Furuta inequality. In 1987, Furuta established an operator inequality which is beautiful and historical extension of the famous Lowner-Heinz inequality.In the first part of the thesis, we mainly introduce the Furuta inequality and some conclusion about it. In order to have a good grip of the Furuta inequality, we consider two examples and derive a satellite of the Furuta inequality by using the method of proving above examples. On the other hand, we simply discuss the remainder problem of the Furuta inequality.In the second part of the thesis, we study some operator function. Firstly, we discuss an operator function via the Furuta type inequality with negative index. And by using the thought of the geometrical structure in the Furuta inequality, we point out its monotonicity in different domain. In addition, we try our best to consider the Grand Furuta inequality with negative index, and obtain a series of conclusions. Secondly, we give and prove a typical operator monotone functionthrough other's idea. Moreover we state the situation in chaotic order.In the last part, we mainly give an application via the Furuta inequality to the generalized spectral geometric mean. The Lowner-Heinz inequality induces the geometric mean of two positive operators in Hilbert space. In 1997, Fielder and Pta'k introduced spectral geometric mean F(A,B) of positive definite matrixes, and discuss its properties. In this note we will introduce the generalized spectral geometric mean E_α(A,B) and to extend their theorems. In addition, we show some comparison theorems on Eα(A,B) by virtue of the Furuta Inequality.