Analysis And Generalization Of Some Problems Via The Furuta Inequality

Operator inequality is an important subject of operator theory. Furutainequality, a famous operator inequality, is proposed by Japanesemathematician Furuta based on the L(?)wner-Heinz inequality. Furutainequality has played important roles in operator inequality, operatorequation and the study of math, physics.The main purpose of this thesis is generalizing the Furuta inequalityby the theory of operator mean and the theory of operator monotonicity,and applying these theories to C^*-algebra.In the first chapter, we introduce the Furuta inequality and itsremainder problem, citing some conclusions of operator mean theory.Many scholars have generalized the Furuta inequality. In this chapter weconsider the generalized Furuta inequality proposed by Bach, Furuta andderive a satellite of this generalized Furuta inequality by changingconditions.The second chapter mainly concentrates on some Furuta inequalitiesin the form of operator power mean. We first introduce the geometric structure of Furuta inequality and then discuss the monotonicity of anoperator power mean function. In addition, we study the monotonicity ofsome operator generalized power mean functions under the order definedbyA²â‰¥(AB² A)^1/2.The third chapter discusses operator monotone function. Firstly, weintroduce its definition. By constructing four groups of operator functions,we demonstrate the monotonicity of these operator functions under thechaotic order and the order defined by A^k≥B^k and thereby concludesome relevant corollaries.In the last chapter, we mainly give an application via the operatormean theory to C^*-algebra, introducing and studyingα-power geometricmean and generalized spectral geometric mean of two positive definiteelements. In 1997, Fielder and Pta'k introduced spectral geometric meanof two positive definite matrixes, and discussed its properties. Afterwards,Lu Lihua and Jiang Jianfei have proposed generalized spectral geometricmean of two positive operators. In this note we will extend thegeneralized spectral geometric mean of two positive operators definiteelements to two positive definite elements.