Researches On Partial Orders Of Operators And Operator Inequalities

In this article, we study four types of operator partial orders, operator inequalities and the characterizations for convex function of commutativity of a C*?algebra. We divide this article into three chapters.The contents of the first chapter are star order, left-star order, right-star order and minus order of operators. This section is the most important one in this article. Very recently, the study of these aspects on matrices attracts much attention of many authors. Gro/3, J.K.Baksalary and Jan Hauke have gotten great development in these aspects. Some results on these problems can be seen in [1], [2], [3], [4]. In these papers, the authors have given the characterizations, properties and relationships of four types of matrix partial orders. Meanwhile, the relationships between partial orders of matrices and their powers also have been given. In this chapter, we give the definitions of four partial orders on complex Hilbert spaces. Using the method of block matrix of operator, which is different from the above authors, we give concise and precise geometric characterizations of four types of operator partial orders. Furthermore, we study properties of four types of operator partial orders and relationships between partial orders of operators and their powers.The contents of the second chapter are operator inequalities and norm inequalities. Researches of inequalities emerged very early in mathematics. Today inequalities play an important role in several mathematical fields. And it provides a more active and attractive research field for mathematics. With the development of operator algebras and operator theory, the theory of inequalities has become interesting problems in functional analysis. E.F.Beckenbach and R.Bellman(see[8]) gave many new conclusions on inequalities in book named inequalities in that time. D.S.Mitrinovic(see[9]) gave several classical inequalities in book named analytic inequalities. J.Bendat and S.Sherman in [6] gave characterizations of monotone and convexity operator functions. F.Hansen and G.K.Perdersen in [5] gave several important results on Jensen's operator inequality and Jensen's trace inequality. T.Ando in [7] gave some important inequalities on geometric and harmonic means. In this chapter, we give some operator inequalities and norm inequalities by means of spectral decomposition and functional calculus. We also give another proof of the norm inequality proved by R.Nakamoto in [19].The content of the third chapter is characterizations for convex function of commu-tativity of a C* - algebra. There exist several characterizations for the commutativity of C* - algebra. One type is the well-known Stinespring theorem. That is, a C*- algebra A is commutative if and only if every positive linear map from C*- algebra A to C*-algebra B becomes completely positive. Another type is based on an operator monotone function on a C*- algebra. Guoxing Ji and J.Tomiyama in [11] gave another characterization for commutativity based on the property of functions exp x and xp which are monotone increasing functions on the positive axis but not operator monotone on M2, the matrix algebra of all complex 2x2 matrices. That is, & C*- algebra A is commutative if and only if there exists a continuous function which is not matrix monotone on MI but operator monotone on a C*- algebra. In this chapter, we give characterizations for convex function of commutativity of a C*- algebra. That is, there exists a continuous convex function which is not matrix convex on M2 but operator convex on a C*- algebra A.