Research On Partial Order And Properties Of Operators

Functional analysis is one of the important research fields of basic mathematics Operator algebra and operator theory are important components of functional analysis Partial order theory in a Hilbert space is an important research object in operator theory.Core partial order,dual core partial order,star partial order,sharp partial order and Drazin order are all important order relations.Firstly,based on the known order relations and characterizations of generalized inverse,the characterizations of operator orders in a Hilbert space are given.Secondly,the mapping of preservering the order relation of operators is studied.Finally,the characterizations of Drazin order on the algebra are studied.The structure of this article is as follows:In the first part,the origin and development of functional analysis,operator theory,operator algebra and order theory are introduced.The domestic andinternational research status of star partial order,sharp partial order,core partial order,dual core partial order and Drazin order,and related preliminary knowledge are given.In the second part,under the condition of the space decomposition of the Hilbert space H=R(Ak)⊕ N(Ak),the matrix characterization of operator Drazin order is given by the matrix characterization of operator Drazin inverse,and then the related properties of operator Drazin order are studied.In the third part,the properties of sharp partial order,core partial order and dual core partial order of bounded linear operators in a Hilbert space are studied,and the characterization of the bounded linear surjective maps that preserve the core partial order is given.In the fourth part,the properties of Drazin inverse and Drazin order of elements in the algebra are studied.By using the properties of the Drazin inverse of idempotent,the characterization of the Drazin order of idempotent in the algebra is given.