Additive Maps Preserving Self-Jordan Product On B(H)

Preserver problems on operator algebras have been one of the most attractive research topics for many scholars for a long time. In recent years, more and more researchers have consisdered preserve problems forcus on preserving certain properties, subsets or relationships of operators. Some preserve problems about special product of operators also arised lots of scholars’attention. So in this paper we discuss the characterizations of linear maps preserving self-Jordan product of two matrices on the matrix algebras and additive bijective maps preserving self-Jordan product of two operators on the operator spaces. Using the properties of maps that preserving self-Jordan product, many characters of those maps can be found. This paper contains three chapters, main content of every chapter as follows:In Chapter1, the signigcation and background of this thesis selecting subject are introduced. In addition, we offer some necessary notations, definitions, and some conclusions in this paper.In Chapter2, the characterizations of a linear map φ on Mn preserving self-Jordan product of matrices is studied. In course of the proof, fristly, we prove that a linear maps φ which preserves the self-Jordan product of two matrices can also preserve the rank of the matrices. Secondly, on the basis of the above steps, we obtain that such maps must have the follow representations:φ(X)=UXU*,(?)X∈Mn or φ(X)=UXTU*,(?)X∈Mn.where U∈Mn is a unitary operator, and XT denotes the transpose of X under a constant basis in Mn.In Chapter3, we discuss the characterizations of the additive bijective maps φ which preserves the self-Jordan product of two operators on operator spaces. In course of discussion, we prove that an additive bijective maps which preserves self-Jordan product of two operators on operator spaces can also preserve the projection of rank-1operators and orthgonality of rank-1operators in both directions in the front several steps. And then we obtain that such maps must be constent times of isomorphisms or anti-isomorphisms on operator spaces by famous Uhlhorn’s theorem and Winger’s theorem.