B (h) To Keep Jordan Area Projection Additive Mapping

In recent years, the descriptions on linear and non-linear maps preserving cer-tain character or computing on operator algebras or operator spaces have been concerned widely for a long time. Some scholars have achieved a series of results in this research areas. For example, the characterization of linear and non-linear surjection preserving operator pairs whose products are projections[6,17] and so on. Jordan product is a kind of special operator product in operator algebras or opera-tor spaces, describing the structure of maps preserving qualities of Jordan product becomes an interesting problem. Such as, additive maps preserving Jordan zero-product of operators[19-21] and maps preserving the spectrum of Jordan products of operators[23] and so on. This text mainly studies linear maps preserving the nonze-ro projections of Jordan products of two operators on Mn and additive bijection preserving the nonzero projections of Jordan products of two operators on operator spaces. It shows that such maps must be the constant times of isomorphisms or anti-isomorphisms on operator spaces. From this, we get some structural theorems of these maps. This paper contains three chapters, main content of every chapter as follows:In chapter1, we mainly introduce some basic definitions and prepared theorems which are always used in this paper.In chapter2, we depict the characterization of linear maps preserving the nonze-ro projections of Jordan product of two operators on operator spaces with dimension greater than or equal to2. When the dimension of the Hilbert space is finite, we research the structure of linear mapping φ on Mn, we prove the linear map φ pre-serving the nonzero projection of Jordan product of two operators on operator spaces is a bijection, from this we can get φ preserves rank-1operators. On the basic of the above steps, we obtain that such mapping must be constant times of isomorphism or anti-isomorphism. When the dimension of the Hilbert space is infinite, let the map φ be a bounded surjection, and φ preserves square nilpotent elements, we obtain that such mapping must be constant times of isomorphism or anti-isomorphism.In chapter3, we discuss that the characterization of an additive bijection φ which preserves the nonzero projections of Jordan product of two operators in both direction on an operator space with the dimension greater than or equal to3, we can get that such map must be constant times of isomorphism or anti-isomorphism by proving that φ preserves projections and their orthogonality.