Research On Preserving The Truncation Of Operator On B(H)

In recent decades,preserving problem is a hot research issue in the field of operator algebras.The research on maps preserving various kinds of partial orders has aroused extensive attention of scholars.Based on a condition in the definition of diamond partial order AA*A=AB*A,the concept of the truncation of operator is introduced.Let A,B?B(H),we say that operator A is a truncation of operator B,if A=PABQA.The truncation reveals a relationship between operators,in this paper,we mainly study two kinds of preserving problems of the truncation of operator on B(H).In the second chapter,we mainly study additive maps preserving the truncation of operators on B(H).Let H be a complex Hilbert space with dimension greater than one,B(H)is the algebra of all bounded linear operators on H.If ?:B(H)?B(H)is an additive surjective map,then ? preserves the truncation of operator(or AA*A=AB*A)in both directions if and only if there exist a nonzero scalar ??C and operators U and V on H which are both unitaries or both anti-unitaries such that ?(T)=?UTV or ?(T)=?UT*V for all T?B(H).We say ? preserves the truncation of sum of two operators in both directions,if ?(A)is the truncation of ?(B)+?(C)if and only if A is the truncation of B+C.In the third chapter,we consider the relation between maps preserving the truncation of sum of operators and additive maps preserving the truncation of operator,and prove that if a bijection ? on B(H)preserves the truncation of sum of two operators in both directions,then there exist a nonzero scalar ? ?C and operators U and V on H which are both unitaries or both anti-unitaries such that cp(T)=?UTV or ?(T)=?UT*V for all T?B(H).Finally,it is proved that if cp preserves the truncation of difference of two operators in both directions,then the characterization of ? is the same as above.