Additive Maps Preserving Two Kinds Of Operator Equations

Linear preserver problems has attracted the attention of many mathematicians in recent decades.It aims to characterize the linear or additive maps on matrix algebra or operator algebra that leave certain properties,functions,subsets or relations invariant.In the study of linear preserving problems,the characterization of additive maps preserving invertibility,idempotents,nilpotents and zero products on matrix algebras and operator algebras has attracted extensive attention of scholars,and a series of profound results have been obtained.The study of these problems is essentially the study of additive maps preserving a class of operator equations.Motivated by the concepts of generalized inverses and star partial orders,this paper characterizes additive maps preserving two kinds of operator equations.Generalized inverse is a very important concept in operator theory,and inner inverse is a very elementary notion in generalized inverse theory.Let B(X)be the algebra of all bounded linear operators on an infinite dimensional complex Banach space X.In chapter 2,we mainly prove that an additive surjective map φ on B(X)preserves inner inverses in both directions if and only if threre exist a scalarα∈{-1,1} and two bijective bounded linear,or conjugate linear operators A,B such that φ(T)=αATA-1 or φ(T)=αBT*B-1 for all T∈B(X).From the definition of inner inverse,it can be seen that the characterization of additive surjective maps preserving inner inverses in both directions is actually the characterization of additive surjective maps preserving operator equation ABA=A in both directions.Star partial order is one of many partial orders on matrix algebra and operator algebra.The operator equation A*A=A*B is of great significance as an equation of the definition of star partial orders.Let H be a complex Hilbert space with dimension greater than 1,B(H)the algebra of all bounded linear operators on H.In chapter 3,we mainly prove that an additive surjective map φ on B(H)preserves A*A=A*B in both directions if and only if there exist a nonzero scalar α∈C and two invertible bounded linear,or invertible bounded conjugate linear operator U and V such that φ(T)=αUTV for all T∈B(X),where U is a unitary or anti-unitary operator.