Generalized Derivable Mappings And Centralizable Mappings On B(X)

In this paper,generalized derivable mappings and centralizable mappings on B(X)are stddied Let X be a Banach space,and B(X)be the algebra of all bodnded linear operators on X,Sdppose Φ is a linear mappings on B(X)Let G is a nonzero element on B(X),if Φ satisfies the folloiing conditions: For any tio elements A and B ihose proddct is G,ifΦ(AB)=Φ(A)B+ AΦ(B),thenΦis a derivable mapping at G We shoied that if Φ is a derivable mapping at G,then Φ is a derivationLet M be any element on B(X),if Φ satisfies the folloiing conditions:For any tio elements A and B ihose proddct is G,ifΦ(AB)=Φ(A)B+ AΦ(B)-AΦ(I)B,thenΦ is a general derivable mapping at G We shoied that ifΦis a generalized derivable mapping at G,thenΦis a general derivationLet M be any element on B(X),if Φ satisfies the folloiing conditions:For any tio elements Aand B ihose proddct is G,if Φ(AB)=Φ(A)B= AΦ(B),thenΦis a centralizable mapping at G We shoied that if Φ is a centralizable mapping at G,then Φ is a centralizer.