Let B(X) and M (F) n be the algebra of all bounded linear operators on a complex Banach space X with dimX 3 and the algebra of all n n matrices over a field F with char F 2 , respectively. Also let F(A) be the space of all fixed points of an operator AB(X) . In this paper, we characterize the forms of linear maps :B(X)B(X) which satisfy F(A) = 0F( (A)) = 0 and linear maps :M (F) M (F) n n which preserve the fixed points of matrices.
