Operator Algebras On Idempotent Operators Preserving Idempotence Linear Mapping

Let X be a Banach space over F where F is either the field of real numbers R or the field of complex numbers C, H be an infinite dimensional Hilbert space. In this paper we study the form of idempotent over X and the similarity of idempotents and projections.Let B(X) denote the algebra of all bounded linear operators on X, and F(X) the algebra of all finite rank operators in B(X). In this paper, we study the linear mappings of F(X) which preserve idempotents. We get the equivalence of the linear mappings of F(X) which preserve idempotents onto the algebra A over F and the Jordan homorphisms. Applying this theorem we get the form of linear mappings of F(X) which preserve idempotents. Furthermore,we also get the corresponding results on B(X) and B{H).