Identity related to additive mappings on standard operator algebras.

Let X be a real or complex Banach space, let L ( X ) be the algebra of all bounded linear operators of X and let A ( X ) ⊆ L ( X ) be a standard operator algebra. Suppose there exists a linear mapping T : A ( X ) → L ( X ) satisfying the relation T ( A n ) = T ( A ) A n −1 − AT ( A n −2 ) A − A n −1 T ( A ) for all A ∈ A ( X ) , where n > 2 is some fixed integer. Then T is of the form: (i) T ( A ) = 0 for all A ∈ F ( X ) and (ii) T ( A ) = BA , for all A ∈ A ( X ) and some B ∈ L ( X ) . [ABSTRACT FROM AUTHOR]