Search

Let R be a prime ring of characteristic different from 2, Qr its right Martindale quotient ring, L a non-central Lie ideal of R (i.e., L , R ≠ 0 ), n ≥ 1 a fixed integer, F and G two generalized skew derivations of R, associated with the same automorphism α of R. If (F (x y) − G (x) y) n = 0 , for any x , y ∈ L , then there exists a ∈ Q r such that , for any x ∈ R , unless when R satisfies the standard polynomial identity s 4 (x 1 , ... , x 4) . [ABSTRACT FROM AUTHOR]

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]