On Certain Identity Related to Jordan

In this paper we prove the following result. LetH be a real or complex Hilbert space, let L(H) be the algebra of all bounded linear operators on H and let A(H) ⊆ L(H) be a standard operator algebra. Suppose we have an additive mapping D : A(H) → L(H) satisfying the relation D(An) = D(A)A∗n−1 + AD(An−2)A∗ + An−1D(A) for all A ∈ A(H) and some fixed integer n > 1. In this case there exists a unique B ∈ L(H) such that D(A) = BA − AB holds for all A ∈ A(H). Throughout, R will represent an associative ring with center Z(R). Given an integer n ≥ 2, a ring R is said to be n−torsion free if for x ∈ R, nx = 0 implies x = 0. An additive mapping x 7→ x∗ on a ring R is called an involution if (xy)∗ = y∗x∗ and x∗∗ = x hold for all x, y ∈ R. A ring equipped with an involution is called a ring with involution or ∗-ring. Recall that a ring R is prime if for a, b ∈ R, aRb = (0) implies that either a = 0 or b = 0, and is semiprime in case aRa = (0) implies a = 0. An additive mapping D : R → R, where R is an arbitrary ∗-ring is called a ∗-derivation in case D(xy) = D(x)y∗ + xD(y) holds for all pairs x, y ∈ R and is called a Jordan ∗-derivation if D(x) = D(x)x∗ + xD(x) is fulfilled for all x ∈ R. It is easy to prove that there are no nonzero ∗-derivations on noncommutative prime ∗-rings (see [1] for the details). Note that the mapping x 7→ ax∗ − xa, where a ∈ R is a fixed element, is a Jordan ∗-derivation; such Jordan ∗-derivations are said to be inner. By our knowledge the concept of Jordan ∗-derivations first appeared in [1]. The study of Jordan ∗-derivations has been motivated by the problem of the representability of quadratic forms by bilinear forms (for the results concerning this problem we refer to [6-10,13,14,16-19,22,23]). It turns 2010 Mathematics Subject Classification. 16W10, 46K15, 39B05.