On Certain Equation Related to Derivations on Standard Operator Algebras and Semiprime Rings
نویسنده
چکیده
In this paper we prove the following result, which is related to a classical result of Chernoff. Let X be a real or complex Banach space, let A(X) be a standard operator algebra on X and let L(X) be an algebra of all bounded linear operators on X. Suppose we have a linear mapping D : A(X) → L(X) satisfying the relation D(Am+n) = D(Am)An + AmD(An) for all A ∈ A(X) and some fixed integers m ≥ 1, n ≥ 1. In this case there exists B ∈ L(X), such that D(A) = AB − BA holds for all A ∈ F(X), where F(X) denotes the ideal of all finite rank operators in L(X). Besides, D(Am) = AmB − BAm is fulfilled for all A ∈ A(X). Throughout, R will represent an associative ring with center Z(R). Given an integer n > 1, a ring R is said to be n−torsion free, if nx = 0, x ∈ R implies x = 0. 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. Let A be an algebra over the real or complex field and let B be a subalgebra of A. A linear mapping D : B → A is called a linear derivation in case D(xy) = D(x)y + xD(y) holds for all pairs x, y ∈ B. In case we have a ring R an additive mapping D : R → R is called a derivation if D(xy) = D(x)y + xD(y) holds for all pairs x, y ∈ R and is called a Jordan derivation in case D(x) = D(x)x + xD(x) is fulfilled for all x ∈ R. A derivation D is inner in case there exists a ∈ R, such that D(x) = ax − xa holds for all x ∈ R. Every derivation is a Jordan derivation. The converse is in general not true. A classical result of Herstein ([5]) asserts that any Jordan derivation on a 2−torsion free prime ring is a derivation. A brief proof 2010 Mathematics Subject Classification. 16N60, 39B05, 46K15.
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