On functional equations related to derivations and bicircular projections

Abstract

In this paper we investigate some functional equations on standard operator algebras and semiprime rings. We prove, for example, the following result, which is related to a classical result of Chernoff. Let X be a real or complex Banach space, let L (X) be the algebra of all bounded linear operators on X and let A (X) ⊂ L (X) be a standard operator algebra. Suppose there exists a linear mapping D : A (X) → L (X) satisfying the relation D(An) = D(A)An−1 + AD(An−2)A+An−1D(A) for all A ∈ A (X), where n > 2 is some fixed integer. In this case D is of the form D(A) = [A,B] for all A ∈ A (X) and some fixed B ∈ L (X) . Some functional equations related to bicircular projections are also investigated. Mathematics subject classification (2010): 16W10, 46K15, 39B05.