On functional equations related to generalized inner derivations on standard operator algebras

Abstract

Our aim is to prove the following result. 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, which possesses the identity operator. Suppose there exists a linear mapping F:𝒜(X)→ℒ(X) satisfying the relation F(An+2)= ∑i=1n+1Ai−1F(A2)An+1−i−∑i=1nAiF(A)An+1−i for all A∈𝒜(X) and some fixed integer n≥1. In this case F is of the form F(A)=AB1+B2A for all A∈𝒜(X) and some fixed B1,B2∈ℒ(X). In particular, F is continuous.