Characterization of Linear Operators on Specific Classes of Algebras

Abstract

Let s, t, u ≥ 1 be fixed integers and K be a Hilbert space. If G, ∂: A → A are linear mappings that fulfill the following algebraic identity 2G(Xs+t+u) = G(Xs)ψ(Xt+u) + γ(Xs)∂(Xt)ψ(Xu) + γ(Xs+t)∂(Xu) + G(Xu)ψ(Xt+s) + γ(Xu)∂(Xt)ψ(Xs) + γ(Xu+t)∂(Xs). Then G will be a generalized (ψ, γ)-derivation having (ψ, γ)-derivation ∂ on A, where ψ and γ are automorphisms of A, a Commutant Subspace Lattice (CSL) subalgebra of a von Neumann algebra on K.