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(X^s+t+u) = G(X^s)ψ(X^t+u) + γ(X^s)∂(X^t)ψ(X^u) + γ(X^s+t)∂(X^u) + G(X^u)ψ(X^t+s) + γ(X^u)∂(X^t)ψ(X^s) + γ(X^u+t)∂(X^s). 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.