Some functional identities characterizing two-sided centralizers and two-sided generalized derivations on triangular algebras

Abstract

Let T be a unital triangular algebra, let n > 1 be an integer, let gamma be an invertible element of Z(T), the center of T, and let Psi, Omega:\mathcal{T}\rightarrow \mathcal{T}$ be additive mappings satisfying \begin{align*} \Psi(X^n) = \gamma X^{n - 1}\Omega(X) = \gamma \Omega(X) X^{n - 1}\end{align*} for all $X \in \mathcal{T}$. If $\Omega(\textbf{1}) \in Z(\mathcal{T})$, then $\Psi$ and $\Omega$ are two-sided centralizers on $\mathcal{T}$ and also $\Psi = \gamma \Omega$. Moreover, using a functional identity, a characterization of two-sided generalized derivations is presented. Some other related results are also discussed.