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

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*} Ψ(X^n) = γX^{n - 1}Ω(X) = γΩ(X) X^{n - 1}\end{align*} for allX \in $\mathcal{T}$$\mathrm{.}If$Ω($\mathbf{\text{1}}$) \in Z($\mathcal{T}$)$,then$Ψ$and$Ω$aretwo-sidedcentralizerson$$\mathcal{T}$$andalso$Ψ= γΩ$. Moreover, using a functional identity, a characterization of two-sided generalized derivations is presented. Some other related results are also discussed.