On right (left) \theta-centralizers on Banach algebras

Let \mathcal{A} be a Banach algebra with unity ۱, and \theta:\mathcal{A}\rightarrow \mathcal{A} be an continuous automorphism. In this paper we characterize a continuous linear map T: \mathcal{A}\rightarrow \mathcal{A} which satisfies one of the following conditions:\[a,b\in \mathcal{A}, \, ab =w\Longrightarrow \theta(a)T(b)=T(w) ,\]\[ a,b \in \mathcal{A}, \, ab =w\Longrightarrow T(a)\theta(b)=T(w) ,\]or\[ a,b \in \mathcal{A}, \, ab =w\Longrightarrow \theta(a)T(b)=T(a)\theta(b)=T(w) ,\]where w\neq ۰ is a left (right) separating point of \mathcal{A}.

Keywords:

Left θ-centralizer , right \theta-centralizer , θ-centralizer , Banach algebra