Let \(\mathcal {A}\) and \(\mathcal {B}\) be standard operator algebras on complex Banach spaces \(\mathcal {X}\) and \(\mathcal {Y}\) , respectively. For \(T\in \mathcal {B}(\mathcal {X})\) , we denote by \(\sigma _{\pi }(T)\) the peripheral spectrum of T, defined by \(\sigma _{\pi }(T)=\{\lambda \in \sigma (T): |\lambda |=r(T)\}\) . The aim of this paper is to describe the maps \(\phi _1, \phi _2: \mathcal {A} \rightarrow \mathcal {B}\) whose ranges contain all operators of rank at most two and which satisfy \( \sigma _{\pi }(TST) = \sigma _{\pi }(\phi _1(T) \phi _2(S) \phi _1(T)) \quad \text {for all } T, S \in \mathcal {A}. \) Furthermore, if \(\mathcal {X}=\mathcal {H}\) and \(\mathcal {Y}=\mathcal {K}\) are two complex Hilbert spaces, we characterize maps \(\phi _{1}, \phi _{2}:\mathcal {A}\rightarrow \mathcal {B}\) whose ranges contain all operators of rank at most two and satisfy \( \sigma _{\pi }(TS^{*}T) = \sigma _{\pi }(\phi _1(T) \phi _2(S)^{*} \phi _1(T)) \quad \text {for all } T, S \in \mathcal {A}. \) We note that several known results follow as immediate consequences.
