Jordan isomorphisms and maps preserving spectra of certain operator products
Volume 184 / 2008
Abstract
Let $\mathcal{A}_1, \mathcal{A}_2$ be (not necessarily unital or closed) standard operator algebras on locally convex spaces $X_1, X_2$, respectively. For $k \ge 2$, consider different products $T_1* \cdots *T_k$ on elements in ${\cal A}_i$, which covers the usual product $T_1* \cdots *T_k = T_1\cdots T_k$ and the Jordan triple product $T_1*T_2 = T_2T_1T_2$. Let ${\mit\Phi} :\mathcal{A}_1\rightarrow\mathcal{A}_2$ be a (not necessarily linear) map satisfying $\sigma({\mit\Phi}(A_1)*\cdots *{\mit\Phi}(A_k)) =\sigma(A_1*\cdots *A_k)$ whenever any one of $A_i$'s has rank at most one. It is shown that if the range of ${\mit\Phi}$ contains all rank one and rank two operators then ${\mit\Phi}$ must be a Jordan isomorphism multiplied by a root of unity. Similar results for self-adjoint operators acting on Hilbert spaces are obtained.
