Maps preserving peripheral spectrum of generalized products of operators

Abstract

Let A1 and A2 be standard operator algebras on complex Banach spaces X1 and X2, respectively. For k⩾2, let (i1,…,im) be a sequence with terms chosen from {1,…,k}, and assume that at least one of the terms in (i1,…,im) appears exactly once. Define the generalized product T1⁎T2⁎⋯⁎Tk=Ti1Ti2⋯Tim on elements in Ai. Let Φ:A1→A2 be a map with the range containing all operators of rank at most two. We show that Φ satisfies that σπ(Φ(A1)⁎⋯⁎Φ(Ak))=σπ(A1⁎⋯⁎Ak) for all A1,…,Ak, where σπ(A) stands for the peripheral spectrum of A, if and only if Φ is an isomorphism or an anti-isomorphism multiplied by an mth root of unity, and the latter case occurs only if the generalized product is quasi-semi Jordan. If X1=H and X2=K are complex Hilbert spaces, we characterize also maps preserving the peripheral spectrum of the skew generalized products, and prove that such maps are of the form A↦cUAU⁎ or A↦cUAtU⁎, where U∈B(H,K) is a unitary operator, c∈{1,−1}.