Maps preserving peripheral spectrum of generalized products of operators

Let A(1) and A(2) be standard operator algebras on complex Banach spaces X-1 and X-2, respectively. For k >= 2, let (i(1), . . . , i(m)) be a sequence with terms chosen from {1, . . . , k}, and assume that at least one of the terms in (i(1), . . . , i(m)) appears exactly once. Define the generalized product T-1*T-2* . . . *T-k = Ti1Ti2 . . .T-im on elements in A(i). Let Phi: A(1) --> A(2) be a map with the range containing all operators of rank at most two. We show that Phi satisfies that sigma(pi)(Phi(A(1)))* . . . * Phi(A(k))) = sigma(pi)(A(1)* . . . *A(k)) for all A(1), . . . , A(k), where sigma(pi)(A) stands for the peripheral spectrum of A, if and only if Phi 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 X-1 = H and X-2 = 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 bar right arrow cU AU* or A bar right arrow cU A(t)U*, where U is an element of B(H, K) is a unitary operator, c is an element of {1, -1}. (C) 2014 Elsevier Inc. All rights reserved.