Let A1β and A2β be standard operator algebras on
complex Banach spaces X1β and X2β, respectively. For kβ₯2, let
(i1β,...,imβ) be a sequence with terms chosen from {1,β¦,k}, and
assume that at least one of the terms in (i1β,β¦,imβ) appears exactly
once. Define the generalized product T1ββT2βββ―βTkβ=Ti1ββTi2βββ―Timββ on elements in Aiβ. Let
Ξ¦:A1ββA2β be a map with the range containing
all operators of rank at most two. We show that Ξ¦ satisfies that
ΟΟβ(Ξ¦(A1β)ββ―βΞ¦(Akβ))=ΟΟβ(A1βββ―βAkβ) for all
A1β,β¦,Akβ, where ΟΟβ(A) stands for the peripheral spectrum of
A, if and only if Ξ¦ 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 X1β=H and X2β=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 Aβ¦cUAUβ or Aβ¦cUAtUβ, where UβB(H,K) is a unitary
operator, cβ{1,β1}.Comment: 17 page