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Maps preserving peripheral spectrum of generalized products of operators
Let and be standard operator algebras on
complex Banach spaces and , respectively. For , let
be a sequence with terms chosen from , and
assume that at least one of the terms in appears exactly
once. Define the generalized product on elements in . Let
be a map with the range containing
all operators of rank at most two. We show that satisfies that
for all
, where stands for the peripheral spectrum of
, if and only if is an isomorphism or an anti-isomorphism multiplied
by an th root of unity, and the latter case occurs only if the generalized
product is quasi-semi Jordan. If and 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 or , where is a unitary
operator, .Comment: 17 page
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