Maps preserving peripheral spectrum of generalized products of operators

2013/05/30 by Wen Zhang, Zhang, Wen, Jinchuan Hou +1
Mathematics · #47A12 #47B49 #47L10 #FOS: Mathematics #Functional Analysis (math.FA) #Operator Algebras (math.OA) #math.FA #math.OA #msc:47A12 #msc:47B49 #msc:47L10

paper · pdf · doi:10.48550/arxiv.1305.7100

17 pages

arxiv created 2013/05/30 · arxiv updated 2013/05/31

Abstract

Let A₁ and A₂ be standard operator algebras on complex Banach spaces X₁ and X₂, respectively. For k≥2, let (i₁,...,i_m) be a sequence with terms chosen from \1,…,k\, and assume that at least one of the terms in (i₁,…,i_m) appears exactly once. Define the generalized product T₁* T₂*⋯* T_k=T_i1T_i2⋯ T_im on elements in A_i. Let Φ:A₁\rightarrowA₂ be a map with the range containing all operators of rank at most two. We show that Φ satisfies that σ_π(Φ(A₁)*⋯*Φ(A_k))=σ_π(A₁*⋯* A_k) for all A₁,…, A_k, where σ_π(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↦ cUA^tU^*, where U\inB(H,K) is a unitary operator, c∈\1,-1\.

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