Maps Preserving Peripheral Spectrum Of Generalized Products Of Operators

Let A and B be standard operator algebras on complex Banach spaces X and Y, respectively. Denote byσ(T), r(T) andσπ(T)={λ∈σ(T)||λ|= r(T)} the spectrum, the spectral radius and the peripheral spectrum of operator T. For k≥2, let (i1,...,im) be a sequence with terms chosen from{1,..., k} such that at least one of the terms in (i1,..., im) appears exactly once, and define the generalized product T1(?)T2(?)…(?)Tk= Ti1Ti2…Tim on elements in A or B. LetΦ:A→B be a map with the range containing all operators of rank at most two. It is shown in this paper thatΦsatisfiesσπ(Φ(A1) (?)…(?)Φ(Ak))=σπ{A1(?)…(?) Ak) for all A1,A2,..., Ak∈A if and only ifΦis a Jordan isomorphism multiplied by an scalarλ∈C withλm= 1. If X= H and Y= K are complex Hilbert spaces, the maps preserving the peripheral spectrum of the skew generalized products are also characterized. It is shown that such maps are of the form A (?) UAU* or A (?) UAtU*, where U∈B(H, K) is a unitary operator and At denotes the transpose of A in an arbitrary but fixed orthonormal basis of H. Furthermore, the same conclusion holds if A and B are replaced by standard real Jordan algebras of self-adjoint operators on complex Hilbert spaces H and K, respectively.