保Taylor联合谱的线性映射

设L(H),Lncom(H)分别是HilbertH上有界算子及n个两两交换的算子组的集合.设T∈Lncom(H),sp(T)表示Taylor联合谱,φi(i=1,2,…,n)是L(H)上满的线性映射且满足φi(Tl)φj(Tk)=φj(Tk)φi(Tl)当且仅当TlTk=TkTl,i,j=1,2,…,n.设T=(T1,T2,…,Tn)∈Lncom(H),φ=(φ1,φ2,…,φn),φ(T)=(φ1(T1),φ2(T2),…,φn(Tn)).文章证明了如果dimH<∞,对任意T=(T1,T2,…Tn)∈Lncom(H),sp(φ(T))=sp(T),则φi=φj,i,j=1,2,…,n.如果dimH=∞,T=(T1,T2,…Tn)∈Lncom(H),sp(φ(T))=sp(T),则φ是自同构或反自同构.

The Linear Maps Preserving the Taylor Joint Spectrum

Let L(H), Lncom(H) be the set of bounded linear operators and n-tuples of commutingoperators on Hilbert space H. For T ∈ Lncom(H), let sp (T)denote the Taylor joint spectrum, and ψi(i = 1, 2, ..., n) be the surjective linear map on L(H) and satisfy ψi(Tl)ψj(Tk) = ψi(Tk)ψi(Tl) if and only ifTlTk = TkTl, i, j= 1, 2, …, n. LetT= (Ti, T2, …, Tn) ∈ Lncom(H) and ψ= (ψ1,ψ2, …,ψn), ψ(T) = (ψ.(T1), ψ2(T2), …, ψn(Tn)). In this paper, we prove that ifdimH＜ ∞, sp(ψ(T)) =sp(T) for allT ∈ Lncom(H), then ψi = ψj. FordimH = ∞, ifψi = ψ, (i = 1, 2, …, n), sp(ψ(T)) =sp(T) for all T ∈ Lncom(H), we prove that ψis either an isomorphism or an anti-isomorphism.