无穷维Hamilton算子纯虚谱的扰动(英文)

Perturbations of Pure Imaginary Spectra of Infinite Dimensional Hamiltonian Operators

研究了Hilbert空间XX中的无穷维Hamilton算子HC=A C0-A*和HF=A FB-A*的纯虚谱的扰动,其中R(B)是闭的.给定算子A,B,证明了∩C∈S(X)σi(HC)=σiπ(A),∪C∈S(X)σi(HC)=σi(A),∩F∈S(X)σi(HF)=σiπ(APR(B)⊥),∪F∈S(X)σi(HF)=σi(APR(B)⊥),其中σi(T),σiπ(T),PM和S(X)分别表示T的纯虚谱,纯虚近似谱,全空间到M的正交投影和X中的所有自伴算子所成之集.

For infinite dimensional Hamiltonian operators HC=[AC0-A＊] and HF=[AFB-A＊] with closed R（B） in the Hilbert space X＋X, the perturbations of pure imaginary spectra are studied. If A,B are given,one sees that ∩C∈S（X）σi（HC）=σiπ（A）,∪C∈S（X）σi（HC）=σi（A）,∩F∈S（X）σi（HF）=σiπ（APR（B）⊥）,∪F∈S（X）σi（HF）=σi（APR（B）⊥）,where σi（T）,σiπ（T）,PM and S（X） denote the pure imaginary spectrum of T, the pure imaginary approximate point spectrum of T,the orthogonal projection （from X） onto M and the set of all self-adjoint operators （not necessarily bounded） in X,respectively.