3×3阶上三角算子矩阵的点谱、剩余谱和连续谱

The Point,Residual and Continuous Spectrum of 3×3 Upper Triangular Operator Matrices

记MD,E,F=[ADE OBE OOC]为Hilbert空间H1■H2■H3上的上三角算子矩阵.我们借助对角元A,B和C的谱性质给出了σ*(MD,E,F)=σ*(A)∪σ*(B)∪σ*(C)对任意D∈B(H2,H1),E∈B(H3,H1),F∈B(H3,H2)均成立的充要条件,其中σ*代表某类特定的谱,如点谱、剩余谱和连续谱等.此外,给出了一些例证.

Let MD,E,F=[ADE OBE OOC]be an upper triangular operator matrix on the Hilbert space H1⊕H2⊕H3.In this paper,we obtain necessary and sufficient conditions ofσ*(MD,E,F)=σ*(A)∪σ*(B)∪σ*(C)for every D∈B(H2,H1),E∈B(H3,H1)and F∈B(H3,H2),in terms of the spectral properties of diagonal entries A,B and C in MD,E,F,whereσ*is the point spectrum,the residual spectrum and the continuous spectrum.Moreover,we construct some examples illustrating our main results.