广义特征值的稳定性及刻画

The characterization and stability of extended eigenvalue

设A是复Hilbert空间H上的有界线性算子.λ∈C,如果存在H上的非零有界线性算子B使得AB=λBA,那么就称λ是A的一个广义特征值.记A的全体广义特征值所构成的集合为Σ(A).利用算子分块的技巧,讨论了上三角算子矩阵的广义特征值的稳定性问题.此外,对H上的正可逆算子A,证得Σ(An/m)=(Σ(A))n/m,其中n,m∈Z,并且m≠0.

Let A be a bounded linear operator on a complex Hilbert space X.λ∈C is called an extended eigenvalue of A if there exists a nonzero operator B on X such that AB=λBA. The set of all extended eigenvalues of A is denoted by ∑（A）. By using the technique of operator block, the extended eigenvalue of upper triangular operator matrices was studied. And more, it is showed that if A≥0 and A is invertible, then ∑（An/m）= （∑（A））n/m, where n,m∈Z, and m≠0.