包含广义逆的算子乘积的谱

给定三个算子A,B,C∈ B(H),其中算子B的值域R(B)是闭的,利用算子矩阵分块技巧给出了∪σ(AB(1))C)=C的充分必要条件,其中σ(D)是算子D ∈B(H)的B(1) ∈B{1}谱,B{1}={X∈B(H):B×B=B}.

Spectra of Operator Products Involving GeneralizedInverses

Let $\mathcal {B(H)}$ denote the set of all bounded linear operatorson a Hilbert space $\mathcal H$. In this note, for three operators$A$, $B$ and $C\in \mathcal {B(H)}$, if the range ${\mathcal R}(B)$of $B$ is closed, then the necessary and sufficient conditions suchthat $\bigcup\limits_{B^{(1)}\in B\{1\}}\sigma(AB^{(1)}C)=C$ is obtained by using block-operator matrix technique, where$\sigma(D)$ is the spectrum of an operator $D\in {\mathcal {B(H)}}$and$B\{1\}:=\{X\in {\mathcal {B(H)}}:BXB=B\}.$