Journal of Operator Theory

Volume 67, Issue 1, Winter 2012 pp. 207-214.

Perturbations of the right and left spectra for operator matrices

Authors: Alatancang Chen (1) and Guojun Hai (2)
Author institution: (1) School of Mathematical Sciences, Inner Mongolia University, Hohhot, 010021, P.R. China
(2) School of Mathematical Sciences, Inner Mongolia University, Hohhot, 010021, P.R. China

Summary: Let ${\mathcal{H}}_{1}$ and ${\mathcal{H}}_{2}$ be separable Hilbert spaces. For given $A\in{\mathcal{B}}({\mathcal{H}}_{1})$ and $C\in {\mathcal{B}}({\mathcal{H}}_{2}, {\mathcal{H}}_{1})$, $M_{(X,Y)}$ denotes an operator acting on ${\mathcal{H}}_{1}\oplus{\mathcal{H}}_{2}$ of the form $M_{(X,Y)}=\left( \begin{smallmatrix} A & C \\ X & Y \\ \end{smallmatrix}\right)$, where $X\in\mathcal{B}(\mathcal{H}_{1}, \mathcal{H}_{2})$ and $Y\in\mathcal{B}(\mathcal{H}_{2})$. In this paper, a necessary and sufficient condition is given for $M_{(X,Y)}$ to be right invertible for some $X\in {\mathcal{B}}({\mathcal{H}}_{1}, {\mathcal{H}}_{2})$ and $Y\in{\mathcal{B}}({\mathcal{H}}_{2})$. In addition, it is shown that if $\dim\mathcal{H}_{2}=\infty$ then $M_{(X,Y)}$ is left invertible for some $X\in {\mathcal{B}}({\mathcal{H}}_{1}, {\mathcal{H}}_{2})$ and $Y\in{\mathcal{B}}({\mathcal{H}}_{2})$; if $\dim\mathcal{H}_{2}<\infty$ then $M_{(X,Y)}$ is left invertible for some $X\in {\mathcal{B}}({\mathcal{H}}_{1}, {\mathcal{H}}_{2})$ and $Y\in{\mathcal{B}}({\mathcal{H}}_{2})$ if and only if ${\mathcal{R}}(A)$ is closed and $\dim\mathcal{N}(A,C)\leqslant \dim{\mathcal{H}}_{2}$.

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