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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