Chen, Alatancang; Hai, Guojun Perturbations of the right and left spectra for operator matrices. (English) Zbl 1261.47010 J. Oper. Theory 67, No. 1, 207-214 (2012). 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 \(\mathcal H_1\otimes\mathcal H_2\) of the form \(M_{(X,Y)}=\left({\begin{matrix} A & C\\ X & Y\end{matrix}}\right)\), 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 \(\mathcal N(A,C)\leq{\text{dim }}\mathcal H_2\). Cited in 13 Documents MSC: 47A10 Spectrum, resolvent 47A55 Perturbation theory of linear operators Keywords:operator matrices; right (left) invertible operator; right (left) spectrum; perturbations of spectra PDFBibTeX XMLCite \textit{A. Chen} and \textit{G. Hai}, J. Oper. Theory 67, No. 1, 207--214 (2012; Zbl 1261.47010)