×

Perturbations of the right and left spectra for operator matrices. (English) Zbl 1261.47010

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 \(\mathcal N(A,C)\leq{\text{dim }}\mathcal H_2\).

MSC:

47A10 Spectrum, resolvent
47A55 Perturbation theory of linear operators
PDFBibTeX XMLCite