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The intersection of essential approximate point spectra of operator matrices. (English) Zbl 1108.47013

Suppose that \({\mathcal K}\) and \({\mathcal H}\) are separable Hilbert spaces. Let \(B({\mathcal K},{\mathcal H})\), \(B_l({\mathcal K},{\mathcal H})\), and \(\text{Inv}({\mathcal K},{\mathcal H})\) be the sets of all bounded linear operators, left invertible bounded linear operators, and invertible bounded linear operators from \({\mathcal K}\) to \({\mathcal H}\), respectively. Let \(\Phi_+\) be the set of all upper semi-Fredholm operators. Consider the sets \(\Phi_+^-=\{T\in\Phi_+:\text{ind}(T)\leq 0\}\) and \(\sigma_{ea}(T)=\{\lambda\in{\mathbb C}:T-\lambda\notin\Phi_+^-\}\). Suppose that \(A\in B({\mathcal H})\) and \(B\in B({\mathcal K})\) are fixed. For \(C\in B({\mathcal K},{\mathcal H})\), \(M_C=\left(\begin{smallmatrix} A & C \\ 0 & B \end{smallmatrix}\right)\in B({\mathcal H}\oplus{\mathcal K})\). In the present paper, the sets \(\bigcap_{C\in B_l({\mathcal K},{\mathcal H})}\sigma_{ea}(M_C)\), \(\bigcap_{C\in \text{Inv}({\mathcal K},{\mathcal H})}\sigma_{ea}(M_C)\), and \(\bigcup_{C\in \text{Inv}({\mathcal K},{\mathcal H})}\sigma_{ea}(M_C)\) are characterized. In particular,
\[ \bigcap_{C\in\text{Inv}({\mathcal K},{\mathcal H})} \sigma_{ea}(M_C) = \bigcap_{C\in B({\mathcal K},{\mathcal H})}\sigma_{ea}(M_C) \cup \{\lambda\in{\mathbb C}: B-\lambda\;\text{is compact}\}. \]

MSC:

47A53 (Semi-) Fredholm operators; index theories
47A10 Spectrum, resolvent
47A55 Perturbation theory of linear operators
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