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The intersection of left (right) spectra of \(2 \times 2\) upper triangular operator matrices. (English) Zbl 1105.47004

Let \(H, K\) be two Hilbert spaces, \(B(H)\) and \(B(H,K)\) be the set of bounded operators on \(H\) and from \(H\) to \(K\), respectively. In the paper under review, the sets \(\bigcup_{C\in B(H, K)}\sigma_l(M_C)\) and \(\bigcup_{C\in \text{Inv}(H, K)}\sigma_l(M_C)\) are compared. Here \(\text{Inv}(H,K)\) stands for the set of invertible operators from \(H\) to \(K\), \(\sigma_l(.)\) denotes the left spectrum and \(M_C \in B(H\oplus K)\) an upper triangular operator matrix with \(A\in B(H), B\in B(K)\) the entries on the diagonal and \(C \in B(H,K)\) the remaining non-vanishing entry.
The main result is \(\bigcup_{C\in B(H, K)}\sigma_l(M_C)= \bigcup_{C\in \text{Inv}(H, K)}\sigma_l(M_C)\cup \{\lambda \in \mathbb C : B-\lambda \: \text{ is compact}\}\). To get this result, one has to assume that both \(H\) and \(K\) are infinite-dimensional (otherwise the right term is always the whole complex plane and the left one is a subset of \(\sigma(A)\cup \sigma(B)\)). Moreover, in the infinite-dimensional case, it is not difficult to see that \(\{\lambda \in \mathbb C : B-\lambda \text{ is compact}\}\) is at most a singleton.

MSC:

47A10 Spectrum, resolvent
47A55 Perturbation theory of linear operators
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References:

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