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Perturbation of spectrums of $2\times 2$ operator matrices. (English) Zbl 0814.47016
Summary: We study the perturbation of spectrums of $2\times 2$ operator matrices such as $M\sb C= [\smallmatrix A & C\\ 0 &B\endsmallmatrix]$ on the Hilbert space $H\oplus K$. For given $A$ and $B$, we prove that $$\bigcap\sb{C\in B(K, H)} \sigma(M\sb C)= \sigma\sb \pi(A)\cup \sigma\sb \delta(B)\cup \{\lambda\in C: n(B- \lambda)\ne d(A- \lambda)\},$$ where $\sigma(T)$, $\sigma\sb \pi(T)$, $\sigma\sb \delta(T)$, $n(T)$, and $d(T)$ denote the spectrum of $T$, approximation point spectrum, defect spectrum, nullity, and deficiency, respectively. Some related results are obtained.

47A55Perturbation theory of linear operators
47A10Spectrum and resolvent of linear operators
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