## 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 $$M_ C= [\begin{smallmatrix} A & C\\ 0 &B\end{smallmatrix}]$$ on the Hilbert space $$H\oplus K$$. For given $$A$$ and $$B$$, we prove that $\bigcap_{C\in B(K, H)} \sigma(M_ C)= \sigma_ \pi(A)\cup \sigma_ \delta(B)\cup \{\lambda\in C: n(B- \lambda)\neq d(A- \lambda)\},$ where $$\sigma(T)$$, $$\sigma_ \pi(T)$$, $$\sigma_ \delta(T)$$, $$n(T)$$, and $$d(T)$$ denote the spectrum of $$T$$, approximation point spectrum, defect spectrum, nullity, and deficiency, respectively. Some related results are obtained.

### MSC:

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