Perturbation of spectrums of \(2\times 2\) operator matrices. (English) Zbl 0814.47016

Du, Hongke; Pan, Jin

Proc. Am. Math. Soc. 121, No. 3, 761-766 (1994).

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

Keywords:

perturbation of spectra; \(2\times 2\) operator matrices; approximation point spectrum; defect spectrum; nullity; deficiency