Perturbation of spectrums of $2\times 2$ operator matrices

Abstract:

In this paper, we study the perturbation of spectrums of $2 \times 2$ operator matrices such as ${M_C} = \left [ {\begin {array}{*{20}{c}} A & C \\ 0 & B \\ \end {array} } \right ]$ on the Hilbert space $H \oplus K$. For given A and B, we prove that \[ \bigcap \limits _{C \in B(K,H)} {\sigma ({M_C}) = {\sigma _\pi }(A) \cup {\sigma _\delta }(B) \cup \{ \lambda \in C:n(B - \lambda ) \ne d(A - } \lambda )\} ,\] 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.