Abstract:

Let$A$,$B$be densely closed linear operators in a separable Hilbert space$X$and$M_{0}=\left( \begin{array} {cc}{A} & {0}\\ {0}& {B} \end{array} \right)$be the corresponding$2\times2$block operator matrices. In this paper, we establish the fine pseudo-spectra of$M_{0}$including the pseudo-point spectrum, the pseudo-residual spectrum, and the pseudo-continuous spectrum under diagonal perturbation, which are, respectively, compared with its point spectrum, residual spectrum, and continuous spectrum. And a concrete example is constructed to justify the proved result. Finally, we obtain the pseudo-point spectrum of$M_{0}$under the upper-triangular perturbation by using the technology of space decomposition.

Key words: Operator matrices, Pseudo-point spectrum, Pseudo-residual spectrum, Pseudo-continuous spectrum, Pseudo-null space