Characterizations of perturbations of spectra of \(2 \times 2\) upper triangular operator matrices (Q428368)

The authors investigate perturbations of the spectrum of a general \(2\times 2\) upper triangular operator matrix \(M_C=\left(\begin{smallmatrix} A&C\\ 0&B\end{smallmatrix}\right)\) acting on the Hilbert space \(\mathcal{H}\oplus \mathcal{K}\), where \(A\in B(\mathcal{H})\), \(B\in B(\mathcal{K})\) and \(C\in B(\mathcal{K},\mathcal{H})\).NEWLINENEWLINE\textit{X. H. Cao}, \textit{M. Z. Guo} and \textit{B. Meng} [Acta Math. Sin., Engl. Ser. 22, No. 1, 169--178 (2006; Zbl 1129.47014)] gave a necessary and sufficient condition for \(M_C\) to be an upper semi-Fredholm operator (resp., a lower semi-Fredholm operator, a Fredholm operator) and characterized the intersection of the upper semi-Fredholm spectrum and the lower semi-Fredholm spectrum of \(M_C\). \textit{X. H. Cao} and \textit{B. Meng} [J. Math. Anal. Appl. 304, No. 2, 759--771 (2005; Zbl 1083.47006)] obtained a necessary and sufficient condition for \(M_C\) to be an upper semi-Weyl operator (resp., a lower semi-Weyl operator, a Weyl operator) and characterized the intersection of the upper semi-Weyl spectrum, the lower semi-Weyl spectrum and the Weyl spectrum of \(M_C\). \textit{I. S. Hwang} and \textit{W. Y. Lee} [Integral Equations Oper. Theory 39, No. 3, 267--276 (2001; Zbl 0986.47004)] provided a necessary and sufficient condition for \(M_C\) to be a left invertible operator and characterized the intersection of the left spectrum, the right spectrum and the spectrum of \(M_C\).NEWLINENEWLINEThe authors of the present paper extend all the results mentioned above by the same techniques. Some counterexamples are presented as well.