The authors investigate perturbations of the spectrum of a general upper triangular operator matrix acting on the Hilbert space , where , and .
X. H. Cao, M. Z. Guo and B. Meng [Acta Math. Sin., Engl. Ser. 22, No. 1, 169–178 (2006; Zbl 1129.47014)] gave a necessary and sufficient condition for 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 . X. H. Cao and B. Meng [J. Math. Anal. Appl. 304, No. 2, 759–771 (2005; Zbl 1083.47006)] obtained a necessary and sufficient condition for 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 . I. S. Hwang and W. Y. Lee [Integral Equations Oper. Theory 39, No. 3, 267–276 (2001; Zbl 0986.47004)] provided a necessary and sufficient condition for to be a left invertible operator and characterized the intersection of the left spectrum, the right spectrum and the spectrum of .
The authors of the present paper extend all the results mentioned above by the same techniques. Some counterexamples are presented as well.