On the equivalence of Browder’s and generalized Browder’s theorem. (English) Zbl 1097.47012

Amouch, M.; Zguitti, H.

Glasg. Math. J. 48, No. 1, 179-185 (2006).

M. Berkani and J. J. Koliha [Acta Sci. Math. 69, No. 1–2, 359–376 (2003; Zbl 1050.47014)] proved that the generalized Browder’s (resp., the generalized \(a\)-Browder’s) theorem implies Browder’s (resp., \(a\)-Browder’s) theorem for a Banach space operator. In the present paper, the authors show that the generalized Browder’s (resp., the generalized \(a\)-Browder’s) theorem holds for a Banach space operator if and only if Browder’s (resp., \(a\)-Browder’s) theorem does. They also give conditions under which the generalized Weyl’s (resp., generalized \(a\)-Weyl’s) theorem is equivalent to Weyl’s (resp., \(a\)-Weyl’s) theorem.

Reviewer: Yufeng Lu (Dalian)

MSC:

47A53	(Semi-) Fredholm operators; index theories
47A10	Spectrum, resolvent
47A11	Local spectral properties of linear operators

Keywords:

Fredholm operator; Browder’s theorem; generalized Browder’s theorem