Let Formula: see text and Formula: see text be Banach spaces over the field Formula: see text and let Formula: see text Formula: see text and Formula: see text In this paper, we investigate some properties of the generalized Fredholm spectrum of the operator matrix Formula: see text where Formula: see text is defined on Formula: see text by Formula: see text Namely, i) We give sufficient conditions under which Formula: see text is generalized Fredholm for every Formula: see text In particular, we prove that Formula: see text when Formula: see text is quasi-nilpotenet and surjective or Formula: see text is quasi-nilpotent and injective. ii) We establish that Formula: see text iii) We prove that if Formula: see text is right invertible, Formula: see text is left invertible and Formula: see text is generalized Fredholm for some Formula: see text then Formula: see text is generalized Fredholm if and only if Formula: see text is generalized Fredholm. iv) At the end of this paper, we provide sufficient conditions for which the inclusion in ii) becomes an equality. Where Formula: see text stands for the generalized Fredholm spectrum.
Amrani et al. (Fri,) studied this question.