On the generalized Fredholm spectrum of upper triangular operator matrices

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-nilpotent 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.