Some spectral properties of linear operators on exotic Banach spaces

Abstract

In this work, we present some results concerning the operators defined on various classes of exotic Banach spaces, containing in particular those studied respectively by V. Ferenczi [7,8] and T. Gowers with B. Maurey [14,15]. We show that, on hereditarily indecomposable or quotient hereditarily indecomposable Banach space X, the set of bounded Fredholm operators is dense in L(X), this gives that the boundary of bounded Fredholm operators is nothing else but the ideal of strictly singular operators if X is hereditarily indecomposable Banach space (resp. the ideal of strictly cosingular operators if X is quotient hereditarily indecomposable Banach space). On the other hand, a comparison between sufficiently rich and exotic Banach spaces is given via some properties of the two maps spectra and Wolf essential spectra.