Projections and isolated points of parts of the spectrum

In this paper, we relate the existence of certain projections, commuting with a bounded linear operator T ∈ L(X) acting on Banach space X, with the generalized Kato decomposition of T. We also relate the existence of these projections with some properties of the quasi-nilpotent part H0(T) and the analytic core K(T). Further results are given for the isolated points of some parts of the spectrum.