On the Isolated Points of the Surjective Spectrum of a Bounded Operator

For a bounded operator T acting on a complex Banach space, we show that if T −λ is not surjective, then λ is an isolated point of the surjective spectrum σsu(T ) of T if and only if X = H0(T−λ)+K(T−λ), where H0(T ) is the quasinilpotent part of T and K(T ) is the analytic core for T . Moreover, we study the operators for which dimK(T ) < ∞. We show that for each of these operators T , there exists a finite set E consisting of Riesz points for T such that 0 ∈ σ(T ) \ E and σ(T ) \ E is connected, and derive some consequences.