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 $T-\lambda$ is not surjective, then $\lambda$ is an isolated point of the surjective spectrum $\sigma_{su}(T)$ of $T$ if and only if $X=H_0(T-\lambda)+K(T-\lambda)$, where $H_0(T)$ is the quasinilpotent part of $T$ and $K(T)$ is the analytic core for $T$. Moreover, we study the operators for which $\dim K(T) < \infty$. We show that for each of these operators $T$, there exists a finite set $E$ consisting of Riesz points for $T$ such that $0\in \sigma (T)\setminus E$ and $\sigma (T)\setminus E$ is connected, and derive some consequences.