Almost invertible operators

Abstract

We prove that a bounded linear operator $T$ is a direct sum of an invertible operator and an operator with at most countable spectrum iff $0\notin\mbox{acc}^{ω_{1}}\,σ(T),$ where ${\omega }_1$ is the smallest uncountable ordinal and $\mbox{acc}^{ω_{1}}\,σ(T)$ is the ${\omega }_1$-th Cantor-Bendixson derivative of $\sigma (T).$