  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
$0\notin\mbox{acc}^{\omega_{1}}\,\sigma(T),$ where $\omega_{1}$ is the smallest
uncountable ordinal and $\mbox{acc}^{\omega_{1}}\,\sigma(T)$ is the
$\omega_{1}$-th Cantor-Bendixson derivative of $\sigma(T).$

Spectral Theory;Functional Analysis