Almost Invertible Operators

This paper aims to provide a thorough characterization of the family of all Cantor-Bendixson derivatives of the spectrum, Browder spectrum, and the Drazin spectrum of bounded linear operators using projections and invariant subspaces. Furthermore, our findings demonstrate that if two commuting operators, R and T , satisfy the conditions that R is Riesz and T is a direct sum of an invertible operator and an operator with an at most countable spectrum, then T + R can also be represented as a direct sum of an invertible operator and an operator with an at most countable spectrum.