Some characterization of Atkinson & Fredholm operators

In this paper, we study the Atkinson operator and Fredholm operator. We showed that every Atkinson operator with a nonpositive index could be represented as a finite-dimensional perturbation of a left-invertible Atkinson operator. Similar theorems can then be proved for Atkinson operators with non-negative index and Fredholm operators with zero and arbitrary index. With the help of these theorems, we gave a unique characterization of the index except for a rational factor.