Atkinson's theorem states: In other words, an operator T ∈ L(H) is Fredholm, in the classical sense, if and only if its projection in the Calkin algebra is invertible.
For the ⇒ implication, express H as the orthogonal direct sum The restriction T : Ker(T)⊥ → Ran(T) is a bijection, and therefore invertible by the open mapping theorem.
So Ker(T) is contained in an eigenspace of C2, which is finite-dimensional (see spectral theory of compact operators).
A more complete treatment of Atkinson's Theorem is in the reference by Arveson: it shows that if B is a Banach space, an operator is Fredholm iff it is invertible modulo a finite rank operator (and that the latter is equivalent to being invertible modulo a compact operator, which is significant in view of Enflo's example of a separable, reflexive Banach space with compact operators that are not norm-limits of finite rank operators).
Note that the hypothesis that Ran(T) is closed is redundant since a space of finite codimension that is also the range of a bounded operator is always closed (see Arveson reference below); this is a consequence of the open-mapping theorem (and is not true if the space is not the range of a bounded operator, for example the kernel of a discontinuous linear functional).