Atkinson theorem ($T\text{ Fredholm}⟺[T]\in \mathcal{Q}(H{)}^{\times }$)

A bounded operator is Fredholm if and only if it has a bounded two-sided inverse modulo compact operators. For the forward implication invert its restriction from the orthogonal kernel complement to its closed range and extend by zero. Conversely a parametrix makes the identity compact on the kernel, forcing finite dimension. A failure of a lower bound on the kernel complement would yield unit vectors $x_n$ with $T{x}_n\to 0$; the compact error then forces a norm-convergent subsequence, whose nonzero limit lies both in the kernel and its complement. This proves closed range; applying the other parametrix identity to the adjoint gives finite cokernel.