= Fredholm index
{c}
{title2=$\operatorname{ind}T=\dim\ker T-\dim\operatorname{coker}T$}
For a <Fredholm operator> between <Hilbert spaces>, the index is the difference of the finite <dimensions> of its <kernel> and <cokernel>. Since the range is closed, its <cokernel> identifies with $\ker T^*$ by <orthogonal projection>. An invertible operator has index zero. Conversely, an index-zero <Fredholm operator> becomes invertible after adding a <finite-rank operator>: choose an isomorphism $J:\ker T\to\ker T^*$, extend it by zero on $(\ker T)^\perp$, and observe that $T+J$ maps the two summands bijectively onto $\operatorname{ran}T\oplus\ker T^*$.
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