nLab Fredholm index

Idea

A Fredholm operator $F \,\colon\, \mathcal{H} \longrightarrow \mathcal{H}$ by definition has finite-dimensional kernel $ker(F)$ and cokernel $coker(F)$. Its index is the difference between these dimensions:

$$ind(F) \;\coloneqq\; dim\big(ker(F)\big) \,-\, dim\big(coker(F)\big) \,.$$

Examples

• charged vacua of free Dirac field in Coulomb background

Related concepts

• index

• index of a Dirac operator

References

Textbook account:

• William Arveson, Section 3.4 of: A Short Course on Spectral Theory, Graduate Texts in Mathematics 209 $[$doi:10.1007/b97227$]$