Toeplitz index theorem

Remarks

Idea of the proof: for continuous $\varphi$ and $\psi$, the operator $T_\varphi T_\psi - T_{\varphi\psi}$ is compact. So $T_{1/\varphi}$ is an inverse of $T_\varphi$ modulo compact operators, and $T_\varphi$ is Fredholm by Atkinson's theorem. The index is constant along homotopies of nonvanishing symbols, and $\varphi$ is homotopic to $z^n$ with $n = \operatorname{wind}(\varphi, 0)$. For $n \ge 0$, $T_{z^n}$ is the $n$-th power of the shift, which is injective with cokernel of dimension $n$, so its index is $-n$; for $n < 0$ one takes adjoints. The hypothesis that $\varphi$ has no zeros is necessary: for continuous $\varphi$, the operator $T_\varphi$ is Fredholm if and only if $\varphi$ has no zeros on $\mathbb{T}$.

Going deeper

For a student

The theorem is the one-dimensional model of the Atiyah–Singer index theorem: an analytic index equals a topological invariant of the symbol. It is also a basic example of index theory for $C^*$-algebras, through the Toeplitz extension $0 \to \mathcal{K} \to \mathcal{T} \to C(\mathbb{T}) \to 0$, whose index map $K_1(C(\mathbb{T})) \to K_0(\mathcal{K}) = \Z$ is the winding number up to a sign convention. Higher-dimensional versions for Toeplitz operators on strictly pseudoconvex domains are due to Boutet de Monvel.