Actually, Toeplitz operators are pseudodifferential operators of order 0. The Atiyah-Singer index theorem can be formulated in sufficient generality that the Toeplitz index theorem is a special case, but it is likely that such a formulation would be proved by reducing everything to the Toeplitz index theorem. It is a general principle in the area that one should use general theory to reduce an index theorem to a single easily calculated example, and the Toeplitz index theorem is a convenient choice of example because it is probably the most elementary index calculation.
One way to see this is to express the index of an elliptic operator as a pairing between a K-theory class and a K-homology class. Using the Kasparov product and other K-homological tools, the index theorem can effectively be reduced to calculating the index pairing between the Bott class in the K-theory group $K^0(D)$, where $D$ is the open unit disk, and the fundamental class in the K-homology group $K_0(D)$ (i.e. the class of the Dirac operator). The Bott class is the imagine under the boundary map $K^1(S^1) \to K^0(D)$ of the class of the unitary $z \mapsto \overline{z}$ on $S^1$, and it turns out that the boundary of the fundamental class of $D$ is the class of the Toeplitz extension. So by the compatibility of the index pairing with boundary maps, the pairing of the Bott class with the fundamental class is the index of the Toeplitz operator $T_{\overline{z}}$, and this is just 1.
You also ask about (other) generalizations of the Toeplitz index theorem. There are a great many results which follow the following basic pattern:
- Begin with a space $X$ equipped with a nice compactification $\overline{X}$.
- Choose a nice family of functions on $X$ (often the kernel of some sort of Laplacian), and form a "Hardy space" $H$ by taking the closure in $L^2(\partial X)$ over their boundary values.
- Given a function $f \in C(\partial X)$, form the operator $T_f = PfP$ where $P: L^2(\partial X) \to H$ is orthogonal projection. $T_f$ is Fredholm in nice cases, and its index is often an interesting invariant.
I'm suppressing a subtlety, which is that it is often necessary to look at matrix valued functions rather than just ordinary functions in order to get something interesting.
For example, one can take holomoprhic functions on strongly pseudoconvex domains in $\mathbb{C}^n$ to obtain a direct generalization of the usual Toeplitz index theorem. Alternatively, one can take the kernel of the Dirac operator on an appropriately chosen Riemannian manifold with boundary.
A good reference for everything I have said is the book "Analytic K-homology" by Higson and Roe - chapters 2 and 11 are particularly relevant.
