I have seen the Gohberg-Krein theorem in the case where the symbol is just a complex-valued mapping. Now there is an obvious generalization of the theorem:
(Gohberg-Krein). Let $\varphi \in M_N(C(\mathbf{T}))$ and let $T_{\varphi}$ be the corresponding Toeplitz operator on $H^2 \otimes \mathbf{C}^N$, where $H^2$ is the Hardy space of the circle. Then $T_{\varphi}$ is a Fredholm operator if, and only if, $\operatorname{det} \varphi$ never vanishes on $\mathbf{T}$ and in this case $$ \operatorname{index}\left(T_{\varphi}\right)=-\operatorname{wn}(\operatorname{det} \varphi) $$ where wn denotes the winding number around the origin.
I was wondering if anyone has a reference for this proof or if there is a simple proof using algebraic topological techniques.