One more proof of the index formula for block Toeplitz operators (Q2835255)

Let \(S^1 = \mathbb{R} / \mathbb{Z}\) and \(g : S^1 \to \mathcal{M}_n(\mathbb{C})\) be a continuous function. It is known that the Toeplitz operator \(T(g)\) generated by the matrix function \(g\) is a bounded linear Fredholm operator if and only if \(\det g(t) \neq 0\) for \(t \in S^1\) and, in this case, the following index formula for \(T(g)\) holds true: NEWLINE\[NEWLINE \operatorname{Ind}T(g) = - W (\det g), NEWLINE\]NEWLINE where \(W(\det g)\) is the winding number of \(\det g\) about the origin (see [\textit{I. C. Gohberg} and \textit{M. G. Krein}, Usp. Mat. Nauk 12, No. 2(74), 43--118 (1957; Zbl 0088.32101)]). The authors present a new, ``direct algebraic and non-topological\("\) proof of this formula.