= Toeplitz index theorem for continuous symbols
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{title2=$\operatorname{ind}T_f=-\operatorname{wind}(f)$}
On the <Hardy space of the circle>, a continuous symbol gives a <Fredholm> Toeplitz operator exactly when it never vanishes. Nonvanishing gives a parametrix $T_{1/f}$ modulo compact semicommutators. Write $f(z)=z^ne^{h(z)}$, where $n$ is its counterclockwise <winding number> and $h$ is continuous. Homotopy through nonvanishing symbols and <local constancy of the Fredholm index> reduce the index to that of the shift $T_{z^n}$, namely $-n$. If $f$ vanishes at a point, normalized Hardy reproducing kernels concentrating there are weakly null unit vectors on which $T_f$ tends to zero, contradicting a Fredholm parametrix.
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