For a continuous nonvanishing complex-valued function $g$ on the real line, several notions of a mean winding number are introduced. We give necessary conditions for a Toeplitz operator with matrix-valued symbol $G$ to be semi-Fredholm in terms of mean winding numbers of $\det G$. The matrix function $G$ is assumed to be continuous on the real line, and no other apriori assumptions on it are made.
"Vector semi-Fredholm Toeplitz operators and mean winding numbers." Nagoya Math. J. 195 57 - 75, 2009.