Index of a singular integral operator

Definition

Let $\Gamma$ be a smooth simple closed curve in $\C$, oriented counterclockwise. Let $a$ and $b$ be Hölder continuous functions on $\Gamma$ with $a(t)^2 - b(t)^2 \ne 0$ for all $t \in \Gamma$, and let $k$ be a Hölder continuous function on $\Gamma \times \Gamma$. Consider the singular integral operator of a singular integral equation of normal type $$K \varphi(t) = a(t) \, \varphi(t) + \frac{b(t)}{\pi i} \, \mathrm{p.v.} \int_\Gamma \frac{\varphi(\tau)}{\tau - t} \, d\tau + \int_\Gamma k(t, \tau) \, \varphi(\tau) \, d\tau, \quad t \in \Gamma.$$ The index of $K$ is the integer $$\kappa = \frac{1}{2\pi} \Big[ \arg \frac{a(t) - b(t)}{a(t) + b(t)} \Big]_\Gamma.$$ Here the bracket is the change of a continuous branch of the argument of $(a - b)/(a + b)$ after one run of $t$ around $\Gamma$. In other words, $\kappa$ is the winding number around $0$ of the closed curve $t \mapsto (a(t) - b(t))/(a(t) + b(t))$.