Local spectra and index of singular integral operators with piecewise continuous coefficients on composed curves

We establish a symbol calculus for deciding whether singular integral operators with piecewise continuous coefficients are Fredholm on the Lebesgue space L^p(Γ, w) where 1 < p < ∞, Γ is a composed Carleson curve, and w is a Muckenhoupt weight in the class A_p(Γ). We also provide index formulas for the operators in the closed algebra of singular integral operators with piecewise continuous matrix-valued coefficients. Our main theorem is based upon three pillars: on the identification of the local spectrum of the Cauchy singular integral operator at the endpoints of simple Carleson arcs, on an appropriate "N projections theorem", and on results of geometric function theory pertaining to the problem of extending Carleson curves and Muckenhoupt weights.