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 opera tors with piecewise continuous coefficients are Fredholm on the Lebesgue sp ace LP(Gamma, w) where 1 < p < infinity, G is a composed Carleson curve, an d w is a Muckenhoupt weight in the class A(p)(Gamma). We also provide index formulas for the operators in the closed algebra of singular integral oper ators with piecewise continuous matrix-valued coefficients. Our main theore m is based upon three pillars: on the identification of the local spectrum of the Cauchy singular integral operator at the endpoints of simple Carleso n area, on an appropriate "N projections theorem", and on results of geomet ric function theory pertaining to the problem of extending Carleson curves and Muckenhoupt weights.