Singular integral operators with piecewise continuous coefficients and their symbols

The algebra generated by one-dimensional singular integral operators with piecewise continuous coefficients is studied. Operators acting on the space $L_p$ with a weight are considered. The contour is assumed to consist of closed and open arcs. The structure of the symbols of the operators considered is elucidated. It is found that the symbol is a matrix-function of the second order depending both on $p$ and on the weight. Criteria that the operator be Fredholm and a formula for its index are established.