The authors study Fredholm properties of operators of the form \[ K_{a}\varphi=\lambda\varphi-P_{+}aP_{-}Q\varphi\tag{1} \] in weighted \(L_{p}\)-space on the real axis, where \(P_{\pm}=1/2(I\pm S)\), \(S\) is the standard operator of singular integration, \(a\) is a piecewise continuous function and \((Q\varphi)(x)=\varphi(-x)\).
These operators can be considered as a particular case of operators from the algebra generated by the operators of multiplication by piecewise continuous functions, the singular operator \(S\) and the Carleman shift operator \(Q\). The general Gohberg-Krupnik theorem on the Fredholm property of operators in this algebra takes a rather complicated form in this special case. The authors use some general abstract theorem given by them earlier to obtain in effective terms the Fredholm conditions and the index formula for the operators (1).