About Riemann matrix operator in the space of smooth vector functions

In the space of vector functions smooth on the unit circle, we consider the matrix operator of linear conjugation generated by the Riemann boundary-value problem. It is assumed that the coefficients of the boundary value problem are smooth matrix functions. The concept of smooth degenerate factorization of the plus and minus types of a smooth matrix function is introduced and studied. In terms of degenerate factorizations, we give necessary and sufficient conditions for the noethericity of the considered Riemann matrix operator in the space of smooth vector functions. For a function smooth on a circle having at most finitely many zeros of finite orders, the concept of a singular index is introduced and studied, generalizing the concept of the index of a non-degenerate continuous function. For the Noetherian matrix Riemann operator, a formula is obtained for calculating the index of this operator, which coincides with the well-known similar formula in the case where the coefficients of the Riemann operator are non-degenerate.