On Factorization of Functional Operators with Reflection on the Real Axis

Problems of factorization of matrix functions are closely connected with the solution of matrix Riemann boundary value problems and with the solution of vector singular integral equations. In this article, we study functional operators with orientation-reversing shift reflection on the real axes. We introduce the concept of multiplicative representation of functional operators with shift and its partial indices. Based on the classical notion of matrix factorization, the correctness of the definitions is shown. A theorem on the relationship between factorization of functional operators with reflection and factorization of the corresponding matrix functions is proven.