Applications of operator equalities to singular integral operators and to Riemann boundary value problems

In the first part of this article (Section 2), we consider a Riemann boundary value problem with shift andpiecewise constant coefficients. In the second part (Section 3), we consider a matrix characteristic singularintegral operators with piecewise constant coefficients of a special structure. Simplicity of the shift and thecoefficients under consideration permits us to obtain conditions under which the Riemann boundary valueproblem admits an unique solution and to study the invertibility of the matrix characteristic singular integraloperators.