A functional-differential operator with involution

The functional-differential operator with involution generating the reflection operator is presented. Operator arises, in particular, in studying the expansions in eigen and associated functions of the integral operator, whose kernel has discontinuities on the lines. The system of eigen and associated functions of the operator forms Riesz bases with parentheses, which further unite the eigen and associated functions. The operator is shown to have improved properties in comparison with the pure differentiation operator in any vector functions with unique components. The operator under consideration also is a functional-differential operator on two edges of the graph and an ordinary first-order differential operator on the third edge.