Singular decomposition of a differential operator on a semiaxis

For an unbounded operator appearing in a natural connection with a boundary value problem on a semiaxis for a system of ordinary differential equations, we establish a decomposition that generalizes the singular decomposition of the finite-dimensional case. The operator is represented as a product of three operators: an isometry, a nonnegative definite diagonal operator, and one more isometry. The spectral matrix of the decomposition is expressed in terms of a solution to some matrix Lourier–Riccati equation.