An open system corresponding to a non-self-adjoint bounded operator was introduced by M. S. Livšic. This open system is of first order and underlies two directions closely related to the theory of non-self-adjoint operators, namely: dilation theory (Nagy–Foias) and scattering theory (Lax–Phillips). It is an open system that leads to the characteristic function of a non-self-adjoint operator, which is the main analytic instrument for obtaining triangular (M. S. Livšic) and functional (Nagy–Foias) models.