Constructions analogous to those from Chap. 1 for non-self-adjoint operators can be obtained by diverse and ambiguous approaches. Firstly, due to the different domains \( \mathfrak{D}(A) \) and \( \mathfrak{D}\left({A}^{\ast}\right) \) of the main operator A, one requires reasonable matching of the corresponding defective subspaces of A and A∗. Secondly, the known research methods use the Cayley transform, and this is not always convenient as it requires the calculation of the resolvent of the operator A, which has its own difficulties. Thirdly, even in the case of dissipativity of A, to use the Lax–Phillips scattering scheme it is required to correctly define the outgoing and incoming subspaces to find the scattering operator giving the main analytic tool for this research, the characteristic function. These circumstances initiated the definition of different classes of non-self-adjoint unbounded operators. Due to this, it is worth noting the works of A. V. Kuzhel [K47–K52], E. R. Tsekanovskii [Ts1–Ts3], A. V. Strauss [S28–S29], B. S. Pavlov [P2–P18], Yu. M. Arlinskii [A25–A28] and others. It is especially worth noting the approach of V. Derkach and M. Malamud [D14], in which an important tool for researching symmetric operators is the boundary values space and the Weyl function corresponding to it (see the Comments to this chapter).