On the functional model for dissipative operators. The coordinate-free approach

The paper is based on seminar talks and is mostly instructional. Dissipative operators are studied by means of their Caley transforms using the coordinate-free model for contractions developed by N. K. Nikolskii and V. I. Vasyunin. It is shown that different known froms of the characteristic functions and self-adjoint dilation can be derived from a general scheme. Also we find new formulae for the dilation and its eigen-functions which generalise those obtained by B. S. Pavlov for the Shrödinger operator on a half-line with a real-valued potential and a nonreal boundary condition. Several examples are considered.