Models of self-adjoint and unitary operators in Pontryagin spaces

This paper represents a revised version of the lectures, delivered by the author at KROMSH-2019. These lectures are devoted to describing a few different ways of constructing a model representation for self-adjoint and unitary operators acting in Pontryagin spaces, and a comparison between them. Two of these models are based on the regularized integral Krein–Langer representation of a numerical sequence generated by the powers of a self-adjoint (in the sense of Pontryagin spaces) operator. The steps to deduce both this representation and the spectral function of the corresponding operator are given. In both models (first of which belongs to the author of this paper), the operator is realized as an operator of multiplication by an independent variable, but the space of functions in which it acts is different for each of the models. The third model, introduced by V. S. Shulman, is based on his own concept of a quasi-vector.