On the Structure of Semigroups of Operators Acting in Spaces with Indefinite Metric

The first part of this paper concludes the cycle of studies in the structure of continuous one-parameter semigroups of operators originated in 2001 in the journal "Nonlinear Analysis" and continued in several other publications. In particular, the following theorem is proved: If h is an (indefinite or definite) complex Krein space and J is the K-semigroup of plus-operators acting in h then any plus-operator F(t) is an element of J, where t >= 0, is a bistrict operator. This theorem permits removing several restrictions imposed on the sets of plus-operators in the preceding papers and thus reinforces the results of these papers. In the second part of the present paper, we consider the heredity problem in discrete one-parametric semigroups. Namely, we study the problem of finding what indefinite properties the generating plus-operator of a semigroup and all its positive integer powers can have only simultaneously and what indefinite properties they have not necessarily simultaneously. In conclusion, we consider several applications to dynamical systems with continuous and discrete time