Abstract: Relatively uniformly continuous linear dynamical systems on vector lattices were introduced and studied by M. Kramar Fijavz, M. Kandic, and J. Gluck in [4,5,6] in order to extend the theory of C0-semigroups to the setting where the topological arguments are not available. In many cases, the underling state space need not to be even a vector lattice but still possesses a structure of an ordered vector space with a generating cone. In [3] some of results of [4,5] are extended to ordered vector spaces by using the technique developed in [1]. The theory of non-topological generators of relatively uniformly continuous positive linear dynamical systems on relatively uniformly complete vector lattices goes back to [6]. It is proved in [2] that, under rather mild assumptions, a positive relatively uniformly continuous linear dynamical system possesses the unique relatively uniformly continuous extension to relative uniform completion of the underlying vector lattice. This allows to drop complete ness assumption in several results obtained in [6]. Thanks. This work is carried in the framework of the State Task of the Sobolev Institute of Mathematics (project no. FWNF-2022-0004). References [1] E. Emelyanov, Collective order convergence and collectively quali ed sets of operators, Siberian Electronic Mathematical Reports, (to appear) (2025). [2] E. Emelyanov, On the extension of one-parameter operator semigroups to completions of Archimedean vector lattices, arxiv.org/pdf/2412.17097 [3] E. Emelyanov, N. Erkursun-Ozcan, S. Gorokhova, Collective order boundedness of sets of operators between ordered vector spaces, arxiv.org/pdf/2410.17030 [4] J. Gluck, M. Kaplin, Order boundedness and order continuity properties of positive operator semigroups, Quaest. Math., 47(S1) (2024), 153168. [5] M. Kandic, M., Kaplin, Relatively uniformly continuous semigroups on vector lattices, J. Math. Anal. Appl., 489 (2020), 124139. [6] M. Kaplin, M. Kramar Fijavz, Generation of relatively uniformly continuous semigroups on vector lattices, Analysis Math., 46 (2020), 293322.

Cite: Emelyanov E.
Linear dynamical systems on ordered vector spaces. Non-topological approach.
Conference Dynamics in Siberia 24 Feb - 1 Mar 2025