On Unbounded Positive *-Representationson Fréchet-Domains

Let D be a Frechet-domain from Op*-algebra, abbreviated F-domain. The present paper is concerned with the study of positive ^representations of L+(D), of the Calkin representation of L+(D) and of bounded sets in ultrapower Du. For this the density property plays an important role. It was introduced by S. Heinrich for locally convex spaces in [2]. In the paper [3] we gave several characterizations of the density property of an F-domain D. In this work we give a characterizati ons of continuity of positive "^representations and Calkin representation of L+(D] by the density property of D. This generalizes the well-known result due to K. Schmiidgen, see [12], Further we describe bounded subsets in ultrapower Z)u. If D has the density property, then every bounded set M c Du has a simple structure: For each bounded set M c Du there exists a bounded set N c D with M c'N u. S. Heinrich proved an analogous result for bounded ultrapowers on locally convex spaces. Acknowledgements I would like to thank Professor Dr. K.-D. Kiirsten for suggesting a problems led to this work and for many helpful discussions. § 1. Preliminaries Throughout the paper, D denotes a dense linear subspace of a Hilbert space H. We denote the norm, unit ball, and the scalar product of H by ||-||, UH and , respectively. For a closable linear operator T on H, let ?, D(T) and || T|| denote the closure, domain, and the norm of T (provided the later exists), respectively. The set of linear operators