Maximal Op*-Algebras on DF-Domains

Abstract

Let $D$ be any dense domain in a Hilbert space and let ${\mathcal{L}}^{+}(D)$ be the maximaximal Op*-algebra of (possibly unbounded) linear operators. In this paper the uniform topology ${\tau }_D$ on ${\mathcal{L}}^{+}(D)$ is investigated for the case where $D$ is a DF-space with respect to the graph topology. As a main result, a characterization of the bounded subsets of $D$ and of the topology ${\tau }_D$ by strongly bounded selfadjoint operators is given. Especially, each bounded subset of $D$ is contained in some bounded ellipsoid. This is applied to approximate the operators in ${\mathcal{L}}^{+}(D)$ by bounded ones.