On the regularity of the partial \(O^*\)-algebras generated by a closed symmetric operator (Q1328906)

Summary: Let \(\mathcal D\) be a given dense domain in a Hilbert space and \(T\) a closed symmetric operator with domain containing \(\mathcal D\). Then the restriction of \(T\) to \(\mathcal D\) generates (algebraically) two partial *-algebras of closable operators (called weak and strong), possibly non-Abelian and nonassociative. We characterize them completely. In particular, we examine under what conditions they are regular, that is, consist of polynomials only, and standard. Simple differential operators provide concrete examples of all the pathologies allowed by the abstract theory.