On the Regularity of the Partial $O^{\ast }$-Algebras Generated by a Closed Symmetric Operator

Abstract

Let be given a dense domain $\mathcal{D}$ in a Hilbert space and a closed symmetric operator $T$ with domain containing $\mathcal{D}$. Then the restriction of $T$ to $\mathcal{D}$ generates (algebraically) two partial ${}^{\ast }$-algebras of closable operators (called weak and strong), possibly nonabelian 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.