JournalsprimsVol. 28 , No. 5DOI 10.2977/prims/1195167935

On the Regularity of the Partial <i>0</i>*-Algebras Generated by a Closed Symmetric Operator

  • Jean-Pierre Antoine

    Université Catholique de Louvain, Louvain-La-Neuve, Belgium
  • Atsushi Inoue

    Hiroshima University, Japan
  • Camillo Trapani

    Università degli Studi di Palermo, Italy
On the Regularity of the Partial <i>0</i>*-Algebras Generated by a Closed Symmetric Operator cover

Abstract

Let be given a dense domain D in a Hilbert space and a closed symmetric operator T with domain containing D. Then the restriction of T to D generates (algebraically) two partial *-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.