Commutants

Abstract

In a broad sense this chapter deals with the commutativity of both single unbounded operators and families of operators. In particular, various notions of commutants of O*-algebras are investigated. Section 7.1 contains some general results on strongly commuting self-adjoint operators. Apart from being of interest in itself, they are used in Sections 7.3, 9.1 and 10.2. In Section 7.2 we define six (in general different) concepts of unbounded and bounded commutants for an O*-algebra, and we discuss the relations between them. The self-adjointness of the O*-algebra implies that the weak and the strong unbounded commutants coincide, but it is not sufficient to ensure a close connection between unbounded and bounded commutants. For this further restrictions are needed. Such a class of O*-algebras which we call strictly self-adjoint O*-algebras is considered in Section 7.3.