On the domains of generalized operators

Abstract

The domains of generalized operatorsT: Φ₋→Φ₊ on rigged Hilbert spaces Φ₋ ⊂H ⊂ Φ₊ are investigated. We introduce an equivalence relation for operators with different domains. Arguments are given for taking Φ₊ to be the weak quasi-completion of Φ₋ and for Φ₋ to be Mackey quasi-complete. For domains of closed symmetric Hilbert space operators we give a representation for Φ₊ and provide certain elements in the equivalence class of the corresponding operator.