Operators in rigged Hilbert spaces: Some spectral properties

A notion of resolvent set for an operator acting in a rigged Hilbert space D⊂H⊂D×D⊂H⊂D× is proposed. This set depends on a family of intermediate locally convex spaces living between DD and D×D×, called interspaces. Some properties of the resolvent set and of the corresponding multivalued resolvent function are derived and some examples are discussed.