Operators in Rigged Hilbert spaces: some spectral properties

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