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A notion of resolvent set for an operator acting in a rigged Hilbert space$\D \subset \H\subset \D^\times$ is proposed. This set depends on a family ofintermediate locally convex spaces living between $\D$ and $\D^\times$, calledinterspaces. Some properties of the resolvent set and of the correspondingmultivalued resolvent function are derived and some examples are discussed.

Operators in Rigged Hilbert spaces: some spectral properties