On the existence of a cyclic vector of some families of operators

Abstract

It is proved that, under some restrictions, a family of selfadjoint commuting operators A=(Aφ) φ ∈ Φ where Φ is a nuclear space, has a cyclic vcctor iff there exists a Hilbert space H ⊂ Φ′ of full operator-valued measure E, where Φ′ is the dual of Φ, E is the joint resolution of the identity of the family A.