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

Abstract

Under certain restrictions, it is proved that a family of self-adjoint commuting operators $A = (A_{\varphi})_{\varphi \in \Phi}$ where $\Phi$ is a nuclear space, possesses a cyclic vector iff there exists a Hubert space $H \subset \Phi'$ of full operator-valued measure $E$, where $\Phi'$ is the space dual to $\Phi$, $E$ is the joint resolution of the identity of the family $A$.