Definition 3

Let A be an algebra over $\mathbf{C}$ and E a complex topological vector space. Let $\pi$ be a representation of A in E. An element $x$ of E is said to be a cyclic vector for $\pi$ if the set of elements $\pi (a)x$ for $a\in A$ is total in E.

Let $u\in \mathscr{L}(E)$ be an endomorphism of E. It is said that $x\in E$ is a cyclic vector for $u$ if it is a cyclic vector for the identical representation of the stellar subalgebra generated by $u$ in $\mathscr{L}(E)$.