Cyclic vector and cyclic unitary representation

Definition

Let $(\pi,\mathcal H)$ be a strongly continuous unitary representation of a group $G$ . The cyclic subspace generated by $\xi\in\mathcal H$ is the closed linear subspace

\mathcal H_\xi=\overline{\operatorname{span}}\{\pi(g)\xi:g\in G\}.

The vector $\xi$ is cyclic if $\mathcal H_\xi=\mathcal H$ . The representation is cyclic if it has at least one cyclic vector. Cyclicity is a density condition: it does not require the orbit itself to equal $\mathcal H$ , nor does it select a unique cyclic vector.

Basic properties

The subspace $\mathcal H_\xi$ is closed and invariant under every $\pi(g)$ , because multiplication by $g$ permutes the orbit. The restriction of $\pi$ to $\mathcal H_\xi$ is therefore a cyclic representation with cyclic vector $\xi$ . A nonzero vector in a one-dimensional representation is cyclic, while the zero vector is cyclic only for the zero Hilbert space.

Coefficients and reconstruction

The diagonal matrix coefficient

\varphi_\xi(g)=\langle\pi(g)\xi,\xi\rangle

is positive-definite and determines the pointed cyclic representation $(\pi,\mathcal H,\xi)$ up to a unique unitary equivalence preserving the distinguished vector. Conversely, the GNS construction builds such a pointed cyclic representation from a positive-definite function Folland, §3.2–3.3.

Conventions and scope

For a representation of an algebra $A$ , one instead closes $\pi(A)\xi$ ; if $A$ is nonunital, conventions may use the unitization or a nondegeneracy hypothesis. The phrase “cyclic subspace” always depends on both the representation and the chosen vector, even when this dependence is suppressed in notation.

References

G. B. Folland, A Course in Abstract Harmonic Analysis, 2nd ed., CRC Press, 2016. DOI record. Relevant: §§3.2–3.3 on cyclic representations and positive-definite functions.