Gårding density theorem - Imported Knowlpedia Content

Statement

Let $G$ be a finite-dimensional Lie group and let $(\pi,\mathcal H)$ be a strongly continuous unitary representation. The Gårding density theorem states that the Gårding subspace

\operatorname{span}\{\pi(f)v:f\in C_c^\infty(G),\ v\in\mathcal H\}

is dense in $\mathcal H$ . Every vector in this subspace is smooth, so the smooth-vector space $\mathcal H^\infty$ is also dense. Thus the unbounded operators of the derived Lie-algebra representation have a common dense invariant domain.

Proof mechanism

Choose nonnegative functions $f_j\in C_c^\infty(G)$ with integral $1$ and supports shrinking to the identity. Strong continuity of the orbit map gives

\lVert\pi(f_j)v-v\rVert \leq\int_G f_j(g)\lVert\pi(g)v-v\rVert\,dg\longrightarrow 0.

Each $\pi(f_j)v$ is smooth because derivatives can be transferred to $f_j$ . This approximate-identity argument is already present in Gårding, pp. 331–332.

Consequences

The theorem makes the derived representation available for every strongly continuous representation, even when most Hilbert-space vectors are not differentiable. It also gives a canonical family of cores for infinitesimal generators. The Dixmier–Malliavin theorem later sharpened density by identifying the entire smooth-vector space with the Gårding subspace.

Conventions and scope

The theorem is sometimes stated for continuous representations on Banach spaces or more general complete locally convex spaces; additional integration and equicontinuity hypotheses then enter. The Hilbert-unitary version stated here needs only strong continuity. It proves density, not that the smooth-vector space is closed in the Hilbert norm; except in special cases, it is not.

References

Lars Gårding, Note on Continuous Representations of Lie Groups, Proceedings of the National Academy of Sciences 33 (1947), 331–332. DOI record. Relevant: the original smoothing and density argument.
Garth Warner, Harmonic Analysis on Semi-Simple Lie Groups I, Springer, 1972. DOI record. Relevant: §4.4 on differentiable vectors and Gårding's theorem.