Analytic vector of a Lie-group representation

Definition

Let $G$ be a finite-dimensional real Lie group and $(\pi,\mathcal H)$ a strongly continuous unitary representation. A vector $v\in\mathcal H$ is an analytic vector if its orbit map

G\longrightarrow\mathcal H,\qquad g\longmapsto\pi(g)v,

is real analytic in a neighborhood of the identity, as a Hilbert-space-valued map. Equivalently, in local exponential coordinates its Taylor series, formed from iterated operators of the derived representation, converges in norm to the orbit map near $0$ . Every analytic vector is smooth, but a smooth vector need not be analytic.

Operator criterion

For a one-parameter group with self-adjoint generator $A$ , a vector $v$ is analytic precisely when $v$ lies in every domain $\mathcal D(A^n)$ and

\sum_{n=0}^{\infty}\frac{t^n}{n!}\lVert A^n v\rVert<\infty

for some $t>0$ . For a Lie group, analogous factorial estimates for a basis of its Lie algebra characterize analytic orbit maps. These estimates connect analytic vectors to Nelson's analytic vector theorem.

Density and examples

Nelson proved that analytic vectors are dense for continuous representations of finite-dimensional Lie groups, using heat-kernel regularization Nelson, pp. 589–595. For translations of $\mathbb R$ on $L^2(\mathbb R)$ , vectors whose Fourier transforms have a square-integrable exponential weight are analytic. A vector with only polynomial Fourier decay can be smooth while failing the factorial convergence test.

Conventions and scope

References

Edward Nelson, Analytic vectors, Annals of Mathematics 70 (1959), 572–615. DOI record. Relevant: §§4–6 on analytic vectors for Lie-group representations and heat-kernel regularization.
Michael Reed and Barry Simon, Methods of Modern Mathematical Physics II: Fourier Analysis, Self-Adjointness, Academic Press, 1975. Publisher record. Relevant: §X.6 on analytic vectors and essential self-adjointness.