Essential skew-adjointness of derived representation operators

Statement

Let $\pi$ be a strongly continuous unitary representation of a Lie group $G$ on $\mathcal H$ , and let $X$ lie in its Lie algebra. On the dense invariant domain $\mathcal H^\infty$ , the derived operator

d\pi(X)v=\left.\frac{d}{dt}\right|_{t=0}\pi(\exp(tX))v

is essentially skew-adjoint. If $A_X$ is the self-adjoint Stone generator determined by $\pi(\exp(tX))=e^{itA_X}$ , then

\overline{d\pi(X)|_{\mathcal H^\infty}}=iA_X.

Equivalently, $-i\,d\pi(X)$ is essentially self-adjoint on $\mathcal H^\infty$ , and its closure is $A_X$ .

Why essential skew-adjointness holds

Stone's theorem first identifies the maximal skew-adjoint generator $iA_X$ of the one-parameter subgroup. Smooth vectors lie in its domain and the derivative there agrees with $d\pi(X)$ . Smoothing arbitrary vectors by convolution against compactly supported smooth functions on $G$ produces a dense invariant subspace of smooth vectors that is a core for $A_X$ . Agreement on this core forces the stated closure Warner, §4.4.

Consequences

The operator $d\pi(X)$ has a unique skew-adjoint extension, so its closure recovers the original subgroup by

\pi(\exp(tX))=\exp\!\left(t\,\overline{d\pi(X)}\right).

There is therefore no ambiguity in passing between the global unitary action along $\exp(\mathbb RX)$ and its infinitesimal action. Spectral theory applies to the self-adjoint operator $A_X=-i\,\overline{d\pi(X)}$ .

Domains and sign conventions

References

Garth Warner, Harmonic Analysis on Semi-Simple Lie Groups I, Grundlehren der mathematischen Wissenschaften 188, Springer, 1972. DOI record. Relevant: §4.4 on differentiable vectors and infinitesimal operators.
Michael Reed and Barry Simon, Methods of Modern Mathematical Physics I: Functional Analysis, Academic Press, 1972. DOI record. Relevant: §VIII.4 on Stone's theorem and generators.