Let $G$ be a semi-simple real Lie group, and $\rho$ a strongly continuous unitary representation of $G$ in a Hilbert space $H$. Let $\mathfrak{g}$ be its Lie algebra and $d\rho$ be the infinitesimal representation associated to $\rho$. Then there exists a dense subspace $\mathcal{D}^\omega(\rho)$ of analytic vectors for $\rho$ and for $d\rho$. Cf. for instance Section 10.4 of the book by Schmüdgen.

On the other hand, for any hermitian elliptic element $x$ in the enveloping algebra $U(\mathfrak{g})$, $d\rho(x)$ is an essentially self-adjoint operator (Section 10.2 in Schmüdgen's book). Therefore it admits a dense subspace $\mathcal{D}^\omega(d\rho(x))$ of analytic vectors.

My first question is: Is $\mathcal{D}^\omega(\rho)$ contained in $\mathcal{D}^\omega(d\rho(x))$? Or in words, is the dense common subspace of analytic vectors for the unitary operators associated to group elements and for the unbounded operators associated to Lie algebra element also a set of analytic vectors for the elliptic elements of the enveloping algebra?

My second question is: Is $\mathcal{D}^\omega(\rho)$ invariant under the action of the 1-parameter group of unitary operators associated with $d\rho(x)$?