Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: On an invariance property of the space of smooth vectors   Let $(\pi, \mathcal H)$ be a continuous unitary representation of the
(infinite dimensional) Lie group $G$ and $\gamma \: \mathbb R \to
\mathrm{Aut}(G)$ define a continuous action of $\mathbb R$ on $G$. Suppose that
$\pi^\#(g,t) = \pi(g) U_t$ defines a continuous unitary representation of the
semidirect product group $G \rtimes_\gamma \mathbb R$. The first main theorem
of the present note provides criteria for the invariance of the space $\mathcal
H^\infty$ of smooth vectors of $\pi$ under the operators $U_f = \int_\mathbb R
f(t)U_t\, dt$ for $f \in L^1(\mathbb R)$, resp., $f \in \mathcal S(\mathbb R)$.
Using this theorem we show that, for suitably defined spectral subspaces
$\mathfrak g_{\mathbb C}(E)$, $E \subseteq \mathbb R$, in the complexified Lie
algebra $\mathfrak g_{\mathbb C}$, and $\mathcal H^\infty(F)$, $F\subseteq
\mathbb R$, for $U$ in $\mathcal H^\infty$, we have \[ \mathsf{d}\pi(\mathfrak
g_{\mathbb C}(E)) \mathcal H^\infty(F) \subseteq \mathcal H^\infty(E + F).\]

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The categories of this paper are: chem-ph
3
The categories of this paper are: physics
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The categories of this paper are: adap-org
4
The categories of this paper are: cond-mat
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The categories of this paper are: econ
11
The categories of this paper are: hep-lat
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The categories of this paper are: acc-phys
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The categories of this paper are: funct-an
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The categories of this paper are: supr-con
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The categories of this paper are: atom-ph
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The categories of this paper are: eess
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The categories of this paper are: gr-qc
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The categories of this paper are: nlin
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The categories of this paper are: solv-int
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The categories of this paper are: quant-ph
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The categories of this paper are: q-fin