# Analytic vector

in the space $V$ of a representation $T$ of a Lie group $G$

A vector $\xi \in V$ for which the mapping $g \mapsto [T(g)](\xi)$ is a real-analytic vector function on $G$ with values in $V$ (cf. Representation theory). If $T$ is a weakly-continuous representation of a Lie group $G$ in a Banach space $V$, then the set $V^{\omega}$ of analytic vectors is dense in $V$ ([1], [2], [3]). This theorem has been generalized to a wide class of representations in locally convex spaces ([5]). It has been proven in [6] that a representation of a connected Lie group $G$ in a Banach space $V$ is uniquely determined by the corresponding representation of the Lie algebra of $G$ in the space $V^{\omega}$.

An analytic vector for an unbounded operator $A$ on a Banach space $V$, defined on a domain $D(A)$, is defined as a vector $$\xi \in \bigcap_{n = 1}^{\infty} D(A^{n})$$ for which the series $$\sum_{n = 1}^{\infty} \frac{x^{n}}{n!} \| {A^{n}}(\xi) \|$$ has a positive radius of convergence. This notion, which was introduced in [2], is a special case of the general concept of an analytic vector; here, the set of points on the real line with the operation of addition plays the role of the Lie group $G$. It was found useful in the theory of operators on Banach spaces and in the theory of elliptic differential operators.

#### References

 [1] P. Cartier, J. Dixmier, “Vecteurs analytiques dans les répresentations de groupes de Lie”, Amer. J. Math., 80 (1958), pp. 131–145. [2] E. Nelson, “Analytical vectors”, Ann. of Math., 70 (1969), pp. 572–615. [3] L. Gårding, “Vecteurs analytiques dans les répresentations des groups”, Bull. Soc. Math. France, 88 (1960), pp. 73–93. [4] P. Cartier, “Vecteurs analytiques”, Sem. Bourbaki 1958/1959, 181 (1959), pp. 12–27. [5] R.T. Moore, “Measurable, continuous and smooth vectors for semigroup and group representations”, Mem. Amer. Math. Soc., 78 (1968). [6] Harish-Chandra, “Representations of a semisimple Lie group on a Banach space I”, Trans. Amer. Math. Soc., 75 (1953), pp. 185–243.

Integrability to the corresponding (simply-connected and connected) Lie group $G$ of a representation of a Lie algebra $\mathfrak{g}$ in a Hilbert space follows from the existence of a dense set of analytic vectors for the ‘Laplacian’ $\Delta$ — the sum of the squares of skew-symmetric representatives, with a common dense invariant domain $D$, of linear generators of $\mathfrak{g}$ (known as Nelson’s criterion ([2])). More practical criteria were developed later, e.g., the existence of common analytic vectors for Lie generators (known as the $\text{FS}^{3}$-criterion ([a2], [a3], [a4])), as well as similar results in more general spaces and a study of separate versus joint analyticity.