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.

Comments

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.

References

[a1]	G. Warner, “Harmonic analysis on semi-simple Lie groups”, 1, Springer (1972).
[a2]	M. Flato, D. Sternheimer, “Deformations of Poisson brackets, separate and joint analyticity in group representations, nonlinear group representations and physical applications”, J.A. Wolf (ed.), M. Cahen (ed.), M. De Wilde (ed.), Harmonic Analysis and Representations of Semisimple Lie Groups, Reidel (1980), pp. 385–448.
[a3]	J. Simon, “On the integrability of representations of finite dimensional real Lie algebras”, Commun. Math. Phys., 28 (1972), pp. 39–46.
[a4]	M. Flato, J. Simon, H. Snellman, D. Sternheimer, “Simple facts about analytic vectors and integrability”, Ann. Scient. Ec. Norm. Sup. Ser. 4, 5 (1972), pp. 423–434.