View source for Analytic vector

''in the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012470/a0124701.png" /> of a representation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012470/a0124702.png" /> of a Lie group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012470/a0124703.png" />''

A vector <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012470/a0124704.png" /> for which the mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012470/a0124705.png" /> is a real-analytic vector function on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012470/a0124706.png" /> with values in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012470/a0124707.png" /> (cf. [[Representation theory|Representation theory]]). If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012470/a0124708.png" /> is a weakly-continuous representation of a Lie group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012470/a0124709.png" /> in a Banach space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012470/a01247010.png" />, then the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012470/a01247011.png" /> of analytic vectors is dense in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012470/a01247012.png" /> [[#References|[1]]], [[#References|[2]]], [[#References|[3]]]. This theorem has been generalized to a wide class of representations in locally convex spaces [[#References|[5]]]. It has been proved [[#References|[6]]] that a representation of a connected Lie group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012470/a01247013.png" /> in a Banach space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012470/a01247014.png" /> is uniquely determined by the corresponding representation of the Lie algebra of the Lie group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012470/a01247015.png" /> in the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012470/a01247016.png" />.

An analytic vector for an unbounded operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012470/a01247017.png" /> on a Banach space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012470/a01247018.png" />, defined on a domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012470/a01247019.png" />, is defined as a vector

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012470/a01247020.png" /></td> </tr></table>

for which the series

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012470/a01247021.png" /></td> </tr></table>

has a positive radius of convergence. This notion, which was introduced in [[#References|[2]]], is a special case of the general concept of an analytic vector; here the set of points on the real line with the addition operation plays the role of the Lie group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012470/a01247022.png" />. It was found useful in the theory of operators on Banach spaces and in the theory of elliptic differential operators.

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  P. Cartier,   J. Dixmier,   "Vecteurs analytiques dans les répresentations de groupes de Lie"  ''Amer. J. Math.'' , '''80'''  (1958)  pp. 131–145</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  E. Nelson,   "Analytical vectors"  ''Ann. of Math.'' , '''70'''  (1969)  pp. 572–615</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  L. Gårding,   "Vecteurs analytiques dans les répresentations des groups"  ''Bull. Soc. Math. France'' , '''88'''  (1960)  pp. 73–93</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  P. Cartier,   "Vecteurs analytiques" , ''Sem. Bourbaki 1958/1959'' , '''181'''  (1959)  pp. 12–27</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  R.T. Moore,   "Measurable, continuous and smooth vectors for semigroup and group representations"  ''Mem. Amer. Math. Soc.'' , '''78'''  (1968)</TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top">  Harish-Chandra,   "Representations of a semisimple Lie group on a Banach space I"  ''Trans. Amer. Math. Soc.'' , '''75'''  (1953)  pp. 185–243</TD></TR></table>



====Comments====
Integrability to the corresponding (simply-connected and connected) Lie group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012470/a01247023.png" /> of a representation of a Lie algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012470/a01247024.png" /> in a Hilbert space follows from the existence of a dense set of analytic vectors for the  "Laplacian"  <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012470/a01247025.png" />, the sum of the squares of skew-symmetric representatives with a common dense invariant domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012470/a01247026.png" />, of linear generators of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012470/a01247027.png" /> (Nelson's criterion, [[#References|[2]]]). More practical criteria were developed later, e.g. the existence of common analytic vectors for Lie generators (the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012470/a01247028.png" /> criterion, [[#References|[a2]]], [[#References|[a3]]], [[#References|[a4]]]), as well as similar results in more general spaces and a study of separate versus joint analyticity.

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  G. Warner,   "Harmonic analysis on semi-simple Lie groups" , '''1''' , Springer  (1972)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  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</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  J. Simon,   "On the integrability of representations of finite dimensional real Lie algebras"  ''Commun. Math. Phys.'' , '''28'''  (1972)  pp. 39–46</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  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</TD></TR></table>