Geodesic
A Nonsmooth Continuous Unitary Representation of a Banach-Lie Group
Journal of Lie theory, Tome 18 (2008) no. 4, pp. 933-936
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We show that the representation of the additive group of the Hilbert space $L^2([0,1],{\mathbb R})$ on $L^2([0,1], {\mathbb C})$ given by the multiplication operators $\pi(f) := e^{if}$ is continuous but its space of smooth vectors is trivial. This example shows that a continuous unitary representation of an infinite dimensional Lie group need not be smooth.
Classification : 22E65, 22E45
Mots-clés : Infinite-dimensional Lie group, unitary representation, smooth vector 
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     author = {D. Beltita and K.-H. Neeb},
     title = {A {Nonsmooth} {Continuous} {Unitary} {Representation} of a {Banach-Lie} {Group}},
     journal = {Journal of Lie theory},
     pages = {933--936},
     year = {2008},
     volume = {18},
     number = {4},
     url = {http://geodesic.mathdoc.fr/item/JLT_2008_18_4_JLT_2008_18_4_a11/}
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D. Beltita; K.-H. Neeb. A Nonsmooth Continuous Unitary Representation of a Banach-Lie Group. Journal of Lie theory, Tome 18 (2008) no. 4, pp. 933-936. http://geodesic.mathdoc.fr/item/JLT_2008_18_4_JLT_2008_18_4_a11/