This note demonstrates a continuous unitary representation with trivial smooth vectors in an infinite-dimensional group, indicating important implications.
In this note we show that the representation of the additive group of the Hilbert space L 2 ([0, 1], R) on L 2 ([0, 1], C) given by the multiplication operators (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.
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Beltita et al. (2008) studied this question.
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