A non-smooth continuous unitary representation of a Banach-Lie group

Details A non-smooth continuous unitary representation of a Banach-Lie group A non-smooth continuous unitary representation of a Banach-Lie group

2008-11-26

arxiv.org/abs/0811.4234v1

Mathematics

Representation Theory

5 pages

Scientific paper

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 $\pi(f) := e^{if}$ 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.

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