Spectra of elements in the range of strongly continuous unitary representations of Lie groups

摘要

The main result proved in this work is that, for a strongly continuous unitary representation of a Lie group, we have the following dichotomy: The representation is norm continuous or on an open neighborhood of the unit (which is the whole group for a weakly exponential Lie group), there is a residual set of elements whose range admits the entire torus as spectrum.