abstract:In mathematics, the adjoint representation (or adjoint action) of a Lie group G is a way of representing the elements of the group as linear transformations of the group's Lie algebra, considered as a vector space. For example, in the case where G is the Lie group of invertible matrices of size n, GL(n), the Lie algebra is the vector space of all (not necessarily invertible) n-by-n matrices.
This paper discusses the integralrepresentationof a class of self-adjointoperators. By applyingsuchrepresentation, it is provedthat the spectrumsofsuchoperators are discrete.