Spectral theory for commutative algebras of differential operators on Lie groups
Abstract
The joint spectral theory of a system of pairwise commuting self-adjoint left-invariant differential operators L_1,...,L_n on a connected Lie group G is studied, under the hypothesis that the algebra generated by them contains a "weighted subcoercive operator" of ter Elst and Robinson (J. Funct. Anal. 157 (1998) 88-163). The joint spectrum of L_1,...,L_n in every unitary representation of G is characterized as the set of the eigenvalues corresponding to a particular class of (generalized) joint eigenfunctions of positive type of L_1,...,L_n. Connections with the theory of Gelfand pairs are established in the case L_1,...,L_n generate the algebra of K-invariant left-invariant differential operators on G for some compact subgroup K of Aut(G).
Keywords
Cite
@article{arxiv.1007.5248,
title = {Spectral theory for commutative algebras of differential operators on Lie groups},
author = {Alessio Martini},
journal= {arXiv preprint arXiv:1007.5248},
year = {2013}
}