Spectral theory for commutative algebras of differential operators on Lie groups

Details Spectral theory for commutative algebras of differential operators on Lie groups Spectral theory for commutative algebras of differential operators on Lie groups

2010-07-29

arxiv.org/abs/1007.5248v1

Mathematics

Functional Analysis

Scientific paper

10.1016/j.jfa.2011.01.008

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).

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