An Invariant Symmetric Non-Selfadjoint Differential Operator

E. G. F. Thomas

Journal of Lie theory, Tome 12 (2002) no. 1, pp. 245-257.

Résumé

Let $D$ be a symmetric left invariant differential operator on a unimodular Lie group $G$ of type $I$. Then we show that $D$ is essentially self-adjoint if and only if for almost all $\pi \in \widehat{G}$, with respect to the Plancherel measure, the operator $\pi(D)$ is essentially self-adjoint. This, in particular, allows one to exhibit a left invariant symmetric differential operator on the Heisenberg group, which is not essentially self-adjoint.