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Journal of Lie Theory 12 (2002), No. 1, 245--257
Copyright Heldermann Verlag 2002

An Invariant Symmetric Non-Selfadjoint Differential Operator

Erik G. F. Thomas
Mathematisch Instituut, Universiteit Groningen, Postbus 800, 9700 AV Groningen, The Netherlands


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.

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