Geodesic
An Invariant Symmetric Non-Selfadjoint Differential Operator
Erik G. F. Thomas1, 2
1Mathematisch Instituut, Universiteit Groningen, Postbus 800, 9700 AV Groningen, The Netherlands
2[
Journal of Lie Theory, Tome 12 (2002) no. 1, pp. 245-257 
Erik G. F. Thomas. An Invariant Symmetric Non-Selfadjoint Differential Operator. Journal of Lie Theory, Tome 12 (2002) no. 1, pp. 245-257. http://geodesic.mathdoc.fr/item/JOLT_2002_12_1_a11/
@article{JOLT_2002_12_1_a11,
     author = {Erik G. F. Thomas},
     title = {An {Invariant} {Symmetric} {Non-Selfadjoint} {Differential} {Operator}},
     journal = {Journal of Lie Theory},
     pages = {245--257},
     year = {2002},
     volume = {12},
     number = {1},
     url = {http://geodesic.mathdoc.fr/item/JOLT_2002_12_1_a11/}
}
TY  - JOUR
AU  - Erik G. F. Thomas
TI  - An Invariant Symmetric Non-Selfadjoint Differential Operator
JO  - Journal of Lie Theory
PY  - 2002
SP  - 245
EP  - 257
VL  - 12
IS  - 1
UR  - http://geodesic.mathdoc.fr/item/JOLT_2002_12_1_a11/
ID  - JOLT_2002_12_1_a11
ER  - 
%0 Journal Article
%A Erik G. F. Thomas
%T An Invariant Symmetric Non-Selfadjoint Differential Operator
%J Journal of Lie Theory
%D 2002
%P 245-257
%V 12
%N 1
%U http://geodesic.mathdoc.fr/item/JOLT_2002_12_1_a11/
%F JOLT_2002_12_1_a11

Voir la notice de l'article provenant de la source Heldermann Verlag

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.