## Une algèbre maximale d’opérateurs pseudo-différentiels. (A maximal algebra of pseudo differential operators).(French)Zbl 0659.35115

The author considers pseudo differential operators P in the class $$L^ 0_{1,1}$$, i.e. $$P=p(x,D)$$ where the symbol p(x,$$\xi)$$ satisfies $| D^{\alpha}_ x D^{\beta}_{\xi} p(x,\xi)| \leq C_{\alpha \beta}(1+| \xi |)^{| \alpha | -| \beta |}.$ It is well known that P is not $$L^ 2$$ continuous, in general. A characterization is given of the largest self-adjoint sub-algebra $${\mathcal A}$$ of $${\mathcal L}(L^ 2)$$ consisting of pseudo differential oprators in $$L^ 0_{1,1}$$; precisely, the author proves that $${\mathcal A}=L^ 0_{1,1}\cap (L^ 0_{1,1})^*$$, i.e. $$P\in {\mathcal A}$$ if and only if the adjoint $$P^*$$ is also well defined as operator in $$L^ 0_{1,1}$$.
Reviewer: L.Rodino

### MSC:

 35S05 Pseudodifferential operators as generalizations of partial differential operators