Norm closure of classical pseudodifferential operators does not contain Hörmander's class

Abstract

The L2-norm closure of the algebra of all zero-order classical (or polyhomogeneous) pseudodifferential operators on a closed manifold $X$ is proven not to contain Hörmander's class ${\Psi }^0(X)$. A set of generators (for that closed algebra) smaller than all zero-order classical pseudos is also described.