On the essential self-adjointness of pseudodifferential operators
A proof is sketched of the essential self-adjointness on the Schwartz space \({\mathcal S}\) of pseudo-differential operators with symbols in \(S^ m_{\lambda,\rho,\delta}\) class, where \(m>0\), \(0\leq \delta <\rho \leq 1\) and \(\lambda\) is a basic weight function in the sense of Kumanogo. The main result says: if p(x,\(\xi)\) satisfies \[ | D^{\alpha}_ x D^{\beta}_{\xi} p(x,\xi)| \leq C_{\alpha \beta}| p(x,\xi)+i| \lambda (x,\xi)^{-\rho | \alpha | +\delta | \beta |}, \] for all \(\alpha\), \(\beta\), then the operator \(p^ w(X,D)\) is essentially selfadjoint on \({\mathcal S}\).
scientific article; zbMATH DE number 4096078