Spectral theory of SG pseudo-differential operators on $L^p(\mathbb R^n)$
To every elliptic SG pseudo-differential operator with positive orders, we associate the minimal and maximal operators on $L^p(\mathbb R^n),\,1< p< \infty,$ and prove that they are equal. The domain of the minimal (= maximal) operator is explicitly computed in terms of a Sobolev space. We prove that an elliptic SG pseudo-differential operator is Fredholm. The essential spectra of elliptic SG pseudo-differential operators with positive orders and bounded SG pseudo-differential operators with orders $0,0$ are computed.