Spectral theory of SG pseudo-differential operators on L-p(R-n)

To every elliptic SG pseudo-differential operator with positive orders, we associate the minimal and maximal operators on L-p(R-n), 1 < p < infinity, 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 axe computed.