Spectral theory of pseudo-differential operators on s1

Abstract

For a bounded pseudo-differential operator with the dense domain C∞(S1) on Lp(S1), the minimal and maximal operator are introduced. An analogue of Agmon-Douglis-Nirenberg [1] is proved and then is used to prove the uniqueness of the closed extension of an elliptic pseudo-differential operator of symbol of positive order. We show the Fredholmness of the minimal operator. The essential spectra of pseudo-differential operators on S1 are described.