Spectral Theory of Pseudo-Differential Operators on $$\mathbb{S}^1$$

Abstract

For a bounded pseudo-differential operator with the dense domain \(C^\infty(\mathbb{S}^1)\) on \(L^p(\mathbb{S}^1)\) , 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 \(\mathbb{S}^1\)are described.