On Some Spectral Properties of Pseudo-differential Operators on $\mathbb {T}$

Abstract

In this paper we use Riesz spectral Theory and Gershgorin Theory to obtain explicit information concerning the spectrum of pseudo-differential operators defined on the unit circle $\mathbb {T} := \mathbb {R}/ 2 \pi \mathbb { Z}$. For symbols in the Hörmander class $S^m_{1 , 0} (\mathbb {T}\times \mathbb {Z})$, we provide a sufficient and necessary condition to ensure that the corresponding pseudo-differential operator is a Riesz operator in $L^p (\mathbb {T})$, $1< p < \infty$, extending in this way compact operators characterisation in Molahajloo (Pseudo-Differ Oper Anal Appl Comput 213:25–29, 2011) and Ghoberg’s lemma in Molahajloo and Wong (J Pseudo-Differ Oper Appl 1(2):183–205, 2011) to $L^p (\mathbb {T})$. We provide an example of a non-compact Riesz pseudo-differential operator in $L^p (\mathbb {T})$, $1<p<2$. Also, for pseudo-differential operators with symbol satisfying some integrability condition, it is defined its associated matrix in terms of the Fourier coefficients of the symbol, and this matrix is used to give necessary and sufficient conditions for $L^2$-boundedness without assuming any regularity on the symbol, and to locate the spectrum of some operators.