# On some spectral properties of pseudo-differential operators on T

## TL;DR

This paper investigates the spectral properties of pseudo-differential operators on the unit circle, providing conditions for Riesz and compact operators, and analyzing their spectra using Fourier coefficients and matrix representations.

## Contribution

It offers new criteria for Riesz and compactness of pseudo-differential operators on the circle, extending classical results to $L^p$ spaces and analyzing spectra without regularity assumptions.

## Key findings

- Characterization of Riesz operators in $L^p$ spaces.
- Example of a non-compact Riesz pseudo-differential operator.
- Necessary and sufficient conditions for $L^2$-boundedness and spectral location.

## 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 H\"ormander 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 and Ghoberg's lemma 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.