  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.
