Non-harmonic $M$-elliptic pseudo differential operators on manifolds

In this article, we introduce and study $M$-elliptic pseudo-differential operators in the framework of non-harmonic analysis of boundary value problems on a manifold $\Omega$ with boundary $\partial \Omega$, introduced by Ruzhansky and Tokmagambetov ( Int. Math. Res. Not. IMRN, (12), 3548-3615, 2016) in terms of a model operator $\mathfrak{L}$. More precisely, we consider a weighted $\mathfrak{L}$-symbol class $M_{\rho, 0, \Lambda}^{m}, m\in \mathbb{R},$ associated to a suitable weight function $\Lambda$ on a countable set $\mathcal{I}$ and study elements of the symbolic calculus for pseudo-differential operators associated with $\mathfrak{L}$-symbol class $M_{\rho, 0, \Lambda}^{m},$ by deriving formulae for the composition, adjoint, and transpose. Using the notion of $M$-ellipticity for symbols belonging to $\mathfrak{L}$-symbol class $M_{\rho, 0, \Lambda}^{m}$, we construct the parametrix of $M$-elliptic pseudo-differential operators. Further, we investigate the minimal and maximal extensions for $M$-elliptic pseudo-differential operators and show that they coincide when the symbol $\sigma\in M_{\rho, 0, \Lambda}^{m},$ is $M$-elliptic. We provide a necessary and sufficient condition to ensure that the pseudo-differential operators $T_{\sigma}$ with symbol in the $\mathfrak{L}$-symbol class $M_{\rho, 0,\Lambda}^{0}$ is a compact operator in $L^{2}(\Omega)$ or a Riesz operator in $L^{p}(\Omega).$ Finally, we prove G\"arding's inequality for pseudo-differential operators associated with symbol from $M_{\rho, 0,\Lambda}^{0}$ in the setting of non-harmonic analysis.