  In this paper, we study the $M$-ellipticity of Fredholm pseudo-differential
operators associated with weighted symbols on $L^p(\mathbb{R}^n)$, $1 < p <
\infty$. We also prove the G\r{a}rding's inequality for $M$-elliptic operators
and the hybrid class of pseudo-differential operators, namely SG$M$-elliptic
operators.
