Minimality of quasi-Banach spaces with applications to pseudo-differential operators

We extend Feichtinger's minimality property concerning smallest non-trivial time-frequency shift invariant Banach spaces, to include quasi-Banach spaces. We also deduce that the sets of non-uniform Gabor series expansions with coefficients in $\ell ^p_{(\omega )}$ in general agrees with the modulation space $M^p_{(\omega )}$ when $p\le 1$. For $p=1$, the latter characterization was in a more general form obtained by Feichtinger in 1981. \par We use these results to prove $\operatorname{Op} (a)$ is a Schatten-$p$ operator from $M^\infty$ to $M^p$ when $a\in M^p$, $0<p\le 1$.