it is valuable to decide whether an operator belongs to ${\mathscr I}_p$ or not, also in the case $p<1$. Here we remark that convenient Schatten-$p$ results with $p<1$ can be found for Hankel and Toeplitz operators in e.g. [@Is], and for pseudo-differential operators on compact Lie groups in e.g. [@DeRu1; @DeRu2; @DeRu3].

In the paper we deduce weighted extensions of –, where in contrast to [@Toft3; @Toft5; @Toft8; @Toft11], the case $p<1$ is included. First we deduce continuity and Schatten-von Neumann properties for suitable types of matrix operators. Thereafter we carry over these results to the case of pseudo-differential operators with symbols in modulation spaces, using Gabor analysis as link, in analogous ways as in e.g. [@GH1; @Gc3; @Gc4; @GrSt; @Toft5; @WaSc].

Here we remark that our analysis is comprehensive compared to [@GH1; @Gc3; @Gc4; @GrSt; @Toft5; @WaSc] because of the absent of local-convexity. The situation is handled by using the Gabor analysis in [@GaSa; @Toft12], for non-local convex modulation spaces, in combination of suitable factorization techniques for matrix operators.

In order to shed some more light we explain some consequences of our investigations. As a special case of Theorem \[thmOpSchatten\] we have $$\tag*{(\ref{ModSchatten1})$'$}
{\operatorname{Op}}(a) \in {\mathscr I}_p
\quad \text{when} 