we introduce the set $s_{t,p}^q$ consisting of all symbols $a$ such that ${\operatorname{Op}}_t(a)$ belongs to ${\mathscr I}_p$ and such that the orthonormal sequences of the eigenfunctions to $|{\operatorname{Op}}_t(a)|$ and $|{\operatorname{Op}}_t(a)^*|$ are bounded sets in the modulation space $M^{2q}$. It follows that $s_{t,p}^q$ is contained in $s_{t,p}$ the set of all symbols $a$ such that ${\operatorname{Op}}_t(a)\in {\mathscr I}_p$.

We prove that ${\mathscr S}$ is continuously embedded in $s_{t,p}^q$, and that $$s_{t,p}^p \cap {\mathscr E}' \subseteq {\mathscr F}L^p\cap {\mathscr E}' \subseteq s_{t,p} \cap {\mathscr E}'$$ for every $p>0$.

Finally we remark that in [@DeRu1; @DeRu2; @DeRu3; @FiRu], Delgado, Fischer, Ruzhansky and Turunen deal with various kinds of continuity and compactness questions for pseudo-differential operators acting on functions defined on suitable Lie groups. In their approach, matrix-valued symbols appear naturally, and several interesting results on matrices are deduced. A part of these investigations are related to the analysis in Section \[sec2\].

Acknowledgement {#acknowledgement .unnumbered}
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The author is very grateful to K. H. Gr[ö]{}chenig, for valuable advices and comments, leading to several improvements of the content and the style. He is also grateful to P. Wahlberg for valuable comments.

Preliminaries {#sec1}
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In this section we recall some facts on Gelfand-Shilov spaces, modulation 