its symbol $a$ belongs to $L^p$. On the other hand, by adding further restrictions on the symbols it is possible to deduce similar sufficient conditions as for Toeplitz operators. For example, if $S(m,g)$ is an appropriate H[ö]{}rmander class parameterized with the Riemannian metric $g$ and weight function $m$ on the phase space, then $$\label{HormSchattenEquiv}
{\{ \, {\operatorname{Op}}_t(a)\, ;\, a\in S(m,g)\, \} }\subseteq {\mathscr I}_p
\qquad \Longleftrightarrow \qquad
m\in L^p.$$ (Cf. Theorems 2.1 and 2.9 in [@BuTo]. See also [@Ho0; @Ho1; @Toft4] for pre-results.)

There are several Schatten-von Neumann results for pseudo-differential operators with symbols in modulation spaces, Besov spaces and Sobolev spaces (cf. [@Toft5] and the references therein). In particular, let $M^{p,q}$ be the classical modulation space with parameters $p,q\in [1,\infty ]$, introduced by Feichtinger in [@Fei1]. Then $$\begin{gathered}
{\operatorname{Op}}(a)  \in {\mathscr I}_p 
\quad  \text{when} \quad
a\in M^{p,p}, \ p\in [1,2],\label{ModSchatten1}
\\[1ex]
{\operatorname{Op}}(a) \, :\, M^{p_1,q_1}\to M^{p_1,q_1} 
\quad  \text{when} \quad
a\in M^{\infty ,1},\ p_1,q_1\in [1,\infty ],
\label{ModCont1}
\intertext{and}
{\operatorname{Op}}(a) \, :\, M^{\infty ,\infty }\to M^{1,1} 
\quad  \text{when} \quad
a\in M^{1 ,1}.
\label{ModCont1B}\end{gathered}$$ The relation was essentially deduced by Gr[ö]{}chenig and Heil, although it seems to be well-known earlier by Feichtinger (cf. [@GH1 Proposition 4.1]). The relation was first proved in 