\mathbf R$, there is a unique $a_2\in {\mathcal S}_s '({\mathbf R^{2d}})$ such that ${\operatorname{Op}}_{t_1}(a_1) = {\operatorname{Op}}_{t_2} (a_2)$. By Section 18.5 in [@Ho1], the relation between $a_1$ and $a_2$ is given by $$\label{calculitransform}
{\operatorname{Op}}_{t_1}(a_1) = {\operatorname{Op}}_{t_2}(a_2)
\quad \Longleftrightarrow \quad
a_2(x,\xi )=e^{i(t_1-t_2){\langle D_x ,D_\xi\rangle}}a_1(x,\xi ).$$

We also recall that ${\operatorname{Op}}_t(a)$ is a rank-one operator, i.e. $$\label{trankone}
{\operatorname{Op}}_t(a)f=(2\pi )^{-d/2}(f,f_2)f_1, \qquad f\in {\mathcal S}_s({\mathbf R^{d}}),$$ for some $f_1,f_2\in {\mathcal S}_s '({\mathbf R^{d}})$, if and only if $a$ is equal to the *$t$-Wigner distribution* $$\label{wignertdef}
W_{f_1,f_2}^{t}(x,\xi ) \equiv {\mathscr F}(f_1(x+t{\, \cdot \, })\overline{f_2(x-(1-t){\, \cdot \, })} )(\xi ),$$ of $f_1$ and $f_2$. If in addition $f_1,f_2\in L^2({\mathbf R^{d}})$, then $W_{f_1,f_2}^{t}$ takes the form $$\label{wignertdef2}
W_{f_1,f_2}^{t}(x,\xi ) = (2\pi )^{-d/2}\int _{{\mathbf R^{d}}} f_1(x+ty)
\overline{f_2(x-(1-t)y)}e^{-{\langle y,\xi\rangle}} \, dy.$$ (Cf. [@BoDoOl1].) Since the Weyl case is of peculiar interests, we also set $W_{f_1,f_2}=W_{f_1,f_2}^{t}$, when $t=1/2$.

Schatten-von Neumann classes
----------------------------

Let ${\mathcal B}(V_1,V_2)$ denote the set of all linear and continuous operators from the quasi-normed space $V_1$ to the quasi-normed space $V_2$, and let ${\Vert {\, \cdot \, }\Vert _{{\mathcal B}(V_1,V_2)}}$ denote corresponding quasi-norm. Let ${\mathscr H}_k$, $k=1,2,3$, be Hilbert spaces and $T\in {\mathcal B}({\mathscr H}_1, {\mathscr H}_2)$. Then the *singular value* of $T$ of order $j\ge 1$ is 