$A$ be the complex matrix $(a(j,k))_{j,k\in \Lambda}$, $p,q\in
(0,\infty ]$, $\omega$ be a map from $\Lambda ^2$ to $\mathbf R_+$, and $$\begin{gathered}
\label{haomegadef}
h_{A,p,\omega }(k) \equiv {\Vert H_{A,\omega}({\, \cdot \, },k)\Vert _{\ell ^p}},
\\[1ex]
\text{where}
\quad
H_{A,\omega}(j,k)=a(j,j-k)\omega (j,j-k).\end{gathered}$$

\[matrixset1\] Let $0<p,q\le \infty$, $\Lambda$ be as in and let $\omega$ be a map from $\Lambda ^2$ to $\mathbf R_+$.

1.  The set $\mathbb U_0(\Lambda )$ consists of matrices $(a(j,k))_{j,k\in \Lambda}$ such that at most finite numbers of $a(j,k)$ are non-zero;

2.  The set $\mathbb U^{p,q}(\omega ,\Lambda )$ consists of all matrices $A=(a(j,k))_{j,k\in \Lambda}$ such that $${\Vert A\Vert _{\mathbb U^{p,q}(\omega ,\Lambda )}} \equiv {\Vert h_{A,p,\omega }\Vert _{\ell ^q(\Lambda )}},$$ is finite, where $h_{A,p,\omega }$ is given by . Furthermore, $\mathbb U^{p,q}_0(\omega ,\Lambda )$ is the completion of $\mathbb U_0(\Lambda )$ under the quasi-norm ${\Vert {\, \cdot \, }\Vert _{\mathbb
    U^{p,q}(\omega ,\Lambda )}}$.

For conveniency we set $\mathbb U^p(\omega ,\Lambda )=\mathbb
U^{p,p}(\omega ,\Lambda )$, and if $\omega =1$ everywhere, then we set $\mathbb U^{p,q}(\Lambda )= \mathbb U^{p,q}(\omega ,
\Lambda )$ and $\mathbb U^{p}(\Lambda )=
\mathbb U^{p}(\omega ,\Lambda )$.

Pseudo-differential operators
-----------------------------

Next we recall some properties in pseudo-differential calculus. Let $s\ge 1/2$, $a\in {\mathcal S}_s 
({\mathbf R^{2d}})$, and $t\in \mathbf R$ be fixed. 