Beals-Cordes-type characterizations of pseudodifferential operators

We show that, if $U$ is the representation of $SO_e(n+1,1)$ on $L^2(S^n)$ given by (2.11), and $P$ is a bounded operator on $L^2(S^n)$, then $P$ belongs to $OPS_{1,0}^0(S^n)$ if and only if

is a $C^\infty$ function on $SO_e(n+1,1)$ with values in the Banach space $\mathcal L(L^2(S^n))$.