Lemma: Null defect of an extension change

Let $s\in {\mathcal{D}}^{\prime }(X)$ with $\mathrm{supp}s\subseteq Z$. Then ${\mathsf{Rec}}_s{|}_H=0$ iff $s\in {\mathcal{S}}_Z(T^{\circ })$.

Proof. By (R1), ${\mathsf{Rec}}_s(a)=0$ iff ${\beta }_{\gamma }(a)s=0$ for all $\gamma$. As $a$ ranges over $H$, $\beta (a)$ ranges over all of $\mathrm{Ann}(T^{\circ }{)}^{M_r}$ by (R2). So ${\mathsf{Rec}}_s{|}_H=0$ iff $gs=0$ for all $g\in \mathrm{Ann}(T^{\circ })$, i.e. iff $⟨s,g\varphi ⟩=0$ for all $g\in \mathrm{Ann}(T^{\circ })$, $\varphi \in \mathcal{D}(X)$. The functions $g\varphi$ are exactly the elements of $\mathrm{Ann}(T^{\circ })\cap \mathcal{D}(X)$: each $g\varphi$ lies there since $\mathrm{Ann}(T^{\circ })$ is an ideal, and each $h\in \mathrm{Ann}(T^{\circ })\cap \mathcal{D}(X)$ equals $h\varphi$ with $\varphi \in \mathcal{D}(X)$, $\varphi =1$ on $\mathrm{supp}h$.