Weak operator-valued tempered distribution on a common domain

Let $S=\mathcal{S}({\mathbb{R}}^4;\mathbb{C})$ with its Schwartz topology, let $H$ be a complex Hilbert space, and let $D\subseteq H$ be a dense complex linear subspace. Use the inner product linear in its first argument and conjugate-linear in its second, as in Real and complex inner-product spaces and their induced length. A weak operator-valued tempered distribution on the common domain $D$ is a complex-linear map $\mathrm{\Phi }:S⟶\mathrm{Lin}(D),$ where $\mathrm{Lin}(D)$ is the vector space of linear maps $D\to D$, such that for every $u,v\in D$ the scalar map $f⟼(\mathrm{\Phi }(f)u,v{)}_H$ is a continuous complex-linear functional on $S$, hence an element of $S^{\mathrm{\prime }}={\mathcal{S}}^{\mathrm{\prime }}({\mathbb{R}}^4)$ by Tempered distribution. In particular, every smeared map preserves the same domain $D$. This definition is weak in the matrix-coefficient sense; it does not define point values $\mathrm{\Phi }(x)$, operators on all of $H$, or self-adjoint closures.