Distribution Theory

As usual, let $\mathcal{S}$ denote the space of Schwartz test functions on ${\mathbb{R}}^n$.

Definition (Operator Valued Distribution)

Let $\mathcal{H}$ be a Hilbert space. An operator valued tempered distribution $\Phi$ (on ${\mathbb{R}}^n$) is a mapping that associates to each test function $f\in \mathcal{S}$ an unbounded linear operator $\Phi (f)$ in $\mathcal{H}$ such that

• there is a dense linear subspace $\mathcal{D}$ of $\mathcal{H}$ that is contained in the domain of all the $\Phi (f)$
• for every fixed pair of vectors $\varphi ,\psi \in \mathcal{D}$ the mapping $f\mapsto ⟨\varphi ,\Phi (f)\psi ⟩$ is a tempered distribution.