Consider the operator-valued generalized function $\phi^{(k)}_{l}:=\phi^{(k)}_{l}$ on space-time $\mathcal{M}$. Now, smooth the operator-valued generalized function with test function $f(x)$ s.t.$$\phi^{(k)}_{l}(f):=\int \!\mathrm{d}^4x\, \phi^{(k)}_{l}(x)f(x)$$ are linear operators in the physical Hilbert space $\mathcal{H}$.
Note: These operators are not assumed to be bounded.
Instead, we suppose that all the operators $\phi^{(k)}_{l}$ have a common domain of definition $\mathcal{D}$, which is a dense linear subspace of $\mathcal{H}$. In order for algebraic operations on the operators to make sense, it is usually further supposed that the operator-valued generalized functions map $\mathcal{D}$ into itself.
It is the last assumption that I will be asking about. In particular what goes awry if the map doesn't take $\mathcal{D}$ into itself? I am interested in both mathematical reasoning, as well as any physical motivation behind this assumption.
