If we equip the Schwartz space $\mathcal{S}$ with its usual Fréchet space topology, then the space of continuous linear functionals $\mathcal{S}^\ast$ is known as the space of Schwartz distributions or tempered distributions. If we equip this space with the strong topology, is there anything we can say about the resulting topological vector space? Evidently, the resulting space will not be a Fréchet space, but perhaps it will have other nice properties. In particular, I am interested in the space of continous linear operators on $\mathcal{S}^\ast$. Is there anything interesting we can say about this space?

Unfortunately, a quick google search did not turn up many sources that dealt with the particulars of the topology on $\mathcal{S}^\ast$, much less the topology on the space of continuous linear operators on $\mathcal{S}^\ast$, so a point in the right direction to a reference would also be greatly appreciated.

EDIT: After thinking about this more deeply, I realize that I am interested in a specific type of operator on $\mathcal{S}^\ast$. $\mathcal{S}$ occurs naturally inside of $L^2$, so after identifying the dual of $L^2$ with itself via the Riesz Representation Theorem, we can in turn regard $\mathcal{S}$ as a subspace of $\mathcal{S}^\ast$. With this in mind, I am interested in the operators on $\mathcal{S}^\ast$ that restrict to operators on $\mathcal{S}$.

The motivation for this question comes from quantum mechanics, where I have in mind the position and momentum operators acting on $\mathcal{S}^\ast$. I am thus interested in the operator algebra they generate. Furthermore, these of course restrict to operators on $\mathcal{S}$, and so I am likewise interseted in the operator algebra of operators on $\mathcal{S}^\ast$ that restrict to operators on $\mathcal{S}$. In particular, I would like to abstractly characterize this space.

As this is the natural space for observables in quantum mechanics (whether physicsts realize it or not), there has to be at least something known about this space. . .