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][1], is there anything we can say about the resulting topological vector space?  Evidently, the resulting space [will not be a Fréchet space][2], 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. . .

  [1]: http://en.wikipedia.org/wiki/Strong_topology_(polar_topology)
  [2]: http://mathoverflow.net/questions/63383/which-frechet-spaces-have-a-dual-that-is-a-frechet-space