  We describe the multiplier algebra of the noncommutative Schwartz space. This
multiplier algebra can be seen as the largest ${}^*$-algebra of unbounded
operators on a separable Hilbert space with the classical Schwartz space of
rapidly decreasing functions as the domain. We show in particular that it is
neither a $\mathcal{Q}$-algebra nor $m$-convex. On the other hand, we prove
that classical tools of functional analysis, for example, the closed graph
theorem, the open mapping theorem or the uniform boundedness principle, are
still available.
