  In this paper we introduce Schwartz operators as a non-commutative analog of
Schwartz functions and provide a detailed discussion of their properties. We
equip them in particular with a number of different (but equivalent) families
of seminorms which turns the space of Schwartz operators into a Frechet space.
The study of the topological dual leads to non-commutative tempered
distributions which are discussed in detail as well. We show in particular that
the latter can be identified with a certain class of quadratic forms, therefore
making operations like products with bounded (and also some unbounded)
operators and quantum harmonic analysis available to objects which are
otherwise too singular for being a Hilbert space operator. Finally we show how
the new methods can be applied by studying operator moment problems and
convergence properties of fluctuation operators.
