  This article can be seen as a sequel to the first author's article ``Chern
classes of the tangent bundle on the Hilbert scheme of points on the affine
plane'', where he calculates the total Chern class of the Hilbert schemes of
points on the affine plane by proving a result on the existence of certain
universal formulas expressing characteristic classes on the Hilbert schemes in
term of Nakajima's creation operators.
  The purpose of this work is (at least) two-fold. First of all, we clarify the
notion of ``universality'' of certain formulas about the cohomology of the
Hilbert schemes by defining a universal algebra of creation operators. This
helps us to reformulate and extend a lot of the first author's previous results
in a very precise manner.
  Secondly, we are able to extend the previously found results by showing how
to calculate any characteristic class of the Hilbert scheme of points on the
affine plane in terms of the creation operators. In particular, we have
included the calculation of the total Segre class and the square root of the
Todd class.
  Using this methods, we have also found a way to calculate any characteristic
class of any tautological sheaf on the Hilbert scheme of points on the affine
plane. This in fact gives another complete description of the ring structure of
the cohomology spaces of the Hilbert schemes of points on the affine plane.
