  This is a survey on a notion of invariant operators, or Fourier multipliers
on Hilbert spaces. This concept is defined with respect to a fixed partition of
the space into a direct sum of finite dimensional subspaces. In particular this
notion can be applied to the important case of $L^2(M)$ where $M$ is a compact
manifold $M$ endowed with a positive measure. The partition in this case comes
from the spectral properties of a a fixed elliptic operator $E$.
