Fourier multipliers in Hilbert spaces

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 $E$. These notes are based on our paper arXiv:1404.6479 (to appear in J. Anal. Math.) and have been prepared for the instructional volume associated to the Summer School on Fourier Integral Operators held in Ouagadougou, Burkina Faso, where the authors took part during 14-26 September 2015