In this note we give an informal description of a possible notion of Fourier multipliers in Hilbert spaces:
Delgado J., Ruzhansky M., Fourier multipliers in Hilbert spaces, in Integral Fourier Operators: Proceedings of a Summer School, Ouagadougou 14–25 September 2015 / Sylvie Paycha; Pierre Clavier (eds.). – Potsdam: Universitätsverlag Potsdam, 2018. link, arxiv, full volume
The full detailed paper on Fourier multipliers in Hilbert spaces and their spectral properties can be found here:
Delgado J., Ruzhansky M., Fourier multipliers, symbols and nuclearity on compact manifolds, J. Anal. Math., to appear, arxiv
Noncommutative phase space analysis 