Fourier multipliers in Hilbert spaces

TL;DR

This survey explores Fourier multipliers on Hilbert spaces, focusing on invariant operators relative to spectral partitions, with applications to $L^2$ spaces on compact manifolds and elliptic operators.

Contribution

It provides a comprehensive overview of Fourier multipliers defined via spectral partitions, highlighting their properties and applications on Hilbert spaces and manifolds.

Findings

Framework for defining invariant operators on Hilbert spaces

Application to spectral analysis on compact manifolds

Connection between Fourier multipliers and elliptic operators

Abstract

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 $E$.