Invariant subspaces of elliptic systems I: pseudodifferential projections

TL;DR

This paper establishes the existence and uniqueness of pseudodifferential projections that decompose an elliptic self-adjoint operator into invariant subspaces, providing explicit formulas and applications to physical examples.

Contribution

It introduces a method to construct and compute pseudodifferential projections that decompose elliptic operators into invariant subspaces, with explicit symbol formulas and applications.

Findings

Existence and uniqueness of orthonormal pseudodifferential projections.

Algorithm for computing full symbols of projections.

Representation of operator functions using these projections.

Abstract

Consider an elliptic self-adjoint pseudodifferential operator $A$ acting on $m$-columns of half-densities on a closed manifold $M$, whose principal symbol is assumed to have simple eigenvalues. We show existence and uniqueness of $m$ orthonormal pseudodifferential projections commuting with the operator $A$ and provide an algorithm for the computation of their full symbols, as well as explicit closed formulae for their subprincipal symbols. Pseudodifferential projections yield a decomposition of $L^2(M)$ into invariant subspaces under the action of $A$ modulo $C^\infty(M)$. Furthermore, they allow us to decompose $A$ into $m$ distinct sign definite pseudodifferential operators. Finally, we represent the modulus and the Heaviside function of the operator $A$ in terms of pseudodifferential projections and discuss physically meaningful examples.