Invariant subspaces of elliptic systems I: pseudodifferential 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.

Fixed typos and edited text in accordance with referees' recommendations

References in corpus (3)

• Invariant subspaces of elliptic systems II: spectral theory
• Diagonalization of elliptic systems via pseudodifferential projections
• Spectral properties of the Neumann-Poincaré operator in 3D elasticity

Cited by in corpus (6)

• Invariant subspaces of elliptic systems II: spectral theory
• Diagonalization of elliptic systems via pseudodifferential projections
• An index theorem on asymptotically static spacetimes with compact Cauchy surface
• Topological obstructions to the diagonalisation of pseudodifferential systems
• Two-term spectral asymptotics in linear elasticity
• Spectral asymptotics for linear elasticity: the case of mixed boundary conditions