Invariant subspaces of elliptic systems II: spectral theory

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 that the spectrum of $A$ decomposes, up to an error with superpolynomial decay, into $m$ distinct series, each associated with one of the eigenvalues of the principal symbol of $A$. These spectral results are then applied to the study of propagation of singularities in hyperbolic systems. The key technical ingredient is the use of the carefully devised pseudodifferential projections introduced in the first part of this work, which decompose $L^2(M)$ into almost-orthogonal almost-invariant subspaces under the action of both $A$ and the hyperbolic evolution.

arXiv commentMinor edits, to appear in Journal of Spectral Theory

References in corpus (8)

• The spectral function of a first order elliptic system
• Global wave parametrices on globally hyperbolic spacetimes
• Invariant subspaces of elliptic systems I: pseudodifferential projections
• Geometric wave propagator on Riemannian manifolds
• Global propagator for the massless Dirac operator and spectral asymptotics
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Cited by in corpus (6)

• Invariant subspaces of elliptic systems I: pseudodifferential projections
• Diagonalization of elliptic systems via pseudodifferential projections
• Global propagator for the massless Dirac operator and spectral asymptotics
• On the Cauchy problem for Friedrichs systems on globally hyperbolic manifolds with timelike boundary
• Topological obstructions to the diagonalisation of pseudodifferential systems
• Two-term spectral asymptotics in linear elasticity