JournalsjstVol. 12, No. 1pp. 301–338

Invariant subspaces of elliptic systems II: Spectral theory

  • Matteo Capoferri

    Cardiff University, UK
  • Dmitri Vassiliev

    University College London, UK
Invariant subspaces of elliptic systems II: Spectral theory cover
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Abstract

Consider an elliptic self-adjoint pseudodifferential operator AA acting on mm-columns of half-densities on a closed manifold MM M, whose principal symbol is assumed to have simple eigenvalues.We show that the spectrum of AA decomposes, up to an error with superpolynomial decay, into mm distinct series, each associated with one of the eigenvalues of the principal symbol of AA. 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 L2(M)L^2 (M) into almost-orthogonal almost-invariant subspaces under the action of both AA and the hyperbolic evolution.

Cite this article

Matteo Capoferri, Dmitri Vassiliev, Invariant subspaces of elliptic systems II: Spectral theory. J. Spectr. Theory 12 (2022), no. 1, pp. 301–338

DOI 10.4171/JST/402