Invariant subspaces of elliptic systems II: Spectral theory

Consider an elliptic self-adjoint pseudodifferential operator A acting on m -columns of half-densities on a closed manifold M 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,...

Full description

Saved in:
Bibliographic Details
Published inJournal of spectral theory Vol. 12; no. 1; pp. 301 - 338
Main Authors Capoferri, Matteo, Vassiliev, Dmitri
Format Journal Article
LanguageEnglish
Published European Mathematical Society Publishing House 01.01.2022
Subjects
Functions, Elliptic
Invariants
Mathematical research
Spectral decomposition (Mathematics)
United Kingdom
Online AccessGet full text
ISSN1664-039X
1664-0403
1664-0403
DOI10.4171/jst/402

Cover

More Information
Summary:Consider an elliptic self-adjoint pseudodifferential operator A acting on m -columns of half-densities on a closed manifold M 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.
ISSN:1664-039X
1664-0403
1664-0403
DOI:10.4171/jst/402