{
  "id": "2103.14325",
  "version": "v1",
  "published": "2021-03-26T08:31:25.000Z",
  "updated": "2021-03-26T08:31:25.000Z",
  "title": "Invariant subspaces of elliptic systems I: pseudodifferential projections",
  "authors": [
    "Matteo Capoferri",
    "Dmitri Vassiliev"
  ],
  "categories": [
    "math.AP",
    "math-ph",
    "math.MP",
    "math.SP"
  ],
  "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.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2021-03-26T08:31:25.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "58J40",
      "47A15",
      "35J46",
      "35J47",
      "35J48",
      "58J05"
    ],
    "keywords": [
      "invariant subspaces",
      "elliptic systems",
      "distinct sign definite pseudodifferential operators",
      "elliptic self-adjoint pseudodifferential operator",
      "orthonormal pseudodifferential projections"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}