TY - JOUR
T1 - Invariant subspaces of elliptic systems I
T2 - Pseudodifferential projections
AU - Capoferri, Matteo
AU - Vassiliev, Dmitri
N1 - Funding Information:
MC was supported by a Leverhulme Trust Research Project Grant RPG-2019-240 and by a Research Grant (Scheme 4) of the London Mathematical Society . Both are gratefully acknowledged.
Publisher Copyright:
© 2022 Elsevier Inc.
PY - 2022/4/15
Y1 - 2022/4/15
N2 - 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 L2(M) into invariant subspaces under the action of A modulo C∞(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.
AB - 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 L2(M) into invariant subspaces under the action of A modulo C∞(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.
KW - Elliptic systems
KW - Invariant subspaces
KW - Pseudodifferential operators on manifolds
KW - Pseudodifferential projections
UR - http://www.scopus.com/inward/record.url?scp=85123946785&partnerID=8YFLogxK
U2 - 10.1016/j.jfa.2022.109402
DO - 10.1016/j.jfa.2022.109402
M3 - Article
AN - SCOPUS:85123946785
VL - 282
JO - Journal of Functional Analysis
JF - Journal of Functional Analysis
SN - 0022-1236
IS - 8
M1 - 109402
ER -