Fredholm conditions for invariant operators: finite abelian groups and boundary value problems
| Autor: | Alexandre Baldare, Victor Nistor, Rémi Côme, Matthias Lesch |
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| Přispěvatelé: | Institut Élie Cartan de Lorraine (IECL), Université de Lorraine (UL)-Centre National de la Recherche Scientifique (CNRS), Mathematisches Institut der Universität Bonn, Rheinische Friedrich-Wilhelms-Universität Bonn, M.L. was partially supported by the Hausdorff Center for Mathematics, Bonn., A.B., R.C., and V.N. have been partially supported by ANR-14-CE25-0012-01 (SINGSTAR), ANR-14-CE25-0012,SINGSTAR,Analyse sur les espaces singuliers et non compacts: une approche par les C*-algèbres(2014) |
| Rok vydání: | 2020 |
| Předmět: |
Pure mathematics
Algebra and Number Theory finite abelian group Group (mathematics) Mathematics - Operator Algebras pseudodifferential operator Vector bundle Boundary (topology) elliptic theory Fredholm operator isotypical component invariant operator Functional Analysis (math.FA) Mathematics - Functional Analysis Sobolev space Mathematics - Analysis of PDEs Operator algebra Compact group Irreducible representation FOS: Mathematics [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP] Abelian group Operator Algebras (math.OA) Analysis of PDEs (math.AP) Mathematics |
| Zdroj: | Journal of Operator Theory Journal of Operator Theory, Theta Foundation, 2021, 85 (1), pp.229-256. ⟨10.7900/jot.2019feb26.2270⟩ |
| ISSN: | 1841-7744 0379-4024 |
| DOI: | 10.7900/jot.2019feb26.2270 |
| Popis: | We answer the question of when an invariant pseudodifferential operator is Fredholm on a fixed, given isotypical component. More precisely, let $\Gamma$ be a compact group acting on a smooth, compact, manifold $M$ without boundary and let $P \in \psi^m(M; E_0, E_1)$ be a $\Gamma$-invariant, classical, pseudodifferential operator acting between sections of two $\Gamma$-equivariant vector bundles $E_0$ and $E_1$. Let $\alpha$ be an irreducible representation of the group $\Gamma$. Then $P$ induces by restriction a map $\pi_\alpha(P) : H^s(M; E_0)_\alpha \to H^{s-m}(M; E_1)_\alpha$ between the $\alpha$-isotypical components of the corresponding Sobolev spaces of sections. We study in this paper conditions on the map $\pi_\alpha(P)$ to be Fredholm. It turns out that the discrete and non-discrete cases are quite different. Additionally, the discrete abelian case, which provides some of the most interesting applications, presents some special features and is much easier than the general case. We thus concentrate in this paper on the case when $\Gamma$ is finite abelian. We prove then that the restriction $\pi_\alpha(P)$ is Fredholm if, and only if, $P$ is "$\alpha$-elliptic", a condition defined in terms of the principal symbol of $P$. If $P$ is elliptic, then $P$ is also $\alpha$-elliptic, but the converse is not true in general. However, if $\Gamma$ acts freely on a dense open subset of $M$, then $P$ is $\alpha$-elliptic for the given fixed $\alpha$ if, and only if, it is elliptic. The proofs are based on the study of the structure of the algebra $\psi^{m}(M; E)^\Gamma$ of classical, $\Gamma$-invariant pseudodifferential operators acting on sections of the vector bundle $E \to M$ and of the structure of its restrictions to the isotypical components of $\Gamma$. These structures are described in terms of the isotropy groups of the action of the group $\Gamma$ on $E \to M$. |
| Databáze: | OpenAIRE |
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