Fredholm conditions for invariant operators: finite abelian groups and\n boundary value problems

2019/11/05 by Alexandre Baldare, Baldare, Alexandre, Rémi Côme +5
Mathematics · #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #Differential Equations and Boundary Problems #FOS: Mathematics #Functional Analysis (math.FA) #Operator Algebras (math.OA) #Spectral Theory in Mathematical Physics

paper · pdf · doi:10.48550/arxiv.1911.02070

openalex publication_date 2019/11/05 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We answer the question of when an invariant pseudodifferential operator is\nFredholm on a fixed, given isotypical component. More precisely, let \Γ\nbe a compact group acting on a smooth, compact, manifold M without boundary\nand let P \∈ \ψ^m(M; E₀, E₁) be a \Γ-invariant, classical,\npseudodifferential operator acting between sections of two \Γ-equivariant\nvector bundles E₀ and E₁. Let \α be an irreducible representation\nof the group \Γ. Then P induces by restriction a map \π_\α(P) :\nH^s(M; E₀)_\α \→ H^s-m(M; E₁)_\α between the \α-isotypical\ncomponents of the corresponding Sobolev spaces of sections. We study in this\npaper conditions on the map \π_\α(P) to be Fredholm. It turns out that\nthe discrete and non-discrete cases are quite different. Additionally, the\ndiscrete abelian case, which provides some of the most interesting\napplications, presents some special features and is much easier than the\ngeneral case. We thus concentrate in this paper on the case when \Γ is\nfinite abelian. We prove then that the restriction \π_\α(P) is Fredholm\nif, and only if, P is "\α-elliptic", a condition defined in terms of\nthe principal symbol of P. If P is elliptic, then P is also\n\α-elliptic, but the converse is not true in general. However, if\n\Γ acts freely on a dense open subset of M, then P is\n\α-elliptic for the given fixed \α if, and only if, it is elliptic.\nThe proofs are based on the study of the structure of the algebra \ψ^m(M;\nE)^\Γ of classical, \Γ-invariant pseudodifferential operators acting\non sections of the vector bundle E \→ M and of the structure of its\nrestrictions to the isotypical components of \Γ. These structures are\ndescribed in terms of the isotropy groups of the action of the group \Γ\non E \→ M.\n

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