Math 2.0

Global analysis, analysis on manifolds Differential operators on manifolds Theorem

Fredholm property of elliptic operators

Builds on Elliptic operator on a compact manifoldDifferential operator between vector bundlesFredholm operatorFredholm indexSobolev spaceParametrixCompact operatorElliptic regularity

Statement

Let MM be a compact smooth manifold without boundary, with a smooth positive density. Let EE and FF be complex vector bundles over MM with Hermitian metrics. Let P ⁣:Γ(E)→Γ(F)P \colon \Gamma(E) \to \Gamma(F) be an elliptic operator of order mm. Let P∗ ⁣:Γ(F)→Γ(E)P^* \colon \Gamma(F) \to \Gamma(E) be its formal adjoint. It is the differential operator with ∫M⟨Pu,v⟩=∫M⟨u,P∗v⟩\int_M \ip{Pu}{v} = \int_M \ip{u}{P^* v} for all smooth sections uu and vv. For s∈ℝs \in \R, let Hs(E)H^s(E) and Hs(F)H^s(F) be the Sobolev spaces of sections.

  1. For every s∈ℝs \in \R, PP extends to a bounded operator Ps ⁣:Hs+m(E)→Hs(F)P_s \colon H^{s+m}(E) \to H^s(F), and PsP_s is a Fredholm operator.
  2. The kernel of PsP_s consists of smooth sections and equals the kernel of PP on Γ(E)\Gamma(E). In particular it is finite-dimensional and does not depend on ss.
  3. The kernel of P∗P^* is a finite-dimensional space of smooth sections. The range of PsP_s is the set of f∈Hs(F)f \in H^s(F) with ∫M⟨f,v⟩=0\int_M \ip{f}{v} = 0 for all vv in the kernel of P∗P^*.
  4. The index of PsP_s equals dim⁡ker⁡P−dim⁡ker⁡P∗\dim \ker P - \dim \ker P^* and does not depend on ss.

Remarks

The proof constructs a parametrix QQ, a pseudodifferential operator of order −m-m built from the inverse of the principal symbol, with QP−IQP - I and PQ−IPQ - I smoothing. Smoothing operators map HsH^s into Hs+1H^{s+1}, and by the Rellich lemma the inclusion Hs+1→HsH^{s+1} \to H^s is compact on a compact manifold. So PsP_s is invertible modulo compact operators, which is Atkinson's characterisation of Fredholm operators. The parametrix also gives elliptic regularity: if PuPu is smooth then uu is smooth, which yields part 2.

Example: on the circle S1=ℝ/2πℤS^1 = \R / 2\pi\Z, let P=−d2/dx2P = -d^2/dx^2. Its kernel and the kernel of P∗=PP^* = P are the constant functions, so the index is 1−1=01 - 1 = 0, and the range consists of the functions with average zero.

Compactness of MM is needed. On ℝ\R, the operator d/dx ⁣:H1(ℝ)→L2(ℝ)d/dx \colon H^1(\R) \to L^2(\R) is elliptic and injective, but its range is dense and not closed, so it is not Fredholm.

Going deeper

For a student

This theorem is the analytic foundation of index theory. The index depends only on the homotopy class of the principal symbol through invertible symbols, which is why it can be computed topologically by the Atiyah–Singer index theorem. It also underlies Hodge theory: applied to the Hodge Laplacian, it gives finite-dimensional spaces of harmonic forms and the orthogonal decomposition of forms.

Reading path

To understand this page from the ground up, read these 13 pages in order. The foundations come first.

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Compact operator, concept, no evidence levelCompact operatorconceptDifferential operator between vector bundles, concept, no evidence levelDifferential operator…conceptElliptic operator, concept, no evidence levelElliptic operatorconceptElliptic operator on a compact manifold, concept, no evidence levelElliptic operator on …conceptFredholm index, concept, no evidence levelFredholm indexconceptFredholm operator, concept, no evidence levelFredholm operatorconceptParametrix, concept, no evidence levelParametrixconceptSmooth function, concept, no evidence levelSmooth functionconceptSobolev space, concept, no evidence levelSobolev spaceconceptWeak derivative, concept, no evidence levelWeak derivativeconceptWeak solution, concept, no evidence levelWeak solutionconceptElliptic regularity, statement, no evidence levelElliptic regularitystatementFredholm property of elliptic operators, statement, no evidence levelFredholm property of …statement
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
  • no evidence level
  1. Compact operator concept
  2. Differential operator between vector bundles concept
  3. Elliptic operator concept
  4. Elliptic operator on a compact manifold concept
  5. Fredholm index concept
  6. Fredholm operator concept
  7. Parametrix concept
  8. Smooth function concept
  9. Sobolev space concept
  10. Weak derivative concept
  11. Weak solution concept
  12. Elliptic regularity statement
  13. Fredholm property of elliptic operators statement

Further reading

  1. Web https://en.wikipedia.org/wiki/Elliptic_operator
About this page
Tag73ANTTKAH3
URIm2://73ANTTKAH3
Kindstatement, theorem
WritingW0 stub
MSC58J05, 47A53, 58J20
StatusThis copy of the site has no build data. Run m2 status 73ANTTKAH3.
AboutElliptic operator on a compact manifold, Fredholm operator
UsesElliptic operator on a compact manifold, Differential operator between vector bundles, Fredholm operator, Fredholm index, Sobolev space, Parametrix, Compact operator, Elliptic regularity

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