ON THE INDEX OF ELLIPTIC CONE OPERATORS. JUAN B. GIL, PAUL LOYA, AND GERARDO MENDOZA. Abstract. We give an explicit index formula for a ...
ON THE INDEX OF ELLIPTIC CONE OPERATORS JUAN B. GIL, PAUL LOYA, AND GERARDO MENDOZA
Abstract. We give an explicit index formula for a b-elliptic scalar cone operator A in terms of the corresponding local index density ωA and a second term that resembles the eta invariant. We show that the ‘eta’ term is the index of a perturbation of the identity by a so-called smoothing Mellin operator constructed from the conormal symbol of A.
1. Introduction Let M be a smooth compact manifold with boundary, let m be a smooth positive b-measure on M . With respect to a suitable choice of a collar neighborhood π : U → ∂M of the boundary and globally defined defining function x we may assume m = x1 dx ⊗ m∂M in U where m∂M is a smooth positive density on ∂M . Let A ∈ x−ν Diff m b (M ) be b-elliptic, ν > 0 (see e.g. Melrose [8] for the notation). Regard A as an unbounded operator A : Cc∞ (M ) ⊂ xµ L2b (M ) → xµ L2b (M ) and denote by Dmin (A) the domain of the closure of A. It is convenient to assume µ = −ν/2; we can always reduce to this case by conjugation with xµ+ν/2 . In this paper, we retake the question of finding the index of the closed extensions of A and give an explicit formula for the index of its closure. To this end we first prove that, for the purpose of index calculations, some significant simplifications can be made. In fact, one can reduce the problem to the case where the operator has coefficients independent of x near ∂M , and even more, one can assume Dmin (A) to be the weighted Sobolev space xν/2 Hbm (M ). These results show that simplifying assumptions made by various authors can indeed be used without lost of generality. Our index formula relies on a factorization theorem proposed by Schulze and proved by Witt [14]. Using this result we conveniently split the index of the given operator A as the sum of two indices; the one expressed in terms of the local index density manufactured from the totally characteristic symbol of A, and the other depending on its conormal symbol. The idea is to factorize A as B(1+H) (modulo a compact operator), where B is a cone pseudodifferential operator without boundary spectrum and H is a so-called smoothing Mellin operator. It turns out that ind B can be locally computed via heat trace asymptotics, and for ind(1 + H) one can use the results by Piazza [11] to express it as an explicit integral which is the winding ˆ number of the curve defined by the Mellin symbol 1 + H(σ) along a prescribed line. Let us remark that our formula is essentially independent of the specific factorization of A. As a matter of fact, the index of B only depends on its homogeneous symbols down to order − dim M which are the same as those of A, and the boundary spectra of 1 + H and A are exactly the same. In the case where the conormal symbol of A satisfies a certain symmetry condition, we recover in a very simple way the already existing formulas (cf. [4, 13]). 2000 Mathematics Subject Classification. Primary 58J20; Secondary 58J05. Key words and phrases. index theorem, manifolds with conic singularities. 1
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JUAN B. GIL, PAUL LOYA, AND GERARDO MENDOZA
2. Invariance properties of the index Let A ∈ x−ν Diff m b (M ) be b-elliptic. By Proposition 3.14 in [5] (cf. also [7]), every closed extension AD on x−ν/2 L2b (M ) is Fredholm and ind AD = ind ADmin + dim D/Dmin . Since dim D/Dmin is completely determined by the boundary spectrum of A, the symbolic information of AD involved in the index is encoded in the index of ADmin . Thus it suffices to study the index of the closure of A. We shall need the following lemma which also establishes the notation. Lemma 1. On Dmin (A), with small enough ε > 0, the operator norm kukA = kukx−ν/2 L2b + kAukx−ν/2 L2b and the norm kukA,ε = kukxν/2−ε L2b + kAukx−ν/2 L2b . are equivalent. Proof. Recall that the embedding xν/2−ε L2b ,→ x−ν/2 L2b is continuous for ε < ν. The equivalence of the norms follows from the continuity of (Dmin (A), k · kA ) ,→ xν/2−ε L2b which is a consequence of the closed graph theorem. ∂ . The operator A = x−ν P is said to have coefficients indepenWrite Dx = −i ∂x dent of x near the boundary if (xDx )P = P (xDx ) near ∂M . Write A = A0 + xA1 with A0 having coefficients independent of x near ∂M . Let ϕ ∈ Cc∞ (R), ϕ = 1 near 0. Furthermore, for τ > 0 let ϕτ = ϕ(x/τ ) and let
A[τ ] = ϕτ A0 + (1 − ϕτ )A. Clearly, A and A[τ ] have the same conormal symbol. Proposition 2. For small enough τ > 0 the operator A[τ ] is also b-elliptic and therefore Dmin (A[τ ] ) = Dmin (A). Moreover, as τ → 0, A[τ ] → A in the graph norm of A. Thus, on Dmin (A), ind A[τ ] = ind A for every small τ > 0. Proof. Let σ(A) denote the totally characteristic principal symbol of A. Then, σ(A[τ ] ) = ϕτ σ(A0 ) + (1 − ϕτ ) σ(A) = ϕτ σ(A) + (1 − ϕτ ) σ(A) − xϕτ σ(A1 ) = σ(A) − τ ϕ˜τ σ(A1 ) with ϕ˜τ = (x/τ )ϕ(x/τ ). Since ϕ˜τ is bounded, τ ϕ˜τ is small for τ small, and thus the invertibility of σ(A) implies that of σ(A) − τ ϕ˜τ σ(A1 ) for such τ . Hence A[τ ] is b-elliptic too. Since A and A[τ ] have the same conormal symbol, part 1 of [5, Proposition 4.1] gives that Dmin (A[τ ] ) = Dmin (A). Further, from the b-ellipticity of A it follows that there is a bounded parametrix Q : xγ Hbs → xγ+ν Hbs+m such that R = I − QA : xγ Hbs → xγ Hb∞
ON THE INDEX OF ELLIPTIC CONE OPERATORS
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is bounded for all s and γ. Write A − A[τ ] = xϕτ A1 = xϕτ A1 QA + xϕτ A1 R = τ ϕ˜τ A1 QA + xϕτ A1 R. Now, A1 Q : x−ν/2 L2b → x−ν/2 L2b is bounded, so if u ∈ Dmin (A), then kτ ϕ˜τ A1 QAukx−ν/2 L2b ≤ c τ kAukx−ν/2 L2b ≤ c τ kukA . Write xϕτ A1 R = τ 1−ε ( xτ )1−ε ϕτ xε A1 R and note that xε A1 R : xν/2−ε L2b → x−ν/2 L2b in continuous. Then, using Lemma 1 we get kxϕτ A1 Rukx−ν/2 L2b ≤ c˜ τ 1−ε kukxν/2−ε L2b ≤ c τ 1−ε kukA . Altogether,1 k(A − A[τ ] )ukx−ν/2 L2b ≤ C τ 1−ε kukA and thus A[τ ] → A as τ → 0.
In general, Dmin is not a Sobolev space. The problem lies in the possible presence of elements of specb (A) along the line =σ = −ν/2. However, for index purposes, one can conveniently reduce the analysis to a slightly modified operator whose closure has a Sobolev space as its domain: Proposition 3. Let A be b-elliptic. Let Aε = xε A, and regard it as an unbounded operator on x−(ν−ε)/2 L2b (M ). If ε > 0 is sufficiently small, then Aε : x(ν−ε)/2 Hbm (M ) → x−(ν−ε)/2 L2b (M ) is Fredholm, and ind Aε = ind ADmin . Proof. Write A = x−ν P with P ∈ Diff m b (M ). Let η > 0 be so small that there is no σ ∈ specb (A) with ν/2 − η ≤ =σ < ν/2 or −ν/2 < =σ ≤ −ν/2 + η. The kernel of A on tempered distributions x−∞ Hb−∞ (M ) is the same as that of P , which we’ll denote K(P ). Recall that Dmax (A) = {u ∈ x−ν/2 L2b | Au ∈ x−ν/2 L2b }. The kernel Kmax (A) of A : Dmax (A) ⊂ x−ν/2 L2b → x−ν/2 L2b consists those elements of K(P ) whose Mellin transforms are holomorphic in =σ ≥ ν/2 since these elements of K(P ) belong to x−ν/2 L2b and Au ∈ x−ν/2 L2b trivially. That is, their Mellin transforms are holomorphic on =σ > ν/2 − η. Thus Kmax (A) = Kmax (Aε ) if 0 < ε < η. On the other hand, the kernel Kmin (A) of A : Dmin (A) ⊂ x−ν/2 L2b → x−ν/2 L2b consists those elements of K(P ) whose Mellin transforms are holomorphic in =σ > −ν/2; indeed in part 1 of [5, Proposition 3.6] it is shown show that Dmin (A) = Dmax (A) ∩ xν/2−η Hbm . Thus if ε < η then Kmin (A) = Kmin (Aε ). Thus dim Kmin (A) = dim Kmin (Aε ). Finally, note that the formal adjoint of A in x−ν/2 L2b is A? = x−ν P ? , where ? P is the formal adjoint of P in L2b , and likewise A?ε = x−ν+ε P ? . Now recall that the Hilbert adjoint of ADmin is A? with domain Dmax (A? ) so the first part of the argument yields dim Kmax (A? ) = dim Kmax (A?ε ). 1Related estimates can also be found in [7, Lemma 1.3.10].
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JUAN B. GIL, PAUL LOYA, AND GERARDO MENDOZA
3. The index formula According to the previous section, we can reduce the computation of the index of the closure of a b-elliptic differential operator A to the case where A has coefficients independent of x near ∂M and such that A : xν/2 Hbm (M ) → x−ν/2 L2b (M )
(1)
is Fredholm. Under these assumptions, we give a formula for the index of A in the spirit of [1, 3, 4, 7, 10, 11, 13]. This formula holds even when A is pseudodifferential. Recently, Witt [14] proved a factorization theorem for operator-valued elliptic Mellin symbols. Using his result, it follows that there is a cone pseudodifferential operator B without boundary spectrum, and a smoothing Mellin operator H, such that A − B(1 + H) is smoothing. This implies ind A = ind B + ind(1 + H). We analyze each term on the right. First, using the Mckean-Singer identity we can write ind B as the difference of the traces of the heat operators for B ∗ B and BB ∗ where B ∗ is the Hilbert space adjoint of B. Since the operator B has no boundary spectrum, by [5] it follows that B ∗ = B ? , the formal adjoint of B. Thus ind B = Tr e−tB
?
B
?
− Tr e−tBB ,
t > 0.
Now according to [6], the right-hand side has an asymptotic expansion as t → 0 1/2m in (with additional log-terms), and the constant term is given by R powers of t ω where ω denotes the local index density of B manufactured from the local B B M totally characteristic symbols of B. Since A and B differ by operators of order −∞, they have the same complete symbols. Thus ωA = ωB and hence, Z ind B = ωA . M
On the other hand, it follows from Piazza [11] that ˆ ind(1 + H) = −ην/2 (0, 1 + H), where (cf. also [9, 10]) ˆ = ην/2 (0, 1 + H)
1 2πi
Z Tr
d −1 ˆ ˆ H(σ) (1 + H(σ)) dσ. dσ
=σ=−ν/2
As a consequence, we obtain the following explicit formula. Theorem 4. The index of (1) is given by Z Z d 1 −1 ˆ ˆ ind A = ωA − Tr H(σ) (1 + H(σ)) dσ. 2πi dσ M =σ=−ν/2
If we require the conormal symbol of A to be symmetric with respect to some line, say =σ = 0, then the index formula becomes much simpler. ˆ ˆ Corollary 5. If A(=σ) = A(−=σ) for every σ ∈ C, then Z X 1 ind A = ωA − N0 − Nτ 2 M 0
