Spectral Asymmetry, Zeta Functions and the Noncommutative Residue

0.33 MB·English

by  Raphael Ponge

SPECTRAL ASYMMETRY, ZETA FUNCTIONS,
AND THE NONCOMMUTATIVE RESIDUE
6
0
0
2 RAPHAE¨LPONGE

n
a Abstract. In this paper we study the spectral asymmetry of (possiblynon-
J selfadjoint)ellipticΨDO’sintermsofthe differenceofzeta functions coming

5 fromdifferentcuttings. Refiningprevious formulasofWodzicki inthecaseof
2 oddclassellipticΨDO’s,ourmainresultshaveseveralconsequenceconcerning
the local independence with respect to the cutting, the regularity at integer

] pointsofetafunctionsandageometricexpressionforthespectralasymmetry
G ofDiracoperatorswhich,inparticular,yieldsanewspectralinterpretationof
D theEinstein-Hilbertactioningravity.
.
h
t
a
m 1. Introduction
[ This paper focuses on the spectral asymmetry of elliptic ΨDO’s. Given a com-

pact Riemannian manifold Mn and a Hermitian bundle E over M, the spectral
8
v asymmetry was first studied by Atiyah-Patodi-Singer [APS1] in the case of a self-
2 adjoint elliptic ΨDO P : C∞(M,E) → C∞(M,E) in terms of the eta function,
0
1
0 (1.1) η(P;s)=TraceP|P|−(s+1), s∈C.
1
This function is meromorphic with at worst simple pole singularities and an im-
3
portant result, due to Atiyah-Patodi-Singer[APS2] and Gilkey ([Gi2], [Gi3]), is its
0
/ regularityat s=0,so that the eta invariantη(P):=η(P,0) is alwayswelldefined.
h
The residues of the eta function at other integer points are also interesting,
t
a e.g., they enter in the index formula of Bru¨ning-Seeley [BS] for first order elliptic
m
operators on a manifold with cone-like singularities
: In [Wo1]–[Wo4]Wodzicki took a different point ofview. Motivatedby anobser-
v
i vationofShubin,helookedatthespectralasymmetryofa(possiblynonselfadjoint)
X ellipticΨDOP :C∞(M,E)→C∞(M,E)oforderm>0intermsofthedifference,
r
a
(1.2) ζ (P;s)−ζ (P;s)=TraceP−s−TraceP−s, s∈C,
θ θ′ θ θ′
of zeta functions coming from different spectral cuttings L = {argλ = θ} and
θ
L = {argλ = θ′} with 0 ≤ θ < θ′ < 2π. In particular, he showed that the
θ′
spectralasymmetryofP wasencodedbythe sectorialprojectionearlierintroduced
by Burak [Bu2] and given by
1
(1.3) Π (P)= λ−1P(P −λ)−1dλ,
θ,θ′
2iπ
ZΓθ,θ′
2000 Mathematics Subject Classification. Primary58J50, 58J42;Secondary58J40.
Keywordsandphrases. Spectralasymmetry,zetaandetafunctions,noncommutativeresidue,
pseudodifferential operators.
ResearchpartiallysupportedbyNSFgrantDMS0409005.
1
where Γ is a contour separating the part of the spectrum of P contained in the
θ,θ′
opensector θ <argλ<θ′ fromthe restofthe spectrum. Moreprecisely, Wodzicki
proved the equality of meromorphic functions,
(1.4) ζ (P;s)−ζ (P;s)=(1−e−2iπs)TraceΠ (P)P−s, s∈C.
θ θ′ θ,θ′ θ
In particular,at everyinteger k ∈Z the function ζ (P;s)−ζ (P;s) is regular and
θ θ′
there we have
(1.5) ordP.lim(ζ (P;s)−ζ (P;s))=2iπResΠ (P)P−k,
θ θ′ θ,θ′
s→k
where Res denotes the noncommutative residue of Wodzicki ([Wo2], [Wo5]) and
Guillemin [Gu1].
Furthermore, Wodzicki proved in [Wo2] that the regular value ζ(P;0) is inde-
pendent of the choice of the cutting L and that the noncommutative residue of a
θ
ΨDOprojectionisalwayszero,whichgeneralizethe vanishingofthe residueatthe
origin of the eta function a selfadjoint elliptic ΨDO.
In this paper, partly motivated by a recent upsurge of interest in the spectral
asymmetry of non-selfadjoint elliptic ΨDO’s ([BK], [Sc]), we prove various results
related to the spectral asymmetry of odd class elliptic ΨDO’s as a consequence of
a refinement of the formulas (1.4)–(1.5) for such operators.
RecallthataΨDOofintegerorderissaidtobeoddclasswhenthehomogeneous
components of its symbol are homogeneous with respect to the dilation by −1. In
particular, the odd class ΨDO’s form an algebra containing all the differential
operators and the parametrices of odd class elliptic ΨDO’s.
Let P : C∞(M,E) → C∞(M,E) be an odd class ΨDO of integer order m ≥ 1
and let L ={argλ=θ} and L ={argλ=θ′} be spectral cuttings for P and its
θ θ′
principal symbol with 0≤θ <θ′ <2π. Our main results are:
(i) If dimM is odd and ordP is even then ζ (P;s) is regular at every integer
θ
point and its value there is independent of the spectral cut L (Theorem 5.1).
θ
(ii) If dimM is even, ordP is odd and the principal symbol of P has all its
eigenvalue in the open cone {θ < argλ < θ′}∪{θ+π < argλ < θ′+π}, then for
any integer k ∈Z we have
(1.6) ordP.lim(ζ (P;s)−ζ (P;s))=iπResP−k.
θ θ′
s→k
In particular,at every integer atwhich they are notsingular the functions ζ (P;s)
θ
and ζ (P;s) take on the same regular value (Theorem 5.2).
θ′
These results are deduced from a careful analysis of the symbol of the sectorial
projectionΠ (P),sothattheproofsarepurelylocalinnature. Itthusfollowsthat
θ,θ′
thetheoremsultimately holdatthelevelofthelocal zetafunctions ζ (P;0)(x)and
θ
ζ (P;0)(x), that is, the densities whose integrals yield the zeta functions ζ (P;s)
θ′ θ
andζ (P;s). Inparticular,weobtainthatifP is anoddclasselliptic ΨDOsastis-
θ′
fyingeither the assumptionsof(i) orthat of(ii), then the regularvalue ζ (P;0)(x)
θ
is independent of the choice of the spectral cutting (Theorem 5.4).
In fact, the independence with respect to the spectral cutting of the regular
values at s=0 of the local zeta functions is not true for generalΨDO’s (see [Wo1,
pp. 130-131]). Therefore, it is interesting to see that this nevertheless can happen
for a wide class of elliptic ΨDO’s.
2
Next, these results have further applications when P is selfadjoint. In this case
weshallusethesubscript↑(resp.↓)torefertoaspectralcutintheupperhalfplane
ℑλ>0 (resp. lower halfplane ℑλ<0).
First, while the aboveresults tellus thatthere aremanyintegerpoints atwhich
there is no spectral asymmetry, they also allow us to single out some points at
which the spectral asymmetry always occurs. For instance, we always have
1
(1.7) lim (ζ (P;s)−ζ (P;s))>0,
s→n i ↑ ↓
when dimM is even and P is a first order selfadjoint elliptic odd class ΨDO (see
Proposition 6.1).
Second, as the eta function η(P;s) can be nicely related to ζ (P;s)−ζ (P;s)
↑ ↓
(see [Sh, p. 114] and Section 6), we can make use of the previous results to study
η(P;s). It is a well known result of Branson-Gilkey [BG] that in even dimension
the eta function of a Dirac operatoris anentire function. We generalize this result
by proving that if ordP and dimM have opposite parities then η(P;s) is regular
at every integer point, so that when P has order 1 and dimM is even the function
η(P;s) is entire (Theorem 6.3).
ThelatterresulthasbeenindependentlyobtainedbyGrubb[Gr]usingadifferent
approach. Furthermore, it allows us to simplify in odd dimension the aforemen-
tioned index formula of Bru¨ning-Seeley [BS] for first order elliptic operators on a
manifold with cone-like singularities (see Remark 6.5).
Third, for Dirac operators our results enable us to express the spectral asym-
metry of these operators in geometric terms. More precisely, assume that M has
even dimension, that E is a Z -graded Clifford module over M equipped with a
2
unitary connection ∇E, and letD/ : C∞(M,E) → C∞(M,E) be the associated
E
Dirac operator. Then in Proposition 7.1 we show that:
- Ateveryintegerthatisnotanevenintegerbetween2andn the zetafunctions
ζ (D/ ;s) and ζ (D/ ;s) are non-singular and take on the same regular value;
↑ E ↓ E
- For k = 2,4,...,n we can express lim (ζ (D/ ;s)−ζ (D/ ;s)) as the inte-
s→k ↑ E ↓ E
gral of a universal polynomial in complete tensorial contractions of the covariant
derivatives of the curvature RM of M and the twisted curvature FE/S/ of E.
As a consequence we get a new spectral interpretation of the Einstein-Hilbert
action I = r (x) g(x)dx, which is an important issue in noncommutative
M M
geometry and we get points at which the spectral asymmetry occurs indepently of
R p
the choice of the Clifford data (E,∇E) (see Proposition 7.2 ).
The paperis organizedasfollows. InSection2werecallthe generalbackground
needed in this paper about complex powers of elliptic operators, the noncommu-
tative residue trace of Wodzicki and Guillemin and the zeta and eta functions of
elliptic ΨDO’s. In Section 3 we gather some of the main facts about the sectorial
projection of an elliptic ΨDO, but we postpone to the Appendix those concerning
its spectral interpretation. In Section 4 we give a detailed review of Wodzicki’s
results on the spectral asymmetry elliptic ΨDO’s needed in this paper. In Sec-
tion5werefinethelatterformulasforoddclassellipticΨDO’sandproveourmain
results. We then specialize these results to the selfadjoint case in Section 6 and to
Dirac operators in Section 7.
Notation. ThroughoutallthispaperweletM denoteacompactRiemannianman-
ifold of dimension n and let E be a Hermitian vector bundle over M of rank r.
3
2. General background
In this sectionwe recallthe main facts about complex powersof elliptic ΨDO’s,
the noncommutativeresidue trace ofWodzicki andGuillemin and the zeta and eta
functions of elliptic ΨDO’s.
2.1. ComplexpowersofellipticΨDO’s. Form∈CweletΨm(M,E)denotethe
spaceof(classical)ΨDO’soforderm onM actingonsectionsofE,i.e.,continuous
operators P :C∞(M,E)→C∞(M,E) such that:
- The distribution kernel of P is smooth off the diagonal of M ×M;
- In any local trivializing chart U ⊂ Rn the operator P is of the form P =
p(x,D)+R,forsomepolyhomogeneoussymbolp(x,ξ)∼ p (x,ξ)ofdegree
j≥0 m−j
mandsomesmoothingoperatorR,wherep(x,D)denotesthelinearoperatorfrom
C∞(U,Cr) to C∞(U,Cr) such that P
c
(2.1) p(x,D)u(x)=(2π)−n eix.ξp(x,ξ)uˆ(ξ)dξ ∀u∈C∞(U,Cr).
c
Z
Let P : C∞(M,E) → C∞(M,E) be an elliptic ΨDO of degree m > 0 with
principal symbol p (x,ξ) and assume that the ray L = {argλ = θ}, 0 ≤ θ < 2π,
m θ
isaspectralcuttingforp ,thatis,p (x,ξ)−λisinvertibleforeveryλ∈L . Then
m m θ
there is a conical neighborhood Λ of L such that any ray contained in Λ is also a
θ
spectral cutting for p . It then follows that P admits an asymptotic resolvent as
m
a parametrix in a suitable class of ΨDO’s with parametrized by Λ (see [Se1], [Sh],
[GS]). This allowsus to showthat, for anyclosedcone Λ′ suchthatΛ′\0⊂Λ and
for R>0 large enough, there exists C >0 such that
Λ′R
(2.2) k(P −λ)−1k ≤C |λ|−1, λ∈Λ′, |λ|≥R.
L(L2(M,E)) Λ′R
Therefore there are infinitely many rays L ={argλ= θ} contained in Λ that are
θ
not through an eigenvalue of P and any such ray is a ray of minimal growth.
On the other hand, (2.2) also implies that the spectrum of P is not C, hence
consistsofanunboundedsetofisolatedeigenvalueswithfinitemultiplicities. Thus,
wecandefine the rootspaceandRiesz projectionassociatedtoλ∈SpP byletting
−1
(2.3) E (P)=∪ ker(P −λ)j and Π (P)= (P −µ)−1dµ,
λ j≥1 λ
2iπ
ZΓ(λ)
where Γ is a direct-oriented circle about λ with a radius small enough so that
(λ)
apart from λ no other element of SpP ∪{0} lies inside Γ .
(λ)
The family {Π (P)} is a family of disjoint projections, in the sense that
λ λ∈SpP
we have Π (P)Π (P)= 0 for λ 6=µ. Moreover, for every λ∈ SpP the root space
λ µ
Eλ(P) has finite dimension and Πλ(P) projects onto Eλ(P) and along Eλ¯(P∗)⊥
(see [RN, Lec¸on 148], [GK, Sect. I.7]). In addition, since P is elliptic Π (P) is a
λ
smoothing operator and E (P) is contained C∞(M,E) (see [Sh, Thm. 8.4]).
λ
Next, assume thatthe rayL is a spectralcutting for both p andP. Thenthe
θ m
family (Pθs)s∈C of complex powers of P associated to Lθ can be defined as follows.
Thanks to (2.2) we define a bounded operator on L2(M,E) by letting
−1
(2.4) Ps = λs(P −λ)−1dλ, ℜs<0,
θ 2iπ θ
ZΓθ
(2.5) Γ ={ρeiθ;∞<ρ≤r}∪{reit;θ ≥t≥θ−2π}∪{ρei(θ−2π);r ≤ρ≤∞},
θ
4
where r >0 is smallenough so that there is no nonzero eigenvalue of P in the disc
|λ|<r and λs =|λ|seisargθλ is defined by means of the continuous determination
θ
of the argument on C\L that takes values in (θ−2π,θ). We then have
θ
(2.6) Ps1+s2 =Ps1Ps2, ℜs <0,
θ θ θ j
(2.7) P−k =P−k, k =1,2,...,
θ
where P−k denote the partial inverse of Pk, that is, the bounded operator that
inverts P on E (Pk∗)⊥ =E (P∗)⊥ and vanishes on E (Pk)=E (P).
0 0 0 0
On the other hand, the ΨDO calculus with parameter allows us to show that
Ps is a ΨDO of order ms and that the family (Ps) is a holomorphic family
θ θ ℜs<0
of ΨDO’s in the sense of [Wo2, 7.14] and [Gu2, p. 189] (see [Se1], [Sh], [GS]).
Therefore, for any s ∈ C we can define Ps as the ΨDO such that Ps = PkPs−k,
θ θ θ
where k is any integer >ℜs.
Thisgivesrisetoaholomorphic1-parametergroupofΨDO’ssuchthatordPs =
θ
ms for any s∈C. In particular, we have P0 =PP−1 =1−Π (P).
θ 0
2.2. Noncommutative residue. The noncommutative residue trace of Wodz-
icki([Wo2],[Wo5])andGuillemin[Gu1]appearsastheresidualtraceonthealgebra
ΨZ(M,E)ofΨDO’sofintegerordersinducedbytheanalyticextensionoftheusual
trace to the class ΨC\Z(M,E) of ΨDO’s of non-integer complex orders. Our exposi-
tion essentially follows that of [KV] and [CM].
First,ifQisinΨint(M,E)=∪ Ψm(M,E)thentherestrictionofitsdistri-
ℜm<−n
butionkerneltothediagonalofM×M anelementk (x,x)ofΓ(M,|Λ|(M)⊗EndE),
Q
thespaceofsmoothEndE-valueddensities. Therefore,theoperatorQistrace-class
and we have TraceQ= tr k (x,x).
M E Q
In fact, as shown in [KV] the map Q → k (x,x) has a unique analytic contin-
Q
uation Q → t (x) to thRe class ΨC\Z(M,E), where analyticity is meant in in the
Q
sense that, for every holomorphic family (Q ) with values in ΨC\Z(M,E), the
z z∈Ω
map z →t (x) is analytic with values in Γ(M,|Λ|(M)⊗EndE).
Qz
Moreover, if Q is in ΨZ(M,E) and (Q ) is a holomorphic family of ΨDO’s
z z∈Ω
defined near z = 0 such that Q = Q and ordQ = z + ordQ, then the map
0 z
z →t (x)hasatworstasimplepolesingularityatz =0insuchwaythatinlocal
Qz
trivializing coordinates we have
(2.8) res t (x)=−(2π)−n q (x,ξ)dn−1ξ,
z=0 Qz −n
Z|ξ|=1
where q (x,ξ) denotes the symbol of degree −n of Q. Since t (x) is a density
−n Qz
we see that we get a well defined EndE-valued density on M by letting
(2.9) c (x)=(2π)−n( q (x,ξ)dn−1ξ).
Q −n
Z|ξ|=1
We can now define the functionals
(2.10) TRQ= tr t (x), Q∈ΨC\Z(M,E),
E Q
ZM
Z
(2.11) ResQ= tr c (x), Q∈Ψ (M,E).
E Q
ZM
Theorem 2.1 ([KV]). 1) The functional TR is the unique analytic continuation
of the usual trace to ΨC\Z(M,E).
5
2) We have TR[Q ,Q ]=0 whenever ordQ +ordQ 6∈Z.
1 2 1 2
3) Let Q∈ΨZ(M,E) and let (Q ) be a holomorphic family of ΨDO’s defined
z z∈Ω
near z =0 such that Q =Q and ordQ =z+ordQ. Then near z =0 the function
0 z
TRQ has at worst a simple pole singularity such that res TRQ =−ResQ.
z z=0 z
The functional Res is the noncommutative residue of Wodzicki and Guillemin.
From Theorem 2.1 we immediately get:
Theorem 2.2 ([Wo2], [Gu1], [Wo5]). 1) The noncommutative residue is a lin-
ear trace on the algebra ΨZ(M,E) which vanishes on differential operators and on
ΨDO’s of integer order ≤−(n+1).
2) We have res TRQP−s =mResQ for any Q∈ΨZ(M,E).
s=0 θ
Notice also that by a well-known result of Wodzicki ([Wo4], [Ka, Prop. 5.4]; see
also [Gu3]) if M is connected and has dimension ≥ 2 then the noncommutative
Z
residue induces the only trace on Ψ (M,E) up to a multiplicative constant.
2.3. Zeta and etafunctions. ThecanonicaltraceTRallowsustodefinethezeta
function of P as the meromorphic function on C given by
(2.12) ζ (P;s)=TRP−s, s∈C.
θ θ
Then from Theorem 2.1 we obtain:
Proposition 2.3. Let Σ = {n−j; j = 0,1,...}\{0}. Then ζ (P;s) is analytic
m θ
outside Σ and on Σ has at worst simple pole singularities such that
(2.13) res ζ (P;s)=mResP−σ, σ ∈Σ.
s=σ θ θ
Notice that (2.13) is true for σ =0 as well, but in this case it gives
(2.14) res ζ (P;s)=ResP0 =Res[1−Π (P)]=0,
s=0 θ θ 0
since Π (P) is a smoothing operator. Thus ζ (P;s) is always regular at s=0.
0 θ
Finally, assume that P is selfadjoint. Then the eta function of P is the mero-
morphic function given by
(2.15) η(P;s)=TRF|P|−s, s∈C,
where F =P|P|−1 is the sign operator of P. Then using Theorem 2.1 we get:
Proposition 2.4. Let Σ = {n−j; j = 0,1,...}. Then η(P;s) is analytic outside
m
Σ and on Σ has at worst simple pole singularities such that
(2.16) res η(P;s)=mResF|P|−σ, σ ∈Σ.
s=σ
Showingthe regularityattheoriginofη(P;s)isamuchmoredifficulttaskthan
for the zeta functions. Indeed, from (2.16) we get
(2.17) res η(P;s)=mResF =m tr c (x),
s=0 E F
ZM
and examples show that c (x) need not vanish locally (see [Gi1]). Therefore,
F
Atiyah-Patodi-Singer [APS2] and Gilkey ([Gi2], [Gi3]) had to rely on global and
K-theoretic arguments to prove:
Theorem 2.5. The function η(P;s) is always regular at s=0.
6
Thisshowsthattheetainvariantη(P):=η(P;0)isalwayswelldefined. Sinceits
appearance as a boundary correcting term in the index formula of Atiyah-Patodi-
Singer[APS1],theetainvarianthasfoundmanyapplicationsandhasbeenextended
to various other settings. We refer to the surveys of Bismut [Bi] and Mu¨ller [Mu¨],
andthe referencestherein,for anoverviewofthe mainresults onthe eta invariant.
3. The sectorial projection of an elliptic ΨDO
Inthissectionwegiveadetailedaccountonthesectorialprojectionofanelliptic
ΨDO introduced by Burak [Bu2].
Let P :C∞(M,E)→C∞(M,E) be an elliptic ΨDO of order m>0 and assume
that L = {argλ = θ} and L = {argλ = θ} are spectral cuttings for both
θ θ′
P and its principal symbol p (x,ξ) with θ < θ′ ≤ θ +2π. In addition, we let
m
Λ and Λ respectively denote the angular sectors θ < argλ < θ′ and
θ,θ′ θ′,θ+2π
θ′ <argλ<θ+2π.
The sectorial projection of P associated to the angular sector Λ is
θ,θ′
1
(3.1) Π (P)= λ−1P(P −λ)−1dλ,
θ,θ′
2iπ
ZΓθ,θ′
(3.2) Γ ={ρeiθ;∞>ρ≥r}∪{reit;θ ≤t≤θ′}∪{ρeiθ′;r ≤ρ<∞},
θ,θ
wherer is smallenoughsothat nonon-zeroeigenvalueofP lies inthe disc |λ|≤r.
Inviewof(2.2)theintegral(3.1)apriori givesrisetoanunboundedoperatoron
L2(M,E) whose domain contains L2 (M,E). We actually get a bounded operator
m
thanks to:
Proposition 3.1. 1) The operator Π (P) is a ΨDO of order ≤ 0, hence is
θ,θ′
bounded on L2(M,E).
2)Thezero’thordersymbolofΠ (P)isthesectorialprojectionΠ (p (x,ξ)),
θ,θ′ θ,θ′ m
i.e., the Riesz projection onto the root space associated to eigenvalues in Λ .
θ,θ′
Proof. LetR = 1 λ−1(P−λ)−1dλ. Thentheargumentsof[Se1,Thm.3]
θ,θ′ 2iπ Γθ,θ′
can be carried through to prove that R is a ΨDO of order ≤ −1. Hence
R θ,θ′
Π (P)=PR is a ΨDO of order ≤0.
θ,θ′ θ,θ′
Next, in some local trivializing coordinates let p(x,ξ) ∼ p (x,ξ) and
j≥0 m−j
r(x,ξ) ∼ r (x,ξ) respectively denote the symbols of P and R , so
j≥0 −1−j P θ,θ′
that Πθ,θ′(PP) has symbol π(x,ξ) ∼ (−αi)!|α|∂ξαp(x,ξ)∂xαr(x,ξ). Furthermore, let
q(x,ξ) ∼ q (x,ξ;λ) be the symbol with parameter of (P −λ)−1. Then
j≥0 −m−j P
by [Se1, Thm. 2] we have
P
1 −1
(3.3) r(x,ξ)= λ−1q(x,ξ;λ)dλ= λ−1q(x,ξ;λ)dλ,
2iπ 2iπ
ZΓθ,θ′ ZΓ(x,ξ)
whereΓ isadirect-orientedboundedcontourcontainedinthesectorΛ which
(x,ξ) θ,θ′
isolates from C\Λ the eigenvalues of p (x,ξ) that lie in Λ .
θ,θ′ m θ,θ′
On the other hand, using the equality,
(3.4) P(P −λ)−1 =1+λ(P −λ)−1,
we see that λ−1(p#q)(x,ξ;λ) =λ−1+q(x,ξ;λ). Thus π(x,ξ) is equal to
−1 −1
(3.5) p#r(x,ξ)= λ−1p#q(x,ξ;λ)dλ= q(x,ξ;λ)dλ.
2iπ 2iπ
ZΓ(x,ξ) ZΓ(x,ξ)
7
Therefore, for j =0,1,... we obtain
−1
(3.6) π (x,ξ)= q (x,ξ;λ)dλ.
−j −m−j
2iπ
ZΓ(x,ξ)
Hence π (x,ξ)= −1 (p (x,ξ)−λ)−1dλ=Π (p (x,ξ)) as desired. (cid:3)
0 2iπ Γ(x,ξ) m θ,θ′ m
Next, the sectorialRroot spaces E (P) and E (P) are
θ,θ′ θ′,θ+2π
(3.7) E (P)=∔ E (P), E (P)=∔ E (P),
θ,θ′ λ∈Λθ,θ′ λ θ′,θ+2π λ∈Λθ′,θ+2π λ
where∔denotesthealgebraicdirectsumandforλ6∈SpP wemaketheconvention
that E (P)=∪ ker(P −λ)k ={0}. Then we have:
λ k≥1
Proposition 3.2. Π (P) is a projection on L2(M,E) which projects onto a sub-
θ,θ′
space containing E (P) and along a subspace containing E (P)∔E (P).
θ,θ′ 0 θ′,θ+2π
Proof. Let L and L be rays with θ < θ < θ′ < θ′ < θ+2π and such that no
θ1 θ2 1 1
eigenvaluesofP andp lieintheangularsectorsθ <argλ<θandθ′ <argλ<θ.
m 1
Thisallowsustoreplaceinthe formula(3.1)forΠ (P)theintegrationoverΓ
θ,θ′ θ,θ′
by that over a contour Γ defined as in (3.2) using θ and θ′ and a radius r
θ1,θ1′ 1 1 1
smaller than that of Γ . Then we have
θ,θ′
−1
(3.8) Π (P)2 = λ−1µ−1P2(P −λ)−1(P −µ)−1dλdµ.
θ,θ′ 4π2
ZΓθ,θ′ ZΓθ1,θ1′
Therefore, by using the identity,
(3.9) (P −λ)−1(P −µ)−1 =(λ−µ)−1[(P −λ)−1−(P −µ)−1],
we deduce that −4π2Π (P)2 is equal to
θ,θ′
P2 µ−1dµ P2 λ−1dµ
(3.10) ( )dλ+ ( )dµ,
λ(P −λ) µ−λ µ(P −µ) λ−µ
ZΓθ,θ′ ZΓθ1,θ1′ ZΓθ1,θ1′ ZΓθ,θ
from which we see that Π (P)2 = 1 λ−2P2(P −λ)−1dλ. Combining this
θ,θ′ 2iπ Γθ,θ′
with (3.4) then gives
R
1 1 P
(3.11) Π (P)2 = λ−2Pdλ+ dλ=Π (P).
θ,θ′ θ,θ′
2iπ 2iπ λ(P −λ)
ZΓθ,θ′ ZΓθ,θ′
Hence Π (P) is a projection.
θ,θ′
Next, let λ ∈ SpP. We may assume that the contour Γ does not intersect
0 (λ0)
Γ . Then thanks to (3.9) we see that 4π2Π (P)Π (P) is equal to
θ,θ′ θ,θ′ λ0
P P dλ
(3.12) dλdµ= ( )dµ.
λ(P −λ)(P −µ) P −µ λ(λ−µ)
ZΓθ,θ′ ZΓ(λ0) ZΓ(λ0) ZΓθ,θ′
Therefore, if λ lies outside Λ then Π (P)Π (P) is zero, while when λ lies
0 θ,θ′ θ,θ′ λ0 0
inside Λ using (3.4) we see that Π (P)Π (P) is equal to
θ,θ′ θ,θ′ λ0
−1 P −1 dµ −1 dµ
(3.13) dµ= + =Π (P).
2iπ µ(P −µ) 2iπ µ 2iπ P −µ λ0
ZΓ(λ0) ZΓ(λ0) ZΓ(λ0)
Since Π (P) has range E (P) it then follows that the range of Π (P) contains
λ0 λ0 θ,θ′
E (P) and its kernel contains E (P)∔E (P). Hence the result. (cid:3)
θ,θ′ 0 θ′,θ+2π
Remark 3.3. Since Propositions 3.1 and 3.2 tell us that Π (P) is a (bounded)
θ,θ′
ΨDO projection, we see that Π (P) has either order 0 or is smoothing.
θ,θ′
8
Proposition 3.4. Let L and L be spectral cuttings for P and its principal
θ1 θ1′
symbol in such way that θ′ ≤θ <θ′ <θ+2π. Then:
1 1
1) The projections Π (P) and Π (P) are disjoint.
θ,θ′ θ1,θ1′
2) We have Π (P) + Π (P) = Π (P) and Π (P) + Π (P) =
θ,θ′ θ′,θ′ θ,θ′ θ,θ′ θ′,θ+2π
1 1
1−Π (P).
0
Proof. First,using(3.9)weseethat4π2Π (P)Π (P)and4π2Π (P)Π (P)
θ,θ′ θ1,θ1′ θ1,θ1′ θ1,θ1′
are both equal to
P2
(3.14) dλdµ=
λµ(P −λ)(P −µ)
ZΓθ,θ′ ZΓθ1,θ1′
P2 µ−1dµ P2 λ−1dµ
( )dλ+ ( )dµ=0.
λ(P −λ) µ−λ µ(P −µ) λ−µ
ZΓθ,θ′ ZΓθ1,θ1′ ZΓθ1,θ1′ ZΓθ,θ
Hence Π (P) and Π (P) are disjoint projections.
θ,θ′ θ1,θ1′
Next, the operator Π (P)+Π (P) is equal to
θ,θ′ θ′,θ′
1
1 P 1 P
(3.15) dλ= dλ=Π (P),
2iπ λ(P −λ) 2iπ λ(P −λ) θ,θ1′
ZΓθ,θ′∪Γθ′,θ1′ ZΓθ,θ1′
since integrating λ−1P(P −λ)−1 along Γ ∪Γ is the same as integrating it
θ,θ′ θ′,θ′
1
alongΓ . Inthespecialcaseθ′ =θ+2π theintegrationalongΓ reducesto
θ,θ1 1 θ,θ+2π
thatalongthesmallcircle|λ|=rwithclockwiseorientation. Therefore,using(3.4)
we see that Π (P)+Π (P) is equal to
θ,θ′ θ′,θ+2π
1 P 1 dλ 1 dλ
(3.16) dλ= + =1−Π (P).
0
2iπ λ(P −λ) 2iπ λ 2iπ P −λ
Z|λ|=r Z|λ|=r Z|λ|=r
The proof is thus complete. (cid:3)
In general, the closures of E (P) and E (P)∔ E (P) don’t yield the
θ,θ′ 0 θ′,θ+2π
wholerangeandthewholekernelofΠ (P)but,asweexplaininAppendix, there
θ,θ′
are special cases where they actually do:
(i) Whenthe principalsymbolofP hasno eigenvalueswithin the angularsector
θ < argλ < θ′, which is equivalent to Π (P) being a smoothing operator (see
θ,θ′
Proposition A.3);
(ii) When P is normal, i.e., commutes with its adjoint, and in particular when
(see Proposition A.5);
(iii)WhenP hasacompletesystemofrootvectors,thatis,thesubspacespanned
by its root vector is dense (see Proposition A.7).
Inthenon-normalcaseitisadifficultissuetodeterminewhetherageneralclosed
unbounded operator on a Hilbert space admits a complete system of root vectors.
Thanks to a criterion due to Dunford-Schwartz [DS] it can be shown that P has a
completesystemofrootvectorswhenits principalsymboladmits spectralcuttings
dividing the complex planes into angular sectors of apertures < 2nπ (see [Ag],
m
[Bu1], [Agr]). Therefore, in this case Π (P) is the projection onto the closure of
θ,θ′
E (P) and along the closure of E (P).
θ,θ′ θ′,θ+2π
In fact, if we content ourselvesby determining the range of Π (P) then it can
θ,θ′
be shown that the range agrees with the closure E (P) when we only require
θ,θ′
9
the principal symbol of P to admit spectral cuttings dividing the angular sector
θ <argλ<θ′ into angular sectors of apertures < 2nπ (see Proposition A.9).
m
Detailed proofs of the above statements are given in Appendix.
4. Zeta functions and spectral asymmetry
In this section, we give a detailed review of the spectral asymmetry formulas of
Wodzicki ([Wo2]–[Wo4]) for elliptic ΨDO’s.
Let P :C∞(M,E)→C∞(M,E) be an elliptic ΨDO of order m>0. Let us first
assume that P is selfadjoint. Then we have:
(4.1) Ps =Π (P)|P|s+e−iπsΠ (P)|P|s, Ps =Π (P)|P|s+eiπsΠ (P)|P|s,
↑ + − ↓ + −
where Π (P) (resp. Π (P)) denotes the orthogonal projections onto the positive
+ −
(resp. negative) eigenspace of P. Hence we have
(4.2) Ps−Ps =(e−iπs−eiπs)Π (P)|P|s =(1−e2iπs)Π (P)Ps.
↑ ↓ − − ↑
Therefore,intheselfadjointcase,thespectralasymmetryofP isencodedbyΠ (P).
−
Suppose now that P is not selfadjoint and let L = {argλ = θ} and L =
θ θ′
{argλ=θ} be spectral cuttings for both P and its principal symbol p (x,ξ) with
m
0 ≤ θ < θ′ < 2π. As observed by Wodzicki ([Wo3], [Wo4]) in this context a
substitute to the projectionΠ (P) is providedby the sectorialprojectionΠ (P)
− θ,θ′
in (3.1). This stems from:
Proposition 4.1 ([Wo3], [Wo4]). For any s∈C we have
(4.3) Ps−Ps =(1−e2iπs)Π (P)Ps.
θ θ′ θ,θ′ θ
Proof. Since in the integral(2.4)defining Ps the value of the argumenthas shifted
θ
of −2π once λ has turned around the circle we have
e2iπs−1 r xseis(θ−2π) θ−2π rseist
(4.4) Ps = d(xeiθ)+ d(reit).
θ 2iπ P −xeiθ P −reit
Z∞ Zθ
Similarly, we have
−e2iπs+1 ∞ xseis(θ′−2π) θ′−2π rseist
(4.5) Ps = d(xeiθ′)+ d(reit).
θ′ 2iπ P −xeiθ′ P −reit
Zr Zθ′
Observe that θ−2π rseist d(reit)− θ′−2π rseist d(reit) is equal to
θ P−reit θ′ P−reit
θ−2π rRseist θ rRseist
(4.6) d(reit)− d(reit)
P −reit P −reit
Zθ′−2π Zθ′
θ−2π rseist
=(1−e2iπs) d(reit).
P −reit
Zθ′−2π
Therefore, the operator Ps−Ps agrees with
θ θ′
e2iπs−1 r xseis(θ−2π) θ′−2π rseist
(4.7) d(xeiθ)+ d(reit)
2iπ P −xeiθ P −reit
Z∞ Zθ−2π
∞ xseis(θ′−2π)
+ d(xeiθ′) .
P −xeiθ′
Zr !
10