Table Of ContentLecture Notes in Mathematics 2323
Alan Carey
Galina Levitina
Index Theory
Beyond
the Fredholm
Case
Lecture Notes in Mathematics
Volume 2323
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Alan Carey (cid:129) Galina Levitina
Index Theory Beyond
the Fredholm Case
AlanCarey GalinaLevitina
SchoolofMathematicsandApplied MathematicalSciencesInstitute
Statistics AustralianNationalUniversity
UniversityofWollongong Canberra,ACT,Australia
Wollongong,NSW,Australia
ISSN0075-8434 ISSN1617-9692 (electronic)
LectureNotesinMathematics
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Preface
These notes are about extending index theory to some examples where non-
Fredholm operators arise. They are far from comprehensive. We have focussed
on one aspect of the problemof what replacesthe notion of spectral flow and the
Fredholmindexwhentheoperatorsinquestionhavezerointheiressentialspectrum.
Most work in this topic stems from the so-called Witten index that is discussed
at length here. The new direction described in these notes is the introduction of
‘spectralflowbeyondtheFredholmcase’.
Spectral flow was introduced by Atiyah-Patodi-Singer [APS76] for elliptic
operators on odd dimensional compact manifolds. They argued that it could be
computed from the Fredholm index of an elliptic operator on a manifold of one
higher dimension. A general proof of this fact was produced by Robbin-Salamon
[RS95]butwasrestrictedtooperatorswithdiscretespectrumasoccurinthestudy
ofellipticoperatorsoncompactmanifolds.
In [GLM+11], a start was made on extending the Robbin-Salamon theorem
to operators with some essential spectrum as occurs on non-compact manifolds.
Thenewingredientintroducedtherewastoexploitscatteringtheoryfollowingthe
fundamental paper [Pus08]. These results do not apply to differential operators
directly, only to pseudo-differentialoperators on manifolds, due to the restrictive
assumption that spectral flow is only considered between an operator and its
perturbationbyarelativelytrace-classoperator.
In these lecture notes, we give an expository account of the main results of
these earlier papers and their generalisation to the study of spectral flow between
anoperatoranda perturbationsatisfyinga higherpth Schattenclassconditionfor
0 ≤ p < ∞, thus allowing differentialoperatorson manifolds of any dimension
d < p +1. To achieve this, we establish an operator trace formula that does not
assumeanyellipticityorFredholmpropertiesatall.Thisoperatortraceformulais
motivatedbyBenameuretal.[BCP+06]andCareyetal.[CGK16].Wethenexplain
someapplicationstoindextheoryandspectralflow.Examplesmaybeobtainedby
using Dirac type operators on L (Rd) for arbitrary d ∈ N (see Sect.7.1). In this
2
setting, Theorem 6.2.2 substantially extends [CGG+16, Theorem 3.5], where the
cased =1wastreated.
v
vi Preface
We now briefly explain the central point of these notes using notation listed
in the Notations. Our discussion focuses on a family {B(t)}t∈R of bounded self-
adjointoperatorsontheHilbertspaceHthatarep-relativetrace-classperturbations
of an unbounded self-adjoint operator A− (see Hypothesis 3.2.5 for the precise
assumptions).DenotebyB+ theuniformnormasymptoteast → ∞ofthefamily
{B(t)}t∈R. Introduce the model operator DA in L2(R;H) by first defining A in
L2(R;H)as(Af)(t)=(A−+B(t))f(t)andthensetting
d
DA = +A, dom(DA)=W1,2(R;H)∩dom(A−).
dt
ThenwithA+ = A−+B+,andAs = A−+sB+,s ∈ [0,1],weobtainthemain
fact,namely,whatwecallthe‘principaltraceformula’forthesemigroupdifference
forallu>0,
(cid:2) (cid:3) (cid:2) (cid:3) ˆ (cid:2) (cid:3)
tr e−uDAD∗A −e−uD∗ADA =− u 1/2 1tr e−uA2s(A+−A−) ds,
π
0
where,asaresultofourassumptions,theoperatordifferenceunderthetraceonthe
left-handsideisindeedtrace-classandtheintegralontheright-handsideconverges.
Toappreciatethesignificanceofthisprincipaltraceformula,thesenotesdevelop
severalapplications.Forthisweexploitsomerecentdevelopmentsinthetheoryof
Krein’s spectral shift function from quantum mechanicalscattering theory. Going
backtothemotivatingworkofPushnitski,wediscusstheproofofageneralisation
ofhisresultthatrelatesthespectralshiftfunctionofthepair(A+,A−)withthatfor
the pair (H ,H ) (see Theorem6.3.1).Thenwe proveresults that relate spectral
2 1
flow along the path {A− + B(t)}∞t=−∞ to the Fredholm or Witten index of DA
in the sense that we establish a theorem of Robbin-Salamon type for sufficiently
regular paths of self-adjoint Fredholm operators joining operators A± that have
some essential spectrum outside zero. An exposition of the theory of the spectral
shiftfunctionisincluded.
As the principaltrace formulaappliesalso to paths of non-Fredholmoperators
joiningA±,weobtainaformulafortheWittenindexunderregularityassumptions
ontherespectivespectralshiftfunctions.Thisleadsustointroduceanotionof‘gen-
eralised spectralflow’ for such pathsand to investigateits properties.Importantly
weconsiderarangeofexamplestowhichthesemoreabstractconsiderationsapply.
Properties of this extended notion of spectral flow are described, and their proofs
exploitthedoubleoperatorintegraltechniquedueoriginallyto[BS66].Weinclude
inthesenotessomebackgroundondoubleoperatorintegralsincludingasimplified
introduction.
Onemaythinkoftheseresultsinthefollowingterms.TheAtiyah-Singerindex
theoremgivesatopologicalexpressionfortheFredholmindexofellipticoperators.
Forthe case of Diracoperatorson evendimensionalcompactmanifolds,one may
formulate the index theorem using a model operator of the same general form as
DA.Thusthe choiceofthis‘modeloperator’isnotarbitrarybutisadaptedtothe
Preface vii
classicalsetting.Wealsonotethatinsupersymmetricquantummechanicsasstudied
byWitten[Wit82],theHamiltoniansinvolvethesametypeofmodeloperator.
Forthegeneraloperatorsweconsiderhere,thatmayhavezerointheiressential
spectrum,thereisnotopologicalformulaforeithertheWittenindexorgeneralised
spectral flow. Rather, what we find is that scattering theory, which is naturally
relevantinunderstandingtheessentialspectrum,providesthroughthespectralshift
function,expressionsforbothofthese quantitiesandalso connectsthem.Inorder
to understandthispointof view,one also needsthe backgroundmaterialincluded
inthesenotesonthespectralshiftfunction,theanalyticapproachtospectralflow,
theWittenindexandthedoubleoperatorintegraltechnique.
These lecture notes are based in part on our recent papers [CLPS22] and
[CGL+22] and also on lectures given at meetings in BIRS, Santiago and Mün-
ster. We have, however, incorporated background from other sources. There are
expository sections in each chapter that also include some survey material from
the literature going back to the origins of the topics covered. Conversely some
technical details are omitted where they may be found in [CLPS22] and do not
addsubstantiallytothediscussion.
Wollongong,NSW,Australia AlanCarey
Canberra,ACT,Australia GalinaLevitina
Acknowledgements
A.C.thankstheErwinSchrödingerInternationalInstituteforMathematicalPhysics
(ESI), Vienna, Austria, for fundingsupportfor this collaborationin the form of a
Research-in Teams project, ‘Scattering Theory and Non-Commutative Analysis’,
forthedurationofJune22toJuly22,2014.G.L.isindebtedtoGeraldTeschland
theESIforakindinvitationtovisittheUniversityofVienna,Austria,foraperiod
of 2 weeks in June/July of 2014. The authors thank BIRS for funding a focussed
researchgrouponthetopicoftheselecturenotesinJune2017andalsogratefully
acknowledgefinancialsupportfromtheAustralianResearchCouncil.A.C. thanks
the Humboldt Stiftung and the University of Münster Mathematics Institute for
support.G.L.acknowledgesthesupportofAustraliangovernmentscholarships.We
thank Fritz Gesztesy, Jens Kaad, Harald Grosse, Denis Potapov, Fedor Sukochev
and Dmitriy Zanin for many conversations on the subject matter of these notes.
Finallywearegratefultotherefereesforcarefullyreadingthemanuscript.
ix
Notations
Herewesummarisenotationthatisusedwithoutcommentinthetext.
For a Banach space X, we denote by L(X) the algebra of all linear bounded
operatorsonX.Wedenoteby1theidentitymappingonX.
In case whenX = H isa separablecomplexHilbertspace H, we usenotation
(cid:8)·(cid:8)fortheuniformnorm.
For an operatorA ∈ L(H) we denote by Re(A) (respectively,Im(A)) the real
(respectively,imaginary)partofA.
We write K(H) for the compact operators on H. The Calkin algebra is the
quotientalgebraL(H)/K(H).
Thecorresponding(cid:3) -basedSchatten–vonNeumannidealsonHaredenotedby
p
L (H),withassociatednormabbreviatedby(cid:8)·(cid:8) ,p≥1.Moreover,tr(A)denotes
p p
theclassicaltraceofatrace-classoperatorA∈L (H).
1
We use the symbols n-lim and s-lim to denote the operator norm limit (i.e.,
convergenceinthetopologyofL(H)),andtheoperatorstronglimit.
If T is a linear operator mapping (a subspace of) a Hilbert space into another,
thendom(T)andker(T)denotethedomainandkernel(i.e.,nullspace)ofT.The
closureofaclosableoperatorS isdenotedbyS.
ThespectrumandresolventsetofaclosedlinearoperatorinHwillbedenotedby
σ(·),andρ(·),respectively.Theessential(respectively,point)spectrumofaclosed
linearoperatorinHisdenotedbyσ (·)(respectively,byσ (·)).
ess p
The spectral projectionsof a self-adjointoperatorS in H we denote by E (·).
S
The space of all (essentially) bounded dE -measurable functions is denoted by
S
L∞(R,dES).
Thenotation[·,·]standsforcommutatoroftwooperators,thatis
[A,B]=AB−BA,
wheneveritisclearhowtodealwithdomainsofpossibleunboundedoperators.
ForaFredholmoperatorT,wedenotebyindex(T)itsFredholmindex.
Unlessexplicitlystatedotherwise,wheneverwewriteL (Rd)(L (0,∞)etc.)
p p
we assume the classical Lebesgue measure on Rd ((0,∞) etc.). The space of all
xi
