Elliptic Boundary Problems for Dirac Operators (Mathematics: Theory & Applications)

MATHEMATICS: THEORY & APPLICATIONS

ELLIPTIC BOUNDARY PROBLEMS FOR DIRAC OPERATORS BERNHELM Booji-BA VNBEK KRZYSZTOF P. WoJcIEcHowsKl

BIRKHA USER

Mathematics: Theory & Applications Editors

Richard V. Kadison Isidore M. Singer

Bernhelm BooB-Bavnbek Krzysztof P. Wojciechowski

Elliptic Boundary Problems

for Dirac Operators

Birkhäuser Boston • Basel • Berlin

1993

Bernhelm BooB-Bavnbek IMFUFA Roskilde University 4000 Roskilde Denmark

Krzysztof P. Wojciechowski Department of Mathematics IUPUI Indianapolis, IN 46202 USA

Library of Congress Cataloging In-Publication Data Booss-Bavnbek, Bernhelm, 1941Elliptic boundary problems for Dirac operators I Bernhelm Booss -Bavnbek, Krzysztof P. Wojciechowski cm. -• (Mathematics) p. Includes bibliographical references and index. ISBN 0-8176-368l-l (H : acid-free). -- ISBN 3-7643-3681-3 (H acid-free) I. Differential equations, Elliptic. 2. Boundary value problems. 3. Dirac equation. I. II. Title. III. Series:

QA377.B686 1993 515'.353--dc2O

Wojciechowski, Krzysztof p., 3953. Mathematics (Boston. Mass.) 93-22006

Birkhäuser Boston 1993

Printed on acid-free paper

CIP

Birkhä user

Copyright is not claimed for works of U.S. Government employees.

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Typeset by authors in Printed and bound by Quinn-Woodbine, Woodbine, NJ

Printed in the U.S.A.

987654321

Contents Preface List of Notations

vii

xv

I. CLIFFORD ALGEBRAS AND Diit&c OPERATORS

1. Clifford Algebras and Clifford Modules 2. Clifford Bundles and Compatible Connections 3. Dirac Operators 4. Dirac Laplacian and Connection Laplacian 5. Eudidean Examples 6. The Classical Dirac (Atiyah-Singer) Operators on Spin Manifolds 7. Dirac Operators and Chirality 8. Unique Continuation Property for Dirac Operators 9. Invertible Doubles 10. Clueing Constructions. Relative Index Theorem

3 10 19

26 29 36 40 43 50 59

PART II. ANALYTICAL AND TOPOLOGICAL TOOLS

11. Sobolev Spaces on Manifolds with Boundary 12. Calderón Projector for Dirac Operators 13. Existence of of Null Space Elements 14. Spectral Projections of Dirac Operators 15. Pseudo-Differential Grassmannians 16. The Homotopy Groups of the Space of Seif-Adjoint Fredhohn Operators A. Elementary Decompositions and Deformations B. The Homotopy Groups of C. 17. The Spectral Flow of Families of Seif-Adjoint Operators A. Continuity of Eigenvalues B. The Spectral Flow on Loops in F. C. Spectral Flow and Index D. Non-Vanishing Spectral Flow

67 75 95 105 111

127 133

138 140 145 157

III. APPLICATIONS

18. Elliptic Boundary Problems and Pseudo-Differential Projections 19. Regularity of Solutions of Elliptic Boundary Problems

163 180

vi

Contents

20. Fredhoim Property of the Operator AR 21. Exchanges on the Boundary: Type Formulas and the Cobordism Theorem for Dirac Operators 22. The Index Theorem for Atiyah-Patodi-Singer Problems A. Preliminary Remarks B. Heat Kernels on the Cylinder C. Duhamel's principle. Heat Kernels on Manifolds with Boundary D. Proof of the Index Formula E. L2-Reformulation F. The Odd-Dimensional Case. A Three-Dimensional Example 23. Some R2marks on the Index of Generalized Atiyah-Patodi-Singer Problems 24. Bojarski's Theorem. General Linear Conjugation Problems 25. Cutting and Pasting of Elliptic Operators 26. Dirac Operators on the Two-Sphere

188 .

205

211 214 231 239

242 248 253 262 276 282

Bibliography

289

Index

303

Preface

Elliptic boundary problems have enjoyed interest recently, especially among C-algebraists and mathematical physicists who want to understand single aspects of the theory, such as the behaviour of Dirac operators and their solution spaces in the case of a non-trivial boundary. However, the theory of elliptic boundary problems by far has not achieved the same status as the theory of elliptic operators on closed (compact, without boundary) manifolds. The latter is nowadays rec-

ognized by many as a mathematical work of art and a very useful technical tool with applications to a multitude of mathematical contexts. Therefore, the theory of elliptic operators on closed manifolds is well-known not only to a small group of specialists in partial differential equations, but also to a broad range of researchers who have specialized in other mathematical topics. Why is the theory of elliptic boundary problems, compared to that on closed manifolds, still lagging behind in popularity? Admittedly, from an analytical point of view, it is a jigsaw puzzle which has more

pieces than does the elliptic theory on closed manifolds. But that is not the only reason. Having worked with the topology, geometry, and analysis of elliptic boundary problems for many years, the authors feel that the existing books on elliptic boundary problems, excellent as they are, present the theory in too great a generality and by means of too intricate a machinery for our intended readership: graduate students in, for example, geometry, topology or mathematical physics, and researchers from fields other than partial differential equations. They will need insight into specific situations, which involve manifolds with boundary and their Dirac operators. Our aim is to give these readers a book that presents the absolute minimum of the necessary machinery and illustrates how the methods work with interesting examples.

To follow our presentation, the reader should be familiar with the basic calculus of pseudo-differential operators on closed manifolds. Knowledge of the first pages of P. Cilkey's book [1984; pp. 1—36] (new edition in preparation) is quite satisfactory for our purpose.

viii

Preface

Our Chapters 17 and 22—26 furthermore presuppose knowledge of the

index theory of elliptic operators on closed manifolds. Once again Gilkey's book provides the reader with all needed material. As an auxiliary reference, we also refer to BooB & Bleecker [1985].

Analytically, the heart of our book consists of deep results proved long ago by A. Calderón and R. Seeley: in our book Theorem 12.4, Theorem 13.1, Corollary 13.7, and Theorem 13.8. Our presentation closely follows, with some simplifications from elementary constructions developed in our previous work, the original work of Seeley (see Seeley [1966], [1969] and Palais [1965a; Seeley's contribution to Chapter 17]). Inevitably, some proofs are relatively long and complicated,

but still we hope that the aforementioned simplifications make the theory of elliptic boundary problems for Dirac operators accessible to a larger audience. In fact, even the theory seems to benefit from those simplifications. They lead, for example, to recognizing that the Cauchy data spaces of a Dirac operator and its adjoint are orthogonal after Clifford multiplication. This is crucial to almost all index calculations presented in the text.

The text itself consists of 26 chapters, divided into 3 parts. The first part — Chapters 1-10 — gives a detailed exposition of Dirac operators acting on sections of bundles of Clifford modules. In the first 7 chapters, we closely follow standard references like Atiyah, Bott & Shapiro (1964], Lawson & Michelsohn [1989], and especially Gillcey [1989] and Branson & Gilkey [1992a]. It is worth mentioning that Chapter 3 distinguishes three different classes of operators. We say that A is an operator of Dirac type, if the principal symbol of A2 is defined by the metric tensor; A is a generalized Dirac operator, if A is the composition of a connection operator and Clifford multiplication; and we call A a compatible Dirac operator, or in short, a Dirac operator, if the connection used to define A is compatible with Clifford multiplication. For simplicity, all results in the book are proved only for Dirac operators, but they also hold for operators of Dirac type. In fact, except for certain index formulas, they hold even for arbitrary linear elliptic operators of first order. The reason why some of our index formulas do not hold for all first-order elliptic operators is that they are computed by reducing the index problem to a problem on the boundary. That works when the unique continuation property (or uniqueness of the Cauchy problem—UCP) for the solutions holds. For general elliptic operators, the lack of UCP implies that not all solu-

Preface

ix

tions have non-trivial traces on the boundary. That would lead to an additional term in our index formulas.

Chapter 8 proves the simplest among different variants of the unique continuation principle for Dirac operators. Let v be a solution of the equation Av = 0, where A is a Dirac operator on a connected

manifold. Assume that v =

0

on some open set; then v =

0 every-

where. The proof closely follows the beautiful exposition in F. Treves [1980], which he based on ideas due to Calderón. Typically, we apply the UCP to a solution v of Ày = 0 with v = 0 on a closed submanifold of codimension 1, so that v vanishes everywhere. The result allows the reducing of index computations for elliptic boundary problems or elliptic transmission problems (general linear conjugation problems) to calculations on the boundary. Chapter 9 shows that any Dirac operator on a compact manifold

with boundary extends to a Dirac operator on a closed manifold. Moreover, the constructed extension is an invertible operator. That greatly simplifies the proof of the Calderón-Seeley theorem, the crucial analytical result of our book (Theorem 12.4).

Chapter 10 applies the glueing construction of Chapter 9 to the relative index theorem and to the excision property for Dirac operators.

The second part of the book — Chapters 11-17 — presents some basic analytical and topological tools that enable us to analyse the nature of the index of elliptic boundary problems. In Chapter 11, we investigate Sobolev spaces on manifolds with boundary. Let v be a section on a compact manifold X with boundary Y and let v belong to the Sobolev space H1. Then the main result is that v, provided

8>

has

a well-defined trace v y, which is an element of the Sobolev

space H8 The crucial analytical result is contained in Chapter 12: Let A be

a Dirac operator on a compact manifold X with boundary Y. First we construct a Poisson type operator K+, mapping sections on Y to (smooth) solutions of A in X \ Y. Then we obtain the Calderón operator by restricting to the boundary Y, where h is a section on Y. We show that is a projection onto the space H(A) := {vly Av = 0 in X \ Y} of Cauchy data of the operator A. Moreover, is a pseudo-differential operator and K+ provides an isomorphism of H(A) with the space of solutions. In the modern literature, the operators and K+ are often constructed modulo smoothing operators. This leads only to approximate projections and

x

Preface

approximate isomorphisms, but it simplifies the construction and is still sufficient to prove the regularity theorems and develop the index theory. However, this is the point where usually extra machinery appears, and generality is achieved at the expense of precision and detail. Therefore we chose a different path. The most important properties of and are collected in Theorem 12.4. The proof of Theorem 12.4 is rounded off in Chapter 13, which, among other results, proves the existence of a well-defined trace on the boundary for any solution of Av = 0. Chapter 11 showed that v has a well-defined trace, if v belongs to the Sobolev space H with s> Here we show that for any s, an element v of H3 with Av = 0 in X\Y has a well-defined trace in on Y. Chapter 14 investigates another operator, which lives on the boundary and is induced by the tangential part B of A. B is a total Dirac operator, and in particular it is seif-adjoint. Then, for any real number a, the 3pectral projection P>a of B onto the interval [a, oo) is a pseudo-differential operator which differs from P+ by a compact operator. For the operator A, these spectral projections define specific boundary conditions first considered by Atiyah, Patodi & Singer in [1975]. We discuss generalized Atiyah-Patodi-Singer problems in the third part of the book. In Chapter 15, we switch to topological tools and introduce the Grassmannian Cr, which is the natural object for studying the index theory of elliptic boundary problems. The Crassmannian is the space of all pseudo-differential projections, which differ from by a compact operator. The idea of studying the infinite-dimensional Grassmannian is due to B. Bojarski [1979). We investigated the topological structure of Cr and found applications to the cutting and pasting of the index of elliptic operators in a series of papers (see Wojciechowski [1981], [1985a], [19861, and BooB & Wojciechowski [1985b], [1986];

more recently, in the context of spectral estimates, BooB & Wojciechowski [1989] and Douglas & Wojciechowski [1991; Appendix B]). The Grassmannian appeared in other contexts in the works of other authors (see, for example, Sato, Miwa & Jimbo [1979], Sato [1981], Segal & Wilson [1985), and Pressley & Segal [1986]). Chapter 15 stud-

ies the topological structure of Gr and shows that, for P E Cr, the operator PP÷ (acting from the range of P÷ to the range of P) is a Fredholm operator and that its index i(P, distinguishes the connected components of Cr. For P, orthogonal, the integer i(P, P÷) can also be interpreted as the spectral flow of a corresponding family {(2P — Id) + — P)} of seif-adjoint operators.

Preface

xi

That is why Chapter 16 studies the topology of the non-trivial component F. of the space of self-adjoint &edholm operators. It is well known that F. is a classifying space for the functor K' (see Atiyah & Singer 119691). To avoid the original and quite complicated proof, we present instead a computation of the homotopy groups of

that space. In particular, the first homotopy group is equal to the integers. This is all we need to know in this book. Chapter 17 is devoted to the study of the spectral flow of a family in F.. It is a nice feature in {Bt}tEsl which is just the class of that it has a straightforward analytical interpretation: the number of the elgenvalues of which change their sign from — to + as t goes along S', minus the number of eigenvalues which change their sign from + to —. We discuss various topological formulas, giving the specis a family of elliptic operators of non-negative tral flow in case order. Such families arise naturally, when elliptic boundary problems or elliptic transmission problems induce specific parametrizations of the tangential parts of Dirac operators in a collar of the boundary, reap. the partitioning submanifold.

That is explained in the third part of the book — Chapters 18—26 — which presents the basic theory of elliptic boundary problems and other applications of the machinery that we have developed. Chapter 18, following the beautiful lectures of Seeley [1969], introduces the concept of an elliptic boundary condition for a Dirac operator A. It is a pseudo-differential operator R (of order 0), which satisfies a specific compatibility condition with the operator A. Following Seeley, we furthermore assume a technical condition, namely that the range of the operator R is closed in suitable Sobolev spaces. This last condition implies, by an argument from M. Birman's and A. Solomyak's work [1982] on elliptic fans, that the orthogonal projections onto the range and the kernel of the operator R are pseudodifferential operators. One consequence is that the boundary integral RP÷ : H(A) range(R) is a Fredhoim operator (Corollary 18.15). In that way it brings the index theory of elliptic boundary problems back to the Grassmannian Cr. Chapter 19 investigates the unbounded operator AR : dom AR —, L2 — the L2-realization of A under the boundary condition R(vly) = is finite-dimensional 0— and shows that the space of solutions of (Corollary 19.2). Moreover, Chapter 19 proves that any solution of AR is smooth up to the boundary and shows the range of the operator AR to be closed.

xii

Preface

The structure of the index of the operator AR is discussed in Chapter 20. First we show that AR is a closed operator in L2 and then we construct the adjoint operator. A nice consequence of Seeley's aforementioned technical condition is the explicit form obtained for the adjoint elliptic boundary condition Q which gives (AR)* = (A*)Q. That way around we show that AR : dom AR L2 is a Fredhoim operator with kernel and cokernel consisting of smooth sections. We reduce the index computation of a global elliptic boundary problem to computing the index of the composition of two projections over the boundary which differ by a compact operator: (0.1)

index AR = dim kerAR — dim = index{RP÷ :

—* rangell}

=

Then we explain the relations between the indices of the realization AR and the full mapping pair (A, R) in the case of local elliptic bound-

ary conditions i.e. when the principal symbol of the boundary condition can be written as the lifting of an isomorphism of the original vector bundles to the cotangent sphere bundle. It is worth mentioning that the boundary conditions we call local elliptic are called elliptic or conditions in the classical literature on partial differential equations. Those boundary problems which we, in accordance with the terminology of global analysis, call global elliptic or simply elliptic were either not treated or are a subclass of what is called injectively elliptic in certain parts of the literature (see e.g. Grubb [1990]). Notice also that local elliptic boundary problems, for which the designation elliptic used to be reserved, play an important but secondary role for Dirac operators where they only appear in odd dimensions or for systems and transmission problems. Chapter 21 uses formula (0.1) to prove the Agranovië-Dynin formula index — index AR2 = index RiP+(R2)*. Next, the Agraformula is used to prove the cobordism theorem for Dirac

operators, which says that the index of a Dirac operator, which bounds, is equal to 0. Historically, the theorem was the basis of the first proof of the Atiyah-Singer index theorem (see Atiyah & Singer [1963) and also Palais [1965a]). That supports the perspective, elaborated in the rest of the book, of looking at the seemingly more elementary index theory of elliptic operators on closed manifolds through the glasses of the seemingly more intricate, but in reality more fundamental, index theory of elliptic boundary problems.

Preface

xiii

The Atiyah-Patodi-Singer index theorem, the index formula for the operator is investigated in Chapters 22 and 23. Recall that

:= Po denotes the spectral projection of the tangential part B of A onto the interval [0, +oo). The operator Ap> finds so many applications in geometry, topology, and number theory that it deserves a separate treatment. (For one approach, aimed specifically at manifolds with corners, see Melrose 11992]). We only prove the index formula for Ap>. In doing so, we follow the original paper by Atiyah, Patodi & Singer [1975], beginning with the Atiyah-Bott formula

= trexp(_t(Ap>)* Ap>) — trexp(—tAp> (Ap>)*). We obtain the corresponding heat kernels by glueing the heat kernels on the closed double X of X with the heat kernels of the corresponding operators on the cylinder Y x [0, +oo). Therefore, in the formula we have a contribution from the interior, a contribution from the cylinder, plus the error term. We take the interior contribution from the local index theorem for Dirac operators, but compute the cylinder contribution. The only problem left is to show that the error disappears, as t approaches 0. Here we use Duhamel's principle, which is explained in detail in Chapter 22C. We could have used the finite propagation speed property, but we chose the more classical approach still used in recent literature. Chapter 22 is rounded off with an L2-reformulation of the results and a discussion of a three-dimensional example treated before by Taubes [19901 and Yoshida [19911. Having proved the Atiyah-Patodi-Singcr formula, we use it together

with the Agranovië-Dynin formula in Chapter 23 to obtain an index formula for Ap, where P is an arbitrary element of the Grassmannian, and to discuss the additivity of the index of Dirac operators with spectral boundary conditions. This allows the writing of the signature of a 4k-dimensional compact oriented manifold with boundary as the true index of an elliptic boundary problem - without the error term coming from the kernel of the tangential part of the signature operator. In Chapter 24, we continue discussing additivity problems and the index of Dirac operators on partitioned manifolds. We prove a theorem conjectured by Bojarski (1979]: Let M be a closed manifold and Y a submanifold of codimension 1, which divides M into two parts M = U X_, where fl X_ = Y. Let denote the restriction of A to Then we have the formula index A = index{(Id = index

:

(A)

H_ (A)1 }

Preface

xiv

where 'P4. denotes the Calderón projector of A4. and C denotes Clifford multiplication by the unit normal vector. We also discuss a formulation of this theorem in the language of Fredholm pairs and then proceed to the general linear conjugation problem. We are looking for s....), which are solutions of the following problem on M couples

A+s+=0 (0.2)

and

As_ = 0 s_=Ts+

inX4. in X_

onY,

where T is a suitable bundle isomorphism. Bojarski's theorem constitutes the special case of T = C. We prove the formula

indexLcp(A,T) = indeXLcp(A,G) + indexl'÷TP4., under a certain consistency assumption on the isomorphism T, where indexLcp(A, T) denotes the index of the general linear conjugation problem (0.2).

Chapter 25 investigates the underlying geometrical problem of surgery — cutting and pasting — of Dirac operators, that is, we cut M along Y and glue it differently by using a diffeomorphism and a corresponding automorphism of the bundle of Clifford modules. Assuming consistency of the data on Y, we obtain a new Dirac operator and come up with a formula for the difference of the indices. The final Chapter 26 illustrates those results, using a simple, but multifaceted example. We investigate the case of M = S2, which is a perfect illustration of all the phenomena discussed in this book. It might be a good idea to start reading this book with a quick look at this chapter.

Many colleagues and students have helped to improve this book, and we heartily thank them all. Our emphasis on Dirac operators and the related geometrical concepts owes much to remarks by Peter Cilkey. Special thanks also to Ron Douglas, who suggested writing this book, to Ulrich Bunke, Jerry Kaminker, Slawomir Klimek, and Anders Madsen for continued and inspiring discussions of many details, to Ingrid Jensen for drawing the figures, to Birthe Wolter for typing, and to Krystyna and Wiadek Uscinowicz for providing us with housing in Szczecin for our monthly meetings some years ago when we began our joint work.

List of Notations a-index, analytical index, 146 A, (total) Dirac operator, 20 partial (chiral, split) Dirac operators, 41 invertible extension to closed double, 51 Aleft, A right, quaternion Dirac operators, 33 Amin, Amax, minimal and maximal extension, 196-198 realizations, 165, 180, 195 AR, A, elliptic operator after cutting and pasting, 277 glued operator, 60 A(N, As, Atiyah-Singer operator, 37 automorphism groups, 268 Aut, A, Atiyah-Patodi-Singer realization, 211 index density, 60 reduced operator of order 0, 139 B', suspension, 148 B, algebra of bounded operators, 129

ch, Chern character, 146 c, Clifford multiplication, 5 c0,. , c3, Clifford matrices, 34 •, P>), spaces of smooth sections C000, on semi-infinite cylinder, 216 C€m, Clifford algebra of Rm, 29 Ct(V, q), Clifford algebra of (V, q), 3 Ct(X), Clifford bundle over X, 10 chiral components, 4 C, Calkin algebra, 129 dA, covariant derivative, 250

Levi-Civita connection, 13 Ds, spin connection, 15 covariant derivative with respect to v, 13 D, 7)', normalized cylindrical Atiyah-Patodi-Singer problems, 215 classical Dirac operator (on spin manifolds), 37 'PA, twisted signature operator, 250 Laplace-Beltrami operator, 21 cylindrical Dirac Laplacians, 222 partial quaternion Dirac operators, 35 5, Dirac distribution, 80, 224

e+, extension by zero operator, 72

List of Notations

erfc, complementary error function, 227 E", bundle after cutting and pasting 277 Eli, EU0,, eUiptic seif-adjoint operators of positive order, 142, 153 ettx, special group of invertible elliptic operators, 112 6tiX invertible elliptic operators, 116 subgroup of (,continuous extension operator of codimension one, 72

£(t,x,y), heat kernel, 59

£, e.,

heat kernels on manifolds with boundary, 214, 237

, heat kernels on the semi-infinite cylinder, 223 ea, ed., heat kernels on the closed double, 231

77B, eta-function, 229

too, deformation retract of

140

Fred2(H), Fred2H÷ (H), spaces of Fredhoim pairs of subspaces, 265 F, space of Fredholin operators, 127 F., spaces of seif-adjoint Fredhoim operators, 128 if F, Thom isomorphism, 150 g, genus of Riemann surface, 249 C, Green's form, 24 c, group of units of Calkin algebra, 129 Q, seif-adjoint units of Calkin algebra, 129

C, C±, C, unitary retracts, 132 special groups of invertible operators, 113 bounded invertible operators, 266 Grassmannian of pseudo-differential projections, 111 connected component of the Grassmannian which contains P.,., 116 full Grassmannian, 133 Gr(H), standard trace operators, 68, 95, 180 r, Gamma-function, 229 r5, Gamma-five matrix, 8 Christoffel symbols, 15 kernel dimensions, 242, 245 Hk, higher Hopf bundle, 283 H, cohomology functors, 272 H'(R'), H8(X), H8(Y), Sobolev spaces, 67, 216 H±(A), H±(A, s), spaces of Cauchy data, 76 spaces of limiting values, 244 h,

index(., .), index of Fredhoim pair, 263 l(I'2, P1), virtual codimension, 119 orthogonal projection onto the range of R, 166

J, reflection operator, 72

List of Notation8

8), spaces of distributional solutions, 76

K(M), Crothendieck group of complex vector bundles, 128

K1(M), higher K-group, 146 Ko(X, Y), relative K-homology, 58 Poisson type operator, 78 K, ideal of compact operators, 129 A(t), symmetric integral, 223 u, y), symmetric heat kernel, 223 £+, continuous extension operators of codimension zero, 72 L2-ker, L2-solution spaces, 243, 244

L2-index, index of L2-solutions on non-compact manifold, 158, 243 LCP(A, 4), linear conjugation problem, 268 CR, linear span, 31 A(V), A(X) = A(TX), exterior algebra, 8, 41 A+, Riesz operator, 96

M', manifold after cutting and pasting, 277 range of the boundary projection symbol, 90, 164, 194 M(N, C), full matrix algebra, 29 NR, orthogonal projection onto the kernel of R, 166 V, covariant differentiation in exterior algebra, 14 connection 1-form, 15 w, standard orientation form, 40 spaces of differential forms, 41

p.4., boundary projection symbol, 79 p, canonical projection onto Calkin algebra, 129 Pa, PJ{v,,

(3.13)

—

=

= From (3.11) and (3.13) we obtain not only (3.14)

=

+div(r)1

at the chosen point x, but in fact (Asj;

=

+ (Si;

for all x' E X, since all expressions in (3.14) are independent of the chosen frame

Stokes' formula (see e.g. Dubrovin, Fomenko & Novikov [1984; 4.26]) gives

IM

div(r)dvol(x)

=

(r; —n) dvolQj) JOM

which proves (a) and (b).

dvol(y) JOM

[I

4. Dirac Laplacian and Connection Laplacian

We discuss the general Bochner identity which gives an expression of the Dirac Laplacian A2 in terms of the connection Laplaci.an D*D and certain bundle endomorphisms.

Now we shall express the Dirac Laplacian A2 in terms of the connection Laplacian D*D and certain bundle endomorphisms. We begin

with the definition of the operator DD. Definition 4.1. Let X be a compact Riemannian manifold and let S be any Ce(X)-module with compatible connection D. The connection Laplacian

D*D:

C°°(X;S)

is defined by taking the trace

DDs := — tr(D2s),

(4.1)

where s E C°°(X; 5) and D2

Coo(X;S* ®S)

C°°(X;TX EITX)

denotes the invariant second derivative given by

:=

(4.2)

—

DDVWS

for any pair of tangent vector fields v and w.

Remarks 4.2. (a) Clearly the operator Dr,.

only

depends on the

value since this is a general property of the covariant derivative. This follows from Also, moreover, only depends on the value

the identity fl2 where

—

fl2

__D —

R denotes the curvature tensor of S.

4. Dirac Laplacian and Connection Laplacian

27

(b) In terms of a local orthonormal frame {v,.1 } for TX we have

DDs =

(4.4)

(c) The concept of the connection Laplacian is of great generality in

differential geometry and not restricted to the case of solely discussed here.

Proposition 4.3. The operator DD : C°°(X; S)

S) has

the principal symbol OrD.D(x,e) =

(4.5)

fore

TXX,

and so DD is elliptic. It is also non-negative and formally selfadjoint. In particular, (DDs1;s2)

(4.6)

JM

(Dsi;Ds2)

for all

E C°°(X; S), provided that either or has support in the interior of X. (As always we assume that X is compact.). Here

(Dsi; Ds2) is defined in terms of local frames {v,j for TX by the expression

(Dsi;Ds2) =

(DV,sl;DVM52)

Proof. We reproduce the main arguments from the proof of Lemma 3.3 and Proposition 3.4 above, replacing left Clifford multiplication by

covariant derivation. Fix an x X and choose a local orthonormal frame {v1,. . . , } for TX with the property that = 0 for all p, &'. Then, at the point x, one has that

DDs =

(4.7) and so as

=

—

since

=

zero-order terms

observed earlier.

From (4.7) we obtain (all inner products are taken at x) that

(D*Dsi;s2) = (DV,Asl;s2) —

= —div(r)+ (Dsi;Ds2)

I. Clifford Algebras and Dirac Operators

28

where r now is the tangent vector field on X defined by the condition (r; w) = 82) for all w E TX. Then

Now the proposition follows by integration.

Theorem 4.4. (The general Bochner identity). Let X be a compact Riemannian manifold (with or without boundary), and let S be a Ct(X)-module with compatible connection. Let A2 denote the Dirac Laplacian and let D*D denote the connection Laplacian. Then

A2 =D*D+1Z. Here IZ is a canonical section of Hom(S, S) defined by the formula

jz,v=1 } is any orthonormal tangent frame at the point in question, is the curvature transformation of S (cf. (4.3) above), and the dot "•" denotes left Clifford multiplication.

where

Proof. We make the same assumptions about x and preceding proof and find at x that A2 =

=

v1,1

= by (4.3).

=

in the

v1,

using (2.12) and

= >v,.,

as

}

+

= —

0

at x

5. Euclidean Examples

Some important examples are discussed: Dirac's search for a square root of the Klein-Gordon operator; the Dirac operator on 11.2 = C considered as mapping even to odd spinors which is exactly the Cauchy-Riemann operator; the Dirac operator on H-valued functions; and the quaternion analogue for the Cauchy-Riemann operator and its expression by Pauli matrices.

The first class of examples we want to consider are Dirac operators on or on a compact connected submanifold X C of codimension 0 with smooth boundary. Let Ctm denote the Clifford algebra of I1.m. It has real dimension Let W be a (complex, N-dimensional) C€m-module, and let c : Ctm Hom(W, W) denote the related representation, i.e. left Clifford multiplication in W with elements of C€m. We study the Euclidean Dirtzc operator A : C°°(X; S) C°°(X; S), where S := X x W is endowed with the trivial (compatible) connection. it is a constant coefficient operator of the form

where all are linear maps : W —. W and from their definition = c(e,1) inherit the relations as (5.1)

for all

ii.

el,... ,em denote the standard basis vectors of If we choose a basis for W, we can identify Hom(W, W) with the full matrix algebra Here

M(N, C), and the c,4's will be represented by elements of the group GL(N, C) because of (5.1) for = ii. Equation (5.1) also implies that A2 = In the coordinates for W it is an N x N diagonal matrix with the (positive) Laplacian = — repeated in the diagonal.

Remarks 5.1. (a) The physics literature distinguishes carefully between Euclidean (elliptic) Dirac operators, which are operators of the type discussed here, and Minkowskian (hyperbolic) Dirac operators,

30

I. Chiford Algebras and Dirac Operators

which are the operators originally investigated by P.A.M. Dirac in the 1920s. Dirac posed the following question: Can the Klein-Gordon operator + m2, where

0 := with gIW denoting the coefficients of the Minkowski metric, be expressed as a product of two first-order differential operators as follows:

[I +m 2?=

1

0

.

\1

.

0

Dirac realized that the coefficients must be matrices which generate the algebra M(4, C) and satisfy the relation (5.1), where the right

side is replaced by with S = R4 x W and W =

In physics, the space C°°(R4; S) is called the space of (4-component) spinors. See e.g. Dubrovin, Fomenko & Novikov [1984; Section 40]. (b) If we only consider irreducible Cim-modules we can determine the following data for each integer m 1: First, find C€m. All C€m are explicitly known, they have real dimension and there is actually a kind of periodicity with period 8; see Lawson & Michelsohn [1989; Section 1.4]. Then determine the irreducible representations C: C€m —, Hom(W, W) along the lines of Lemma 1.3. They are uniquely determined (up to isomorphisms) if m is even, i.e. m = 2n; then the complex dimension of W is N = 2's; if m is odd, m = 2n +1, we have two non-equivalent irreducible representations, each module again of complex dimension N = r. Each vector of the canonical basis {e,j of am provides an endomorphism cM = of W which satisfies the relations in (5.1) (as, in the opposite direction, each system of m endomorphisms of W defines a representation of Ctm, if it satisfies C4

(5.1)). Now choose a basis for W and calculate c,, which is an N x N matrix, the entries of which are linear combinations of the partial derivatives 0/Oxi,... , 0/t9Xm. This matrix, which is enormous for higher m, has the property that its square is where E denotes the N x N identity matrix. Finally, the principal symbol of the Euclidean Dirac operator restricted to unit cotangent vectors provides a generator aA(x,.) : 5m_I GL(N, C) of the homotopy group 7rm_i(GL(N,C)) Z form even and 2N m. As Lawson & Michelsohn [1989, p. 120] put it: "This explicit form is seldom, if ever, useful. It is always simpler to use the structure of

5. Euclidean Examples

31

the Clifford module." Nevertheless, below we shall derive the explicit forms in the cases m = 1, .. . , 4 for three reasons: (1) to illustrate and exercise the basic concepts and constructions of Clifford algebras, Clifford modules, and Dirac operators;

(2) to provide the basic bricks for all geometrically defined operators. This may sound strange, since Euclidean Dirac operators do not immediately extend to elliptic operators over closed manifolds — that is inhibited by the characteristic form of the principal symbols which have non-vanishing local index

for even m. However, later in this book, we shall show how to build up arbitrary classical operators by pasting Euclidean Dirac operators together, see Chapters 24 — 26; (3) and to ease reading the literature, where some calculations are carried out through ingenious manipulations with these matrices, see e.g. Goldstone & Jaffe [1983], Schmidt & Binzer [1987], Schmidt [1987], or Kori [1993].

Letm= 1,sothatC€1 =Cn{1,ei}= C. ThenW=Candc(1)= 1. There are up to equivalence two irreducible representations, depending on whether c maps e1 into i or —i, and these two representations

are not equivalent, since F: z operator is just

is not C-linear. Then the Dirac

(5.2)

The Z2-grading Ct1 = Cit Ctj gives the splitting in real and imaginary parts which the Dirac operator interchanges. Let m = 2, so that dim Ct2 = = 4 and in fact Ct2 = £R{1,el,e2,ele2} = H. The decomposition of H into cEB C is natural and corresponds to the Z2-grading with = CR{1, eje2} and C4 = Ca{ei, e2}. We are looking for a complex vector space W such that Cl2 Homc(W, W). Then the complex dimension of W has to be 24m = 2. Actually W := C C is an irreducible C€2-module with Clifford multiplication given by

c(1) :=

/1

o\ i)'

c(ei)

:=

(0

—i\ 0

c(e2) :=

/0

i

0

With the notation of Example 1.4 we can write down the two basic operators on AC2 Cl2:

p=c(ei)+ic(e2)

and

I. Clifford Algebras and Dirac Operators

32

with the relations

p• p = (c(ei)

+ i

c(e2))(c(ei) + i c(e2))

= —c(1)

+ic(ei)c(e2) +ic(e2)c(ei) +i2c(e2)2

=0

and

= —2(c(1)+ic(ei)c(e2)) so

-4 as required in Example 1.4d. Then

p•

=

—2(c(1)

+ ic(ei) c(e2))

and

1.

=

= (c(ei)

—

ic(e2))

form a basis for W, and Clifford multiplication with p and does not lead out of the subspace W of C€2. Indeed, we have p• W C W, since

p' (p

=

0

and trivially p. (1

= (p

and an invariant description of

and W, determine the ±1-

elgenspaces of the orientation form

= i c(ei) c(e2). For the ease of presentation we omit the c in c(e,) and write just e3 for Clifford multiplication with e3. Then we get

= hence

—2iele2(1

= —2(1 +iele2)

generates W÷; and

= iele2(eI hence

+ieie2) =

j3 generates W.

—

ie2)

=

—(e1 —

ie2)

=

=

5. Euclidean Examp'es

Correspondingly the Dirac operator A = has the form (5.3)

33

(8/Ox,) + e2 (8/Ox2)

e1

—i), := 0/Oxi + jO/Ox2

where 8/Oz := 8/Ox, — jO/Ox2 = 28, and

28 is twice the standard Cauchy-Riemann operator. Thus the Dirac operator on R2 = C, considered as mapping even to odd spinors, is exactly the (double) Cauchy-Riemann operator.

Our choice of c(ei) and c(e2) is by no means canonical. There are actually three matrices, the Euclidean analogues of the classical Pauli matrices Note.

(i o\

)'

c2:=c(ei),

ando3:=c(e2),

which generate the full matrix algebra M(2, C) and satisfy the relations (5.1). Every two of them together generate an irreducible in complex form. By Lemma 1.3 these reprerepresentation of sentations are equivalent. This is easily checked, e.g. for the two representations c3 : Ct2 M(2, C) defined by

c1:=c =

Then

trary a

F'

forjt=1,2.

and for

= 1,2 if F =

(a

for arbi-

0.

Letm=3. W=

Clearly we have two non-equivalent representations of H on H, namely, left quaternion multiplication and right quaternion multiplication. We begin with left quaternion multiplication which gives the following expression for the Dirac operator on H-valued functions on R3: (5.4)

.8 =1— Ox,

.0 0x2

0 Ox3

If we write q H = R4 in the form q = a + bi + ci + dk with real coefficients a, b, c, d and the canonical basis {1, i,j, k}, we can write left quaternion multiplication as 4 x 4 real matrices (corresponding to

1. Clifford Algebras and Dirac Operators

34

L(i)q = i q =

—b

+ ai — dj + ck, L(j)q =

...,

and

L(k)q =...) and

obtain

/0 I

Aleft

101

=1

0

—02 03

03

0

03 02 —01

0

—02

:= If we choose right quaternion multiplication, we obtain a different Dirac operator where

(5.6)

Aright

=

+

+

which can be written in real coordinates (corresponding to R(i)q =

q•i=—b+ai—dj+ck,R(j)q=...,andR(k)q=...)as

/0

—03

—03

0

1

0

Now decompose H = C ® C and rewrite

q= a+ bi+ cj + dk = ((a+ bi), (c+ di)). Then right quaternion multiplication may be described by complex

2 x 2 matrices. Check that R(i) = a1, R(j) = a2, and R(k) = O'3, where

are

the Pauli matrices defined above.

Letm=4. ThenCt4 The representation C: Ct4 Homc( W, W) is uniquely determined up to isomorphisms. So we can just play around with the Pauli matrices until we get complex 4 x 4 matrices c4 = c(e,4) for /L = 0,1,2,3 which satisfy the Clifford relations (5.1). Clearly the following matrices will cO= (?

and

(°

forp=1,2,3.

Then the Dirac operator can be expressed with respect to the splitting

(5.9)

5. Eudlidean Examples

35

where

o

— 0

0

.0 0

.0 .

0

k

Note the analogy with dimension two. For a thorough exploitation of the analogy, see e.g. Delanghe, Sommen & Souëek [1992], where quaternionic analysis and other higher dimensional analogies of complex analysis are described. An analogy of holomorphic differential forms in higher dimensions builds on formal similarities between the Cauchy-Riemann operator and the other Euclidean Dirac operators.

6. The Classical Dirac (Atiyah-Singer) Operators on Spin Manifolds

We consider a spin manifold with a spin structure on its tangent bundle, and a spinor bundle endowed with its canonical connection. We formulate the Lichnerowicz vanishing theorem.

One might argue that this chapter is merely of historical interest now that the fact is established that, for any bundle S of complex there always exists a connection which is left modules over compatible with Clifford multiplication and extends the Levi-Civita connection on X to S; see Chapter 2. Since that is true for any bundle of Clifford modules over a Riemannian manifold X, this somewhat removes the prominence of spin manifolds in earlier work in index theory and the exclusiveness of spin manifolds in admitting non-Euclidean Dirac operators. Nevertheless, for various calculations, one can take great advantage of the symmetry of spin manifolds, of the associated spinor bundles, and of the related Dirac (Atiyah-Singer) operators. Introductions to

the subject can nowadays be found in many places. We especially refer to Gilkey (1984; 3.2 and 3.3] or Lawson & Michelsohn [1989; 11.2, 11.5, and 11.8]. Here it suffices to explain the relation to our previous introduction of isomorphism classes of irreducible Clifford modules. We shall begin with an intuitive argument. Let X be a compact oriented Riemannian manifold of even dimensionm = 2n. For every point x E X, we have three different objects: of real dimension m, (1) the tangent space (2) the Clifford algebra C1(TXZ) of real dimension which contains

(3) and, by Lemma 1.3 (up to isomorphism), the uniquely determined irreducible complex of complex dimension 2", which we shall denote by Whereas the families {TXZ}XEX and {Ce(TXX)}XEX carry natural topological and differential structures which make them bundles, namely, TX and Ct(X), the parametrized family {SX}XEX admits the structure of a continuous and smooth bundle only if X is a spin man-

6. The Classical Dirac (Atiyah-Singer) Operator

37

ifold. In other words, a principal bundle Pspin(TX) over X is given which has Spin(m) as structural group and is (roughly speaking) a double covering of the principal bundle Ps0(TX) of orthogonal oriented frames, or (more precisely) accompanied by an oriented vector bundle isomorphism

TM

Xp Rm,

denotes the double cover. where p: Given such a spin structure we make, of the parametrized family {SX}XEM, the associated bundle as by regarding modules and identifying t)

(x,

(x, s, ic), x E U4 fl U,ç,

are suitable coordinate transformations for the spin manifold X. Then the associated bundle carries a natural Hermitian structure satisfying (2.2) and is endowed with a natural compatible connection induced by the Levi-Civita connection for TX. where

:

fl

Definition 6.1. (a) For spin manifolds of even dimension, one has a (unique) irreducible complex spinor bundle for any choice of a spin structure. We shall denote this bundle by The Dirac operator in this case will be written (b) More generally, a (complex) spinor bundle, associated with TX, is a bundle S of modules over C€(X) of the form

S=

Psp1,,(TX)

M,

where M is a complex left module for the complexified Clifford algebra C€(Rm) 0 C. Then, once again, the Hermitian structure of S is compatible with Clifford multiplication (in the sense of equation (2.2)) and S carries a natural compatible connection as explained, for example, in Lawson & Michelsohn [1989; 11.4]. The Dirac operator will in this case be called (following Lawson & Michelsohn) the Atiyah-Singer operator. We write A or As. The classical necessary and sufficient topological conditions for a Riemannian manifold to admit a spin structure is derived and discussed, for example, in Lawson & Michelsohn, namely, the vanishing

I. Clifford Algebras and Dirac Operators

38

of the second Stiefel-Whitney class. The most elementary examples are:

• All 2-connected manifolds (like homotopy spheres and simplyconnected Lie groups) carry a unique spin structure. • Any manifold, whose tangent bundle is stably parallelizable

trivial after adding a suitable trivial bundle), is spin. This includes any Lie group and any orientable manifold of dimension

any local orthonormal tangent frame at the point in the curvature transformation of S; see equation question and

where

is

(4.3).

Corollary 6.3. We have (6.3)

kerAS=O

for any spinor bundle associated with TX, if the underlying manifold is spin and closed (compact and without boundary) and if the scalar

curvature is 0 and > 0 at some point. Theorem 6.2 is an immediate consequence of the general Bochner identity (Theorem 4.4), and the corollary follows by integration of (6.1). For details, cf. Lawson & Michelsohn [1989; p. 161].

6. The Classical Dirac (Atiyah-Singer) Operator

39

Examples 6.4. (a) We can see C€(X) as a bundle of left modules over itself by left Clifford multiplication and endow S := Ct(X) with its canonical Riemannian connection. The Dirac operator in this case is a square root of the classical Hodge Laplacian. (b) By tensoring we can generate ever new examples from the basic examples. More precisely: Let S be a (complex) C€(X)-module over X (not necessarily spin) with compatible connection DS, and let E be an arbitrary Hermitian vector bundle with connection DE over X. Then the tensor product S 0 E is again a C€(X)-module and is equipped with a compatible connection.

7. Dirac Operators and Chirality

We emphasize the decomposition of a Ct(X)-bundle S = S and the related splitting of Dirac operators. It is illuminating to treat the signature operator and other geometrically defined operators in this context.

Now we are back to our general situation, where X is a Riemannian oriented (not necessarily spin) manifold of dimension m and S a C€(X)-module with compatible connection. We want to globalize the Z2-grading explained in Remark 1.le and decompose the Dirac operator. We do that in a series of lemmata. Let w be a section of C€(X) given at any point x E X by (7.1)

is a positively oriented orthonornial basis of where This provides a well-defined global section of Ce(X). Since w is independent of the choice of basis, for any x , we may choose a local frame {eM} such that Depeyiz = 0 for all v. Hence we have the following lemma:

Lemma 7.1. On any oriented Riemannian X of dimension m, we have a canonically defined section w of Ct(X) which satisfies the following relations:

=

(7.2) (7.3)

(7.4)

wv =

for any section v of TX,

Dw=0.

Now define a bundle automorphism multiplication (7.5)

:=

: S —' S by left Clifford s.

It follows from Lemma 7.1 that, for even-dimensional X, the bundle S splits: (7.6)

with v 5+ C S and v• S C St where spanned by the eigensections of ±i, if m 2 mod 4, or ±1 if m

are the subbundles corresponding to the eigenvalue

0 mod 4.

7. Dirac Operators and Chirality

Lemma 7.2. For the (total) Dirac operator A

41

:

C°°(X; 5)

C°°(X; 5) we have

= From this lemma it follows that the Dirac operator splits in even dimensions. The components

(7.7)

:

AICOO(x;S±)

:

C°°(X;

C°°(X;

are well-defined elliptic operators. That construction provides interesting indices, because in difference to A the operators A± are not formally seif-adjoint. We collect our results:

Theorem 7.3. Let X be a Riemannian, oriented manifold with or without boundary and let S be a Ct(X)-module with compatible connection. If dim X is even, the bundle S and the total Dirac operator A decompose to

1)

and

and the operator A— is formally adjoint to the operator At

Examples 7.4. (a) One of the most prominent Dirac operators is the signature operator. Let X be a compact oriented, Riemannian manifold (with or without boundary) of dimension 4k. Usually one defines the signature operator via differential forms: Recall that the operator A := d + d* acts on the space := C°°(X; A(TX)) of all differential forms. Denoting by the ±1-eigenspaces of the involu-

tion r:

—'

11, defined by

3s

it follows that A interchanges 11' and defines an operator

*

E

and hence by restriction

which is called the signature operator. This corresponds to the preceding definition given in (7.7) when we considered the basic case of ce(x)-module bundles, namely,

S := C€(X) =

=:

S.

Here is identified with A(T*X). Moreover, the Z2-grading is given by the volume element once again.

42

1. Clifford Algebras and Dirac Operators

(b) To get the required form

close to the boundary Y of X, assume that X is isometric to a product

near the boundary. Identifying the restriction of to Y with the space 12(Y) = COO(A(T*Y)) of all differential forms on Y, one finds that

Bs=

where the sign in the formula depends on the parity p of the differential

form; see Atiyah, Patodi & Singer [1975; (4.6)] and Gilkey [1984; pp. 261 if J. One sees that B is a self-adjoint operator on Y which * preserves the parity of forms on Y and commutes with 8 so that B splits into B = B°" and Be%c is isomorphic to Bo&I. We get

:=

where s =

—

E C°°(X; V) and V :=

see also

Branson & Gilkey [1992a, Example 2].

(c) For Dirac operators coupled to vector potentials equal to Dirac operators with coefficients in an auxiliary bundle, we refer to Example 6.4b and Atiyah & Singer [1984]. (d) A simple construction of Dirac operators on S2 with coefficients in line bundles is given in Chapter 26.

Remarks 7.5. (a) Instead of our choice of the orientation section w in (7.1) we could have taken a normalized orientation 'r like in our note to Lemma 1.3. This makes only one difference, namely, r2 = 1 instead of the more complicated formula (7.2) for Then our subbundles become the ±1 eigenspaces of r. (b) From Lemma 1.3 we obtain that the smallest fibre dimension for any total Dirac operator over an even-dimensional manifold X is the

dimension of the irreducible module, i.e. N = 2m/2 for dim X = m. Therefore we obtain

as minimal fibre dimension for the split operators. In fact there is no first-order elliptic differential operator over an open subset of defined on a smaller number of complex valued functions; see also Atiyah [1970].

8. Unique Continuation Property for Dirac Operators

We give a direct proof of the unique continuation property of a Dirac operator by exploiting its simple product decomposition.

Definition 8.1. An operator A over a smooth manifold X (with or without boundary) has the unique continuation property, if any solution s of As = 0, which vanishes on an open subset of X also vanishes on the whole connected component of the manifold.

All classical (Euclidean) Dirac operators eo ipso satisfy the unique continuation property when they are not coupled to a vector potential (that is, not tensored with an auxiliary coefficient bundle; see Chapter 5 and Example 6.4b above). This follows immediately from the Cauchy-Kovalevskaya theory of elliptic equations with analytic coefficients, since the Eudidean Dirac operators are analytic and can even be written as operators with constant coefficients.

It is a remarkable property of all other operators of Dirac type that they also obey the unique continuation principle. In recent years several authors have exploited this property; see, for example, Roze [1970], Jaffe & Taubes [1980], KaIf [1981], Simon [1982], Freed & Uhlenbeck [1984], Donaldson and Kronheimer [1990], and Taubes [19901.

Usually reference is given to the Aronszajn-Cordes uniqueness theorem for second order differential operators A = EIQI It follows (see Figure 8.1) that

(8.2) suppv C [0,

x

and

suppAv C

x

The proof of Theorem 8.2 is split into two lemmas. First we show that (8.1) ensures that s also vanishes in the annular region u < Next the inequality (8.1) is proven. U Lemma 8.5. If (8.1) holds for any R> 0 sufficiently large, then s

is equal to Oforu
0 such that (9.12)

ll(TY1vII

0, then = (because (T)' is bounded) and we would have

x := fl—.oo lim

and

E

y = Tx.

Finally, by eventually passing to a subsequence, we may assume that

> n and This yields

< hull + 1.

—

=

> n,

hence (T)1 is not bounded.

Proposition 9.5. Let A : C°°(M; E)

C°°(M : F) be a bijective elliptic pseudo-differentzal operator of order m over a closed manifold M. Then A—' is a pseudo-differential operator of order —m. Proof. Since A' = (A*A)_1A*, we can assume that A is a selfadjoint and non-negative operator on C°°(M; E). Now A is a Fredholm operator; hence for any k the extension

A has a closed range. By the preceding Proposition 9.4, this means that the inverse operator of A(k) is a bounded operator

E)

:

Hk(M; E).

Let Q denote a parametrix of A (9.13)

QA=Id+K,

where K is a smoothing operator. We multiply (9.13) by A' from the right which yields the following formula for A1: (9.14)

A' = Q — KA'.

The operator KA' is a smoothing operator, hence it has a smooth kernel and A' is a pseudo-differential operator of order —rn []

58

I. Clifford Algebras and Dirac Operators

Remarks 9.6. (a) The operator is not a seif-adjoint operator though it is isomorphic to A = (A+) by (9.11). To be precise, let J : X X denote the isometry which interchanges the two factors X1 and X2 of X by mapping xof the one copy into J(x) := x of the other copy. The pullback J (S+)is equal to S and we can extend which on sections is given by the J to a bundle automorphism of formula

J(s1,82) := (82,81). We have the following formula:

= A-J which gives the required isomorphism.

(b) The construction of operators of the form A U A* was introduced in Wojciechowski [1985b], where the analytical realization of the relative cycles for the K-homology group Ko(X, Y) is discussed. See Douglas & Wojciechowski [1989] for a detailed exposition of this aspect of the theory. There the odd-dimensional case is also treated and operators of the form B U (—B) are constructed, where B is a formally seif-adjoint Dirac operator. Our construction is also applied in some further work, where we discuss spectral invariants of seif-adjoint operators and boundary problems; see Boos & Wojciechowski [1989], Douglas & Wojciechowski [1991], Klimek & Wojciechowski [1992], [1993], and Wojciechowski [1993].

We can apply the glueing construction in a more general context as well. We only need (1) a partitioning of a closed manifold M = U U V into two submanifolds with common boundary V , and (2) Dirac operators A1 on U and A2 on V which are equal on i.e. with common Green form C and boundary Dirac operator B. on M. The details Then we can always construct the operator A1 are left to the reader; see also Wojciechowski [1985b].

10. Glueing Constructions. Relative Index Theorem

We explain the relative index theorem of Cheeger, Gromov & Lawson and offer an analytical interpretation which also works in the odd-dimensional case. We also give a simple proof of a special case of the excision principle for indices.

Here we want to give an analytical interpretation of the relative index theorem of Cheeger-Cromov-Lawson (see Gromov & Lawson [19831 and Lawson & Michelsohn (1987}). We set aside some definitions and technical details, since there are quite a few excellent expositions of the theorem including also various K-theoretical interpretations (see also Borisov, Muller & Schrader [1988], Julg [1988), and Bunke [1992aJ, (1992b]). We only want to offer a simple analytical interpretation that works in the odd-dimensional case as well. The situation is as follows: We have E1 and E2 bundles of Clifford modules over X1 and X1 respectively, where X1 and X2 are two complete Riemannian manifolds (not necessarily compact). Assume

that there exist K, compact subsets of X,, such that there exists E1 IXI\Ki —' E21x2\K2 a unitary isomorphism of Clifford bundles.

Let / denote an isometry of X1 \ K1 onto X2 \ K2 covered by 4' This assumption implies that the corresponding Dirac operators I), are equal outside of K,, j = 1,2. Now assume that dim X, is even. Then the following fact is straightforward.

Proposition 10.1.

Let 5,(t; x, y) denote the kernel of the operator

Then

(10.1)

index(D1,D2) := urn

t—.o

[J-'1strEi(t;x,x)dvoli— J

is a well-defined integer which we call the relative index of the opera-

tors D1 and 1)2. Here the symmetric trace is defined by str e -ti)2I = tr e

as usual.

I

I

—

tr e

i

i

I. Clifford Algebras and Dirac Operators

62

We can also apply our glueing construction to prove the excision principle for indices. We obtain an explicit operator expression for the differences. Following Donaldson & Kronheimer [1990; Section 7.11 (see also Seeley [1965, Appendixj) we consider two sets of data

(M; U, V, D1, D2)

and

(N; W, Z, A1, A2),

where M is a closed Riemannian manifold (compact and without boundary) which is the union of two open submanifolds M = U U V. Let D : C°°(M;E,) —. C°°(M;F2), i =1,2 denote a pair of elliptic differential operators over M. Assume that there exist bundle isomorphisms and

such that

= over We make corresponding assumptions about the second set of data, the second quintuple, i.e. roughly speaking: N = W U Z and A1 is D2

isomorphic to A2 over Z. (See Figure 10.3.) DiIv = D21v

4" U

w z

Fig. 10.3 The excision data: two quintuples

10. Clueing Constructions. Relative Index Theorem

In addition to Seeley and Donaldson & Kronheimer, we make two further assumptions. First consider only (compatible) Dirac operators. Second, assume that U \ V has a smooth boundary Y and that all structures are products in a bicollar neighbourhood of Y. This gives the following variant of the relative index theorem:

Proposition 10.4.

Let a quintuple (M; U, V, D1, D2) be given, i.e. a

closed Riemannian manifold M = U U V given as the union of two open 8ubmanifolds and two Dirac operators D1, D2 over M which coincide under suitable bundle isomorphisms over V. We assume that U \ V has a smooth boundary Y and that all stn&ctures are products in a bicollar neighbourhood of Y. Then we have

U D,u,

index D1 — index 1.32 = index

denotes the restriction

where

j = 1,2,

:= DjIu\v, and double

denotes

U of U \

V

by glueing

Dirac operator obtained over the closed the pieces together along Y.

the

Now assume that we have two quintuples as above:

(M; U, V, D1, D2)

(N; W, Z, A1, A2).

and

Assume also that there exists a diffeomorphism r : U —. V by suitable bundle isomorphisms such that .1

Theorem 10.5. Under the property for

r..#A

7Th

preceding

covered

assumptions we get the excision

indices

index A1 — index A2 = index D1 — index D2. Proof.

indexAi — indexA2

= index 7.

•

Prop. 10.4.

=

indexAi,w UA2,w

* U Dv.,

Prop. 10.4.

=

index D1 — index D2.

D

Remark 10.6. The explicit description of the index differences is due to two circumstances, namely that we deal with Dirac operators and not arbitrary elliptic differential operators, and that we restrict ourselves to the smooth, relatively compact case.

PART II

Analytical and Topological Tools

11. Sobolev Spaces on Manifolds with Boundary

We investigate the continuity properties of taking traces in Sobolev spaces over manifolds with boundary.

Let X be a smooth compact n-dimensional manifold with boundary Y and fix a Riemannian structure on X. Use the Riemannian structure to construct a collar neighbourhood N = Y x I of the boundary Y and denote the (inward) normal coordinate by t E I = [0,1). Actually, we may assume that X is a submanifold of a closed (= compact, without boundary) smooth manifold M, e.g. M = X Uy X, the closed double of X. We extend the identification Y x [0, 1) = N to an imbedding '1' : Y x [—1,1] '—i M.

Definition 11.1. The following chains of Sobolev spaces are naturally associated with our data (s R, s 0): (a) The space H8(M) consists of all complex valued L2-functions u over M which yield elements of in local coordinates. More precisely: Let with open in M and : —p R'2 be a smooth atlas for M. Then, for any smooth function over M with compact support contained completely in there exists an element such that = o E (b) The space H8(Y) is defined in the same way. (c) The space H(X) consists of the restrictions u e H8(M)}, where r+ : L2(M) L2(X) denotes the restriction operator u i—. ulx.

The preceding concepts extend to s 1/2 by continuity. It provides a continuous linear map which is uniformly bounded for t belonging to any compact subinterval of (—1, 1). For given u E H8(M), the

'Vt

H8(M) —+

11. Sobolev Spaces on Manifolds with Boundary

=

mapping t i—' (—1, 1) into

Ut

69

is a continuous mapping from the interval

The delicacy of the trace theorem is due to the fact that we cannot in general apply the bounded Sobolev embedding C which is valid only for s > n/2. A special feature of Theorem 11.4 is the emphasis on the dependence and independence of all constants of the normal parameter t. The main ingredient of the proof is the following lemma:

Lemma 11.5. ForeachtER,sER, ands>1/2 we have (11.5) IIftIIa_1/2 where

J_oo(1+T2)_SdT

is the restriction off on

Proof. First assume t = all y E (11.6)

because

0.

for f E

x {t}.

Note that fo(y) =

f°°00 J(y, t)dt for

In fact /0(y)

= =

J

J

2ir Rn—I

JJ

e_itTf(,7, t) dt dr dire

—00 —00

of the Fourier inversion formula for

f

2ir

r) drj dr dt.

It"

(Since f has compact support, the multiple integrals converge absolutely and the order of integration can be inverted.) Therefore, 2

00

IJo(y)12 = (27r)_2

J(y,

(J

If(y,t)R1 + v2 + t2)8/2(l+ y2 + t2)_8/2dt) 2 IJ(y, t)12(1 + y2 +

j(i

+ y2 +

Since 3> 1/2, the second integral exists and we can apply the HolderSchwarz inequality. Now if we make the substitution t = (1+ y2)'/2r we get

f(1 + y2 +

=

(1

+ y2)8+l/2 J_oo(1 + r2 )_8dr.

II. Analytical and Thpological Tools

70

Therefore,

(1 +

(2ir)_2J

(1

+T2)_8dTj

IJ(y,t)12(1

+y2 + t2)dt,

and integrating with respect to y gives the desired result for the restriction f '-+ ft for t = 0. For general t we make the dilatation g(y, s) := f(y, s+t) and receive (1 +

lift 118—1/2 = 11901Is—l/2

hf 118(j (1 +

=

the dilatation of the variable yields only a phase shift of the Fourier transform: because

= = = for

=

E

f(y, s + t) ds dy

J

f(y,

J

dy

eteTJ(e)

hence

=

and lIgD.

= 111118.

Proof of Theorem 11.4. We suppose that H3(M) and We choose are equipped with fixed norms and an open covering {Uj of Y, an atlas of charts of onto open of unity. Let and a corresponding C°° partition subsets of t belong to a compact subinterval J of (—1,1). We choose a real C°° function x which is equal 1 on J and has support in (—1,1).

Then for any s> 1/2 and uE C°°(M) we have lI14tIIHa_.1/2(Y) < C1

where Cl does not depend on it. Also

= c2IX(t)'I

11. Sobolev Spaces on Manifolds with Boundary

71

where t) := Now, using Lemma 11.5, note that (11.9)

Corollary 11.3 here says (11.10)

IIV&IIH.(Rfl) C4IIUIIH.(M).

with All the inequalities together imply IIUtIIHa_1/2(y) c independent of t and u. Therefore the mapping u Ut can be extended to all of H(M), s> by continuity, and yields, for each

t, a continuous map of H(M) into H''/2(Y) which is uniformly bounded for t varying in any compact subinterval of (—1,1). To finish our proof we shall show that the parametrization t '—' Ut for any given is a continuous mapping from (—1,1) into U E H'(M). This follows from the fact that the function (y, t) ut(y) is an infinitely differentiable function in Y x (—1,1) for u E and from the earlier derived continuity properties. fl

shall now give various reformulations and extensions of the trace theorem. We

Corollary 11.6. Theorem 11.4 remains true, if we replace the closed manifold M by the manifold X with boundary. More precisely: For X of the s > 1/2 and a given parametrization 'I' : Y x [0, 1] collar of Y in X there exists a natural separately continuous mapping which is uniformly bounded on any H°(X) x 10,1) compact subinterval of [0, 1).

For the proof of this corollary and for later use, we recall various restrictions, extensions, and reflections in function spaces over manifolds with boundary. Not all of the mappings are natural; some depend on the choice of smoothing functions and local coordinates. Definition 11.7. We distinguish the following natural mappings (for simplicity s 0): (a) The restriction operator : H8(M)

(11.11)

is given by u

uIx.

H3(X)

II. AnaJytical and Topological Tools

72

(b) The extension by zero operator

L2(M) is defined

L2(X)

by

Iu

(11.12)

onX

onM\X.

Then the composition e+r+ yields the cut-off operator. (c) The reflection operator J : u(y,t) .— u(y, —t)

(11.13)

is only well-defined for functions with support close to Y. It can be defined for the whole function spaces by multiplication with a smooth function x which is equal to 1 near Y and vanishes outside a neighbourhood of Y. (d) A continuous extension operator of codimension 0

forkENLJ{O} is given by graded globalized reflection t)

(11.14)

ift>O

( u(y,t)

if t

is given close to Y in local coordinates by (11.15)

t) := X(t)

J

—

tip)

where x is a smoothing function equal to 1 close to Y and vanishing outside a neighbourhood of Y, and is C°° with compact support on a coordinate patch of Y. Note that 0. —, (f q5)g as The claimed properties of the natural maps follow immediately from the definition. Then Theorem 11.4 implies Corollary 11.6 by the con: H8(X) H8(M). tinuous extension

11. Sobolev Spaces on Manifolds with Boundary

73

Note that the definitions and statements of this section remain valid, if we replace the function spaces by spaces of sections in arbitrary smooth Hermitian vector bundles. By local trivializations we obtain:

Corollary 11.8. Given a bundle E over X we can extend E over M. Near Y we can represent it as x (—1,1) so that sections u of E can be written in the form ut(y) = u(y, t) with y e Y, t E (—1,1) and ut(y) E EJy = EIyx{o} EIyx{t}. Then, for s> 1/2, the trace map y: H'(E) x [0,1) —*

H'"t2(EIy)

is well-defined, separately continuous and uniformly bounded on any compact subinterval of [0,1). We end this chapter with a brief discussion of a few examples.

Examples 11.9. Let

denote the n-dimensional torus is generated by the complete orthonormal sys:= tem of C°° functions {eV}VEZn with (b) For arbitrary s, the space Ha(Tvz) consists of the completion of under the norm (a) The space

:=

Iü(v)12(1

+

11,12)8,

where the zi-th Fourier coefficient is given by

:=

J

In other words, any u belongs to exactly when the converges absolutely to u in the II• norm. sequence E (c) Let y i-4 (y, 0) be the natural embedding of x T' 118

S'. Then for any u E H8(T") and arbitrary s, r, the restriction is well-defined and belongs to

:=

1

exactly when

(1 +

O fort — belongs to ker(A, s) and it follows that A(u — v) = 0 in X \ Y and (u — v)Iy = 0, hence u — v = 0 because the Dirac operators fulfill the unique continuation property. This ends the proof of statement (a) of Theorem 12.4 for s> The general case needs additional arguments and will be discussed in the next chapter.

= (b.1) First we want to show that the limits exist, are uniformly approached, and in fact provide a pseudodifferential operator P± of order 0 with the claimed principal symbol. It suffices to consider the local situation. Since the Green form C is invertible, it may be neglected. Thus we consider operators of the are form with := where the locally supported real valued C°°-functions on X of a C°°-partition of unity suited to a finite covering of X with open sets. Distinguish three situations. (i) When the support of does not intersect Y, then all is trivial since = 0. (ii) When and has a smooth kernel as integral op: H8(Y) erator and C°°(M) is continuous for any 8 < 0 and therefore for all s. From the well-known Sobolev lemma (or from the stronger Theorem 11.4) we then have that converges uniformly as e 0+ and the limit yields an operator of orhave disjoint support, M#1 A—'

der —00.

have support in an open set U which is contained in the strip {ItI < 1}. We choose coordinates is trivial. Unfortunately, no diand assume that x: U (iii) Suppose now that

and

rect approach is available for investigating will lead to an operator T But a slight modification of with well-defined traces Then the investigation of the correcting terms will yield the claimed convergence, pseudo-differential and symbol properties. More precisely, choose a C°°-function ço with supp C U and 1 =: V. Then is a pseudo-differential on supp(l&I +

operator with the (total) symbol given explicitly by C_k with c_1= (a1)' on V. Here a1 = u1(A) denotes the principal symbol of A. We denote by ck(x, D) the homogeneous pseudo-differential

12. Calderón Projector for Dirac Operators

83

operator of order k over U given there in the local coordinates x by

ck(x, D)u(x) =

(12.14) Set

fork =

Ck :=

J

T :=

(12.15)

Then for any k0 EN

—l,—2

C_k

—

k In fact (am ®g; v)

=

Jf

(am(t)g(y); v(y, t)) dydt

0)dy

for v E H8(X) as m — oo. Now recall

á1(r, ij)

=

f

f =f =

Then we obtain (12.17)

= lim = lim

m—'oo

f =

t)

J J

t;

e*t1â(r/m)c_k(y, t;

r1)drdi1

T)dT)

II. Analytical and Topological Tools

84

Now we replace the integral

for any E by an integral over a finite path r(q) in 0). In a certain sense this replacement is the decisive step in establishing the Calderón projection, since it provides the uniform boundedness of the integrands which permits passing inside the integrals. Then for (12.17), this yields the expression (12.18)

t)

=

eitrc_k(y, t;

J

r)dr1 )

(

= as m —i f Before drawing conclusions from (12.18), return to the replacement

since

of

â(r/m) converges boundedly to â(0) =

Let q(i-), r E R denote the integrand of

q(r) :=

t;

00.

1

in (12.17)

r).

We are going to show that q

• is continuous on R (for t 0), • extends to a holomorphic function in IZR := {i- E C R} for some R, 0, In • and suffices q(r) = for some N, when r —' oo in 11R• Then one obtains I

p00

J

—oo

pR

q(r)dr

=

j

—R

pO

q(r)dr —

J

q(Re8)RiezOdO

which is sometimes emphasized by denoting both sides of the equation by the expression q(r)dr. We shall do likewise. First find the necessary holomorphic extension of c_k(y, t; 'r) in an appropriate region of 0). All the ck are rational functions of r, hence meromorphic. In fact, their poles are the roots of a1. More generally we can argue as follows: From the definition of (c_k) k=l,... and the symbol composition formula for pseudo-differential operators, we obtain the following re-

12. Calderón Projector for Dirac Operators

lations for (y,t) c_1a1 c_ 1a0 +

c_2a1 —z

= Id

0a1

Oc_1 0a1

.

=0

—

3=2

c_kal+c_k÷iao+...c_la_k+2—z

h—

Oc_k+i 0a1

.02c_k÷202a1 +*

i.e. C_k can

be written as

(12.19) aj1 x (sums and

products

of derivatives of a1, a0,... ,a_k+2 and of c_l,c_2,... ,c_k+1).

Here

ak denotes the asymptotic expansion of the (total) symbol

of A. Now ai(y, t;z7, ,-) is linear in (ij, r) and non-singular for 1zi12+r2 > 0,

and r E R. Allowing r to be complex, there is a compact set Z in the complex plane C, not intersecting the real axis, such that for I'iI = 1 and (y, t) E supp the complex values of r, for which a1 is has a holomorphic extension not invertible, lie in Z. Thus = to C \ Z for such and (y, t). Then by induction it follows from (12.19) that all C_k and their derivatives have such extensions, each of which is bounded by some polynomial in In. See also Figure 12.2. Let denote the part of Z which belongs to > 0}. We choose a positive real R such that C {InI ( R}. For any 77 E define a closed path '1 E

r(77) := 0 ({r I ml max(1; R1771)}

0})

with counter clockwise orientation and a closed path r0(77) as indicated in Figure 12.3. Our choice of R ensures that the integrand ettT&(r/m)c_k(y, t;77, r) of our real integral in (12.17) is holomorphic for r in the region which is enclosed by r0(77) for all natural m and k. Let us denote

II. Analytical and Topological Tools

86

Fig. 12.2 The complex values of r for which al (y, t;

r) is not invertible

•. .

.

S S

S

S

S

S

.

S T

Fig. 12.3 The closed paths

and

the joint arc of the contours and above by r1, and the segment of close to 00 by r2. Note that is bounded for

t 0 and

0 and that

decays exponentially as In

—+ co

in {E(r) O}. From all this we get

12. Caiderón Projector for Dirac Operators

(00

87

fRIII

j.oO

J...=J...+J...=J ."+J".+ ...

...

r(,7)

since we can move the segment r2 arbitrarily close to oo and thereby let the suppressed integrals go to zero without changing R(v) and the contour by the contour integral over the finite path Having replaced obtain a dominated convergence for the integrands (bounded we for all m by a common polynomial in ftI). We may therefore pass inside the integrals. This proves (12.18). t) in (12.18) is C°° for t > 0, and all its It is clear that derivatives in y and t extend continuously to {t 0}. This gives the in L2(Y;S+Iy) fore 0+. uniform convergence of From (12.18) it also follows that the hereby well-defined operator is a pseudo-differential operator of order 1— k, since its symbol P—k(Y,'7) :=2ir-1-J

c_k(y,O;?1,T)dr

in the integral representation (

f

=

urn

P—k(Y,

and hence belongs

is homogeneous of degree 1 — k in i7 for large

to the standard symbol class Si_k. More precisely, we have for real

8>

1,

=

f

=

r(817)

f

since C...k remains homogeneous in its holomorphic extension,

= 3—k+I

f

c_k(O,y;r/s,r1)s'dr = 3—k+i

r(a17)

=

3—k+1

f

c_k(0, y; r', ii)dr',

fl {IrI = 81—k 2irp..k(y, ii).

since

f

c_k(0,y;T',71)dT'

r=

is the boundary of

0) and therefore equal

r

II. Analytical and Topological Tools

88

(It is not crucial to the argument that s—'r(sij) = but only that the two paths are 0-homologic — form a boundary — in the region where the integrand is analytic.) In our investigation of let us see how far we have gone. First we saw that the corrected operator T = —

C....k of (12.15) is an operator of order —k0 and

that

is well-defined for g E C°°(Y; E C°(Y; Of course also extends on all Sobolev spaces and is pseudodifferential. This follows by the same argument, as in deriving (12.16),

replacing T by Dl', where D is any differential operator of order k, Ic E N. Then for sufficiently large Ic0, namely, Ic0> 1 + k + n/2, once again Sobolev's theorem and the continuity of —'

G is continuous from H y) to G°(S). converge uniformly as e —' 0. Hence is welldefined on all Sobolev spaces. Then we proved that for any Ic show that also

I

Hence

P-k =

urn

is a well-defined pseudo-differential operator over Y of order —k + 1. This shows that the sections converge uniformly to limits

+>

urn

ke

Here we used the observation from the beginning of our proof of part

(a), namely, that Kg belongs to L2(X; S). This permits the introduction of by bounded convergence. From Green's formula (Proposition 3.4), we obtain

J = JItI>c

(Kg;(A)v) (AK9; v)+J

)—J

;

II. Analytical and Thpological Tools

= {t = ±e) denotes the two submanifolds of X of codimension 1 defined by the normal coordinate t = ±e. Since AK9 vanishes outside of Y, the two integrals and ft<e vanish and we obtain where

(12.22)

J(G •g; vly) = urn J e—.O+

= j(C.

(C. + P_g); vly).

inside the integrals because all sections are C°°, the domain Y is compact, and all Y±e may be identified with Y. Since C is invertible and {vIy I v E C°°(X)} = C°°(Y), equation (12.22) proves P+ + 1'— = Id. No new arguments are needed to show range C H+. We proved already that and that AK,.g = is supE in fact, in the interior of ported in Y and therefore =0 One could pass

E ker(A,s) for all s 0 if9 E

0

Remarks 12.5. (a) One observes that the principal symbol of the Calderón projector P÷ has locally (over the connected components of Y) constant rank for all non-zero covectors (y, Consider the subspace Ms,>,, of spanned by the eigenvectors corresponding to with positive real part. Since A the eigenvalues of b(y, : —' takes the form + B) close to Y, we get a(t, y; r,

= C(y)(ir + b(y, ii))

for the principal symbol of A close to Y as noticed above. The el-

lipticity of A therefore implies that b(y, has no purely imaginary eigenvalues. Hence no eigenvalues can pass from the one complex halfplane into the other, and the dimension of cannot change locally.

(b) Now consider the space (12.23)

12. Ca3derón Projector for Dirac Operators

91

The space consists of those solutions of the given system of ordinary If differential equations which are exponentially decreasing on we identify the solutions with their initial values, we obtain the space M> vu,.

Applying the preceding theorem, we shall give a simple geometric proof of the following result, observed in BooB & Wojciechowski [1985]. It answers the question of how to describe the kernel of the Poisson operator (12.24)

is bijective, By Theorem 12.4a, K+IH+(A,s) : H÷(A,s) —i and by Theorem 12.4b the closures of H÷ (A) and H_ (A) are complementary. This complementarity relies on our construction of the invertible double A and remains valid for all invertible Dirac oper-

ators over a partitioned manifold, but clearly breaks down when a given Dirac operator over a partitioned manifold is not invertible. On the contrary, the following corollary expresses the kernel of (12.24) intrinsically in terms of the given manifold with boundary and provides a short exact sequence for elliptic equations on manifolds with boundary, namely, (12.25)

0—' G_I(H+(A*,s)) '—. H84

ker+(A,s) —÷0.

Corollary 12.6. Under the assumptions of the preceding theorem, the subspaces and onal and complementary.

of C°°

are orthog-

Proof. First extend the Dirac operator A = : C°°(X; S) to the operator A = A over the closed double. Denote by P(A) and 2(A) the Calderón projections of L2(Y; ontotheclosureofH÷(A) = H+(A)andofH_(A) = G'(H+(A)) = Then and and the corollary follows from Theorem 12.4b. Another interesting consequence of Theorem 12.4 is that it provides

with the operator

a left-paraxnetrix for A. The point is

II. Analytical and Topological Tools

92

I

Id —K+-)o

Fig. 12.4 Sketch of

A' Af = I

—

of

that of course A'A =

Id over the closed manifold X, but there appears a correction term K+1o coming from the boundary data when

we restrict the operation to the manifold X with boundary Y. For a rough visualization see Figure 12.4. —, C°°(X; S-) Lemma 12.7. For any Dirac operator A : C°°(X; over a compact manifold X with smooth boundary Y, the operator

—.

(12.26) 3 well-defined and we have

(12.27)

r+A_1e+A = Id—K+'yo

Proof. Consider the operator of (12.26) on the distributional level.

12. Calderón Projector for Dirac Operators

Let f E

and

hE

93

We have

(12.28)

= f(Af;

=

)*e+h)dX

= =

ty)dy

— (G70f;

= = ((Id

—

It was pointed out that P(A) is just a projection, i.e. a not necessarily orthogonal idempotent. Actually one can always replace P(A) by the corresponding orthogonal projection in the case of product structure near the boundary and when A is an operator of Dirac type (namely with unitary G and seif-adjoint tangential part B). The orthogonal projection obtained is a pseudo-differential operator with principal symbol equal to the principal symbol of 1'(A). Lemma 12.8. (a) Let P be a projection in a separable Hubert space. Then Port

PP

+ (Id —P)(Id —P))1

is an orthogonal projection onto the range of P. (b) If P is a pseudo-differential projection with orthogonal principal symbol p, we have (TO(Port) = hence Port — P

is

= I',

a compact operator (operator of order —1).

Note. In the terminology to be introduced in Chapter 15, this means that P and Port belong to the same Grassmannian Grp. They

also belong to the same connected component of that PortlrangeP = Plrangep = Id, so index{PPort : range Port

cf. Theorem 15.12 below.

range P} = 0,

The reason is

II. Analytical and Topological Tools

94

Proof. (a): First {PP* + (Id _P*)(Id —P)} is invertible. In fact, pp. +(Id—P)(Id—P) >0, since

({PP + (Id—P)(IcJ—P)}u;u) =

+ II(Id—P)u112 0 with equality, if and only if Pu = ts and = 0, hence 0 (Pwu) = (PPu;u) = IIPuII2 = Hull2 and so u = 0. Next observe the vanishing of the commutator [PP; —P)} + (Id

and hence of the commutator [PP; {PP + (Id —P)(Id —P)}']. This implies

= {(pp') + ((Id_P*)(Id_P))*}_l(PP*)* =

= + (Id —P)(Id Then we find (keep track of the brackets {... }): (Port)2 =

{PP + (Id

=

—P)}' P)J — (Id

+ (Id = {[...J —

—P)} {... }

P)(Id P)} {... = {[Id —(Id —P)(Id —P)] [PP + (Id (Id

+ (Id

= {Id _(Id

Port

Port —P)]}

—P)}' Port

—P)} Port

= Port — (Id = Port — 0

Port.

..

{(Id P)P} P* (PPm + (Id —Pt)(Id

=

Now apparently PPort =

so we only have to show that Port P =

P. Let v = Pu. Then {PPt + (Id Port(v) =

(Pu)

+ (Id —P)(Id —P))' (Pu) = Pu— (Id—P)(Id—P)P{PP' —Pu. = Pu — (Id —Pt)(Id —P)

(b): If P is a pseudo-differential operator, then Port, too; and if P has an orthogonal symbol, then our construction does not change the principal symbol. U

13. Existence of Traces of Null Space Elements

The establishing of the Calderón projector for Dirac operators over compact manifolds with boundary is rounded off by showing that all elements of ker+(A, s), even the distributional solutions, have well-defined traces (boundary values).

We still have not completed the proof of Theorem 12.4a. The argument presented in Chapter 12 works only in case s > To apply the closing argument of the proof of Theorem 12.4a for s we have to show the existence of the trace in for v E (A, 8). This is the subject of the next theorem which is crucial to the whole theory of elliptic boundary problems. We adopt the notation of the preceding chapters, especially • the construction of the invertible double A of chapter 9, • the parametrization {Ye}tEJ of a collar neighbourhood of the boundary Y, the restriction operator r+, and for s > the trace operator-y : xI of Chapter 11, • and for arbitrary real s the null space ker(A, s) and the Poisson type operator : —. ker÷(A, s) of Chapter 12.

Theorem 13.1. Let v E ker÷(A, s) (then in particular v is smooth in X \ Y, Av = 0 in X \ Y, and for e > 0, the trace is a welldefined element of converge to an element 9 E

Then, as e —' 0÷, the sections

to a section 'i) E H3(; S+) v (see Definition 11.7d). Then w := Ai3 is an element

Proof. First we extend v E H8(X;

such that of H8_1(i;S_) which vanishes on X \ Y. Therefore it is enough to prove that, for any distribution w E Ht(X; Sj with w = 0 in X \ Y, lim

exists in

II. Analytical and Topological Tools

Use a partition of unity to localize the problem. One has to investigate the limit lim 'Ye(McpA1M,1,W),

and are smooth functions with compact support contained in a neighbourhood of a given point y E Y. Hence consider the folwhere lowing problem: Split = R. x where

n-

— 1/

— U—

It is a pseudo-differential Let Q denote the operator operator of order —1 in R'. Then, to prove Theorem 13.1, one must only show:

(with values in CN) vanishes for is a well-defined element of

(13.2) Assume that w

U > 0.

Then

To prove (13.2) we use the operator

:= in denotes the Laplacian — and i/i + is a positive_pseudo-differential operator of order 1 with total symbol a(y, = We have the following classical + lemma.

Here

Lemma 13.2. For any s E R, the operator A+ gives an i.somorphism of Banach spaces

—' Proof. To prove that A+ extends to a bounded operator from H3(RTh) it is enough to show that i/i + extends to such a mapping. Let / E H8(R'). We may assume s > 0 since the general result then follows by duality. We wnte (x, = (u, y; r, x TB?' with y = (xi,... We have and (= (Ci,...

f

+

13. Existence of Traces of Null Space Elements

97

hence

IN/i +

= 1k/i +

+

+ J lf(e)l2(1 + =J

+

=

11111.,

thus proving the continuity. Also the proof that A.,. is an isomorphism

is trivial. Notice that (A+)* = Ou+V11 + =: A_. Then the operator (A+)A+ _(O)2 + + 1 = 1+ denotes the is an isomorphism between H8 and H8_2; here Laplacian in R'1. El The reason for using the At-operator to prove (13.2) is the following statement. Proposition 13.3. Let w be a distribution in with support in the half-space {u O}. Then for any t E R, the distribution also has support in {u ( O}. Proof. Let (w, f) denote the duality pairing between elements of 8' and S. Then for A...., defined as above, we have = (w,M..f). It suffices to show that At..f vanishes for u 0, if f E S vanishes for

u e for some e> 0. The point is that for such an f

J(r,

f(u, y) dydu

= .f has a holomorphic extension to the half plane {c(r) 0. We have

Qw =

+

—

= [Q,

[Q,

(w) E

and by the trace theorem 11.4

urn i'e[Q, A÷JA'(w) exists in trace

The same argument gives the existence of the urn

e

Now

= (—& + =

+

+ v'l +

II. Analytical and Topological Tools

100

Since the operator l'o

(v'i +

+

acts in tangential direction we have

+

=

=

+

('yeQA'(w)) =

We are left with the term

= =

vu

+

(7OQA;'(w)).

We have

+

The second sununand is smooth (for a suitable choice of and The first summand actually belongs to the null space of A on X÷ \ Y, since

= Here, let Xj, denote the two halves of the closed double X of X = Xi..

It follows from Proposition 13.3 that A' (to) has support in u 0, hence h := has support in X_. Then A'h belongs to H1(X; S+) and has a well-defined trace in Y. We also have

0= urn

= lim

+

= c ( urn 'ye(Ou + B)(A'h) =

G

( urn

+ B-yo(A'h)

which implies (13.7)

lim

= —B'yo(A'h),

and the theorem is proved for s = —1. The argument works for any 8> — To prove statement (13.2) for arbitrary s we argue by induction, using step by step the lifting argument Qw QA' (to) of (13.6). To explain the induction, let us repeat the arguments: Let v be a distribution which is a solution of Av = 0 in X \ Y. As

in the last part of the proof for s =

—1, we obtain that the traces 10((OuYcv) exist for any k 1, if the trace yov exists. The point is that we have

(13.8)

= 0,

13. Existence of 'fraces of Null Space Elements

and that we can solve that equation to obtain a formula for 70((Ou)Icv)

in terms of

I 0 j < k} as in (13.7) for k = 1. In particular, let w E H8(R") with supp w C {u 0} and let k be sufficiently large, namely k> —s — Then it follows from the preceding argument that the distribution E

has a well-defined trace

E

H8+4(RTh1).

Now use that result for the induction. To explain the induction, let us begin once again with a special case. Let w E HS(Rfl) with suppw C {u 0}, but now 8> Then

+

Qw = [ci,

= =

A÷]A',

+

+

By Lemma 13.4, the first summand belongs to hence it has a trace by the general trace theorem 11.4. By the same argument QA2w has a trace. The third summand is equal to —

+

—

The traces are well defined, since gentially, it follows that

2(9utJl

and

+ i) QA2w.

+

as explained above, exists. And, since acts tanis well defined, and we are left

with the trace of All we need to show is that QA'w has a well-defined trace in since then our method of solving (13.8) for the normal derivatives yields that The same is true for has a well-defined trace in %/i +

and hence for A÷QA'. Now consider

QA'w = QA'w —

+

= [Q, A+JA2w + The first summand belongs to and hence has a well-defined trace in With regard to the second summand, we nohence also tice that QA2w has a well-defined trace in

11. Analytical and Topological Tools

102

by our (13.8) argument. This shows the existence of the

trace of in For general s, use the formula k

Qw =

The first summand is in hence it has for k> —s— a has trace, and we argue as before to show that then each a well-defined trace in [J

show that the traces established in the preceding theorem are and the reasonable with respect to the Poisson type operator Calderón projector 'P+, we prove Proposition 13.5. Let v E ker÷(A,s) and r := 'yo(v). Then To

v=K÷(r)

(13.9)

and

Now let s be arbitrary Proof. The proposition is proved for s> s). Let x E X \ Y and choose e so that and v E

:= X \ {[0,2ej x Y}.

E

The distribution v is C°° in Choose

hence v =

in

and t/' in C°° (X) so that

and

(13.10)

Then

v(x) =

(13.11)

The operator

is a smoothing operator, and we have a conver-

gence in

—,

S—)

where t is a negative number s. This shows that for any e and any xE

=

v(x) = Therefore we have

v = K+r

and

P+r =

urn

= lim

'YcV =

r.

0

To complete the proof of Theorem 12.4a, we need two more results.

13. Existence of

of Null Space Elements

Lemma 13.6. Let r E H8_4(Y;S+Iy) and P÷r = r. Then there exists v e H8(X;

such that Av =0 in X \ Y and

'yov= urn In other words, range(P÷) C

Proof. Let =

denote a sequence of smooth sections such that S+Iy). Then the previous estimate r in is a Cauchy sequence in H8(X; (12.12) implies that hence it has a limit and of course

urn K+ (ru) = K+r = v,

v—,00

and once again we have

r = 2+r = urn '7eV. e—O+

Recall

that

(A, s) denotes the closure of

Corollary 13.7. The range of and the operator injective, when restricted to

(A) in H8

Then

is the space Ha_i

in :

s).

Proof. Assume that we are given r1 = 1'+ri and r2 = P+r2 in =v=

such that

Then

urn

This corollary ends the proof of Theorem 12.4. Next we use Theorem 12.4 and Lemma 12.7 to show the existence of traces of other distributional sections not necessarily belonging to the null space.

Theorem 13.8. Let v E Ht(X; t, s real, and s > — in

Then

S) with

the trace of v on Y is well-defined

Proof. If t > s + 1, we have t> from Theorem 11.4. Assume t < s

and the assertion simply follows is + 1. Then w

well-defined in and we have Aw = Av in H8(X;S); hence — w E ker+ (A, t). Theorem 12.4 implies that there exists a

II. Analytical and Topological Tools

104

unique

r=

E

Ht_4 (Y;

such

a well-defined trace in we have

that = v — w and w has C Ht_4(Y;S+ty). Therefore

'yov= urn We have one important case which has already been established in the second part of the proof of Theorem 13.1.

Corollary 13.9. Let v E

and Ày

L2(X;S). Then v

has a well-defined trace in H4(Y; Remark 13.10. With Theorem 13.8 and its corollaries we sharpen the trace theorem of Chapter 11, which gave a trace in Ht 4(Y) for u Ht only if t> Theorem 13.8 shows that the assumption t> is dispensible, if Au E H8 for some 3> — In Chapters 19 and 20 below, it turns out that the theorem and its corollaries are decisive for regularity and closedness of (global) elliptic boundary value problems.

14. Spectral Projections of Dirac Operators

We account for the construction and the basic properties of the spectral projections associated with the tangential part of a Dirac operator.

In this chapter we discuss one more pseudo-differential projection which plays an important role in the theory of elliptic boundary problems for Dirac operators. Unlike the Caiderón projection, which is defined by global data of a Dirac operator A over a manifold X with boundary Y, we shall employ only the tangential part B for the construction of the spectral projection. In this chapter we may therefore forget about X and A and just consider arbitrary pseudo-differential elliptic self-adjoint operators over a closed manifold Y. Let B : C°°(Y; V)

C°°(Y; V)

be such an operator acting on sections of a Hermitian vector bundle V over Y. Assume (though it is not necessary) that B is of first order. it is well-known that B has a discrete spectrum contained in R numbered like (14.1)

and one can find an orthonormal basis {CJ}JEZ of L2(Y; V) consisting of (smooth) eigenfunctions of B (i.e. Be3 = j E Z), see called e.g. Gilkey [1984; Lemma 1.6.3]. Such a system e,}3€z a spectral decomposition of L2(Y; V) generated by B, or, in short, a spectral resolution of B.

Definition 14.1. For a given self-adjoint elliptic operator B and for any real a we shall denote by L2(Y; V) —' L2(Y; V) the spectral projection, that is, the orthogonal projection of L2(Y; V) onto the subspace spanned by {e1 I A, a}, where {A3; e3 },€z is a spectral resolution of B.

II. Analytical and Topological Tools

106

Note. We shall write P>a(B), when we want to stress the dependence on B; we shall omit the a when a = 0; and we shall adopt the notation P< := Id — Hence

Po(B)

P< :=

and

= Id —

It is well-known that the spectral projection P>a is a pseudodifferential operator of order zero (see e.g. Atiyah, Patodi & Singer [1976, p. 48], Baum & Douglas [1981, Proposition 2.4], or BooI3 & Wojciechowski [1985, Lemma 2.2]). We shall repeat the arguments as this proposition is basic for the following:

Proposition 14.2. The spectral projection P>a is a pseudo-differential operator of order zero for any real number a. The principal symbol of P>o does not depend on a, and for any T*Y\{0} it is equal to the orthogonal projection :

—i

of V1, onto the direct sum of the eigenspaces of the automorphism :

corresponding to the positive eigenvalues. Here b denotes the principal

symbol of the operator B.

Proof. For any pair of real numbers a < b, the operator P>a — Pb is the projection onto the direct sum of eigenspaces corresponding to the eigenvalues from the interval [a, b). This is a finite-dimensional space with a basis consisting of smooth sections; hence is an operator with smooth kernel. Thus it suffices to — prove the proposition for a = 0. Let B' denote the bounded operator (in L2(Y; V), i.e. operator of order zero) (14.2)

B' := (Id+B2)112B.

It has a spectral decomposition (14.3)

;e3

I.. V/i +

3EZ

14. Spectral Projections of Dirac Operators

of instead of II is Auo to two facts, The namely that — the new spectrum is contained in the interval (—1,1), — and there still exists e > 0 such that specB'rl(—2e,2€) C {0}.

(14.4)

It is well-known from spectral theory, cf. e.g. Kato [1976, Section 111.6.4], that the operators and P< = = Po(B) = P have the integral representation (14.5)

(B'—A)'dA,

2irz

and

(B'—A)1dA,

27r2

where (14.6)

:= {(1 + e)ett + 1},

and

r< := {(1 — e)eit

and e is chosen so small that [—e,0) fl spec(B') =

0, see

—

1}

Figure 14.1.

C

Fig. 14.1 Separation of the spectrum of B'

It is easy to determine the full symbol of the operator (the resolvent) (B'—A)' for A E C\spec(B') in any local chart. Its asymptotic expansion has the form P—, (y, A), where the Pj are rational with poles at A = ±1 (see e.g. Gilkey functions of A of order —j [1984, p. 51f]). In particular we find for the principal symbol of B' b'(y,

= p÷(y, ,j) — (id

(y, 7) = p+

—

p_.

II. Analytical and Topological Tools

This yields (b'

—

A)'(y,ii) = [(b' —

= 1

=

1

1 1

1

+

This shows that the principal symbol of P> is equal to 2irt

(14.8)

=

:J{

dA — 1

1

= p+(y,tñ. In the same way we compute (14.9)

Ck(y,fl) =

_L 2irz

Jr> pk(y,n;A)dA.

Let Ck denote an operator with symbol standard argument, one obtains that P> —Ck

cL(y, 77). Then, by the is an operator of order

—k — 1.

Corollary 14.3. Let us assume that the operator B is the tangential a, Pa(B) — P÷(A) is part of a Dirac operator A. Then, for any an operator of order —1.

Proof. The operators Pa(B) and symbol.

have the same principal []

Remarks 14.4. (a) To prove that P(B) is a pseudo-differential operator, an alternative argument may be summarized as follows:

Let B = UBIBI be the (unique) polar decomposition of B, see e.g. Kato [1976, Section VI.2.7]. Here IBI denotes the non-negative square root of B2 = B*B. It is an elliptic pseudo-differential operator of the same order as B by Seeley's theorem on (complex) powers of pseudo-differential operators. The operator UB is a partial isometry

14. Spectral Projections of Dirac Operators

from L2(Y; V) in itself with a decomposition of L2(Y; V) into the three eigenspaces m+, m_, and mo with eigenvalues equal 1, —1, and 0. These eigenspaces are generated by the eigenfunctions of B corresponding to positive, negative, and zero elgenvalues (the last being the kernel of B). The space mo = ker B is finite-dimensional and consists of C°°sections. Let k denote the L2(Y; V) orthogonal projection onto ker B,

kg :=

(14.10)

where e1,. . . , ej is an orthonormal basis of ker B. This shows that k can be defined by a C°°-kernel, so k is a pseudo-differential operator of order —oo, i.e. a smoothing operator. We observe that

P(B)=

(14.11)

Therefore, in order to show that P (B) is a pseudo-differential operator of order zero, it suffices to show that UB is a pseudo-differential operator of order zero. have the same kernel. Hence Notice that the operators B and IBI+k is a pseudo-differential operator of the same order as B, namely of order 1, elliptic and invertible, and (IBI + is an (elliptic) pseudo-differential operator of order —1. Then we can write UB = Un(IBI + k)(IBI + k)' = UBIBI(IBI

since UBk = of order 0.

0.

+ k)' = B(IBI +

This shows that UB is a pseudo-differential operator

(b) Notice that the Calderón projector of a Dirac operator over a manifold with boundary and the spectral projection P> (B) of its tangential part coincide for the Cauchy-Riemann operator over the disc (see Example 21.1 below). One might ask whether that coincidence is a peculiarity of dimension two. For higher dimensions and certain symmetric spaces, harmonic analysis e.g. establishes related 1-1 correspondences, namely between the space of harmonic forms over a symmetric n-dimensional domain with boundary and the Hardy space of harmonic eigenforms to non-negative eigenvalues over the i-dimensional Silov boundary.

110

IL Analytical and Topological Thols

More specifically, we always have, by definition, the inclusion

kerB C rangeP>(B), whereas the question

kerB

range 1'÷(A)

is non-trivial: Roughly speaking, a harmonic form over the 1-codimensional boundary might have too many conditions to satisfy to be extendable over the whole manifold. Nevertheless, Kori [1993; Theorem 5.10] obtains = P(B) for the Dirac operator over the 4dimensional ball with boundary S3 in a suitable metric. (c) A more general variant of spectral projections of elliptic operators is discussed in Wojciechowski [1985a; Sections 2—4], where the index

theory for families of operators with two rays of minimal growth is discussed.

15. Pseudo-Differential Grassmannians

The homotopy groups of the space of pseudo-differential projections with given principal symbol are computed. Criteria are given for two projections belonging to the same connected component.

The next tool needed to explain what the index of global elliptic boundary problems determines is the Grasamannian of pseudodifferential projections. We shall consider the set Gr9,. of all pseudoand with differential projections P with principal symbol equal to the topology given by the standard norm topology in the space of all bounded operators acting on the space L2(Y; Say). (In this chapter we only assume that is a Herinitian bundle and do not care whether originates from a full or split bundle of Clifford modules.) 0 and We assume that is an orthogonal projection and that Id or, equivalently, that dim ker P = oo = dim range P. We caland it turns out that the closure culate the homotopy groups of of this space is a classifying space for the K° functor; but we shall only exploit the fact that this space has countably many components. Remarks 15.1. (a) The homotopy groups of the corresponding total (3rassmannian in a separable Hilbert space (the closure of are well-known (see e.g. Wojciechowski [1981], BooB & Wojciechowski [1982/1985], Wojciechowski [1985a], and Pressley & Segal [1986]); they only differ by the choice of equivalence classes, namely, whether

one takes classes modulo operators of finite rank, modulo compact operators, or modulo operators of Hilbert-Schmidt type. (This makes no difference to the respective homotopy groups, as shown by Palais [1965bJ). Fuglede [1976] observed earlier that the total Crassmannian consists of infinitely many connected components. The point of our

presentation is that we have to restrict ourselves to the pseudodifferential Grassmannian. (b) We assume here that the term projection denotes an idempotent (P2 = P). However, as base point for our homotopy groups, we choose an orthogonal projection = = Pg). We can do that without loss of generality, as explained in Lemma 15.11 below. More specifically, we let P.÷ be an orthogonal projection onto a subspace of

II. Analytical and Topological Tools

112

is the L2(Y; Sly). In the context of this book, a natural choice of orthogonal projection onto the space H+ (A) of Cauchy data of a given Dirac operator A. As explained in Lemma 12.8, the projection P÷ is a pseudo-differential operator and has the same principal symbol as the Calderón

projector constructed in that chapter. The reason is that

in our case the principal symbol orthogonal.

C) of the Calderôn projector is

We introduce a group of invertible elliptic operators on Sly: (15.1)

:= {g g is an elliptic invertible operator of order 0 on Sly with principal symbol equal to I

The key to the whole problem is the following elementary lemma:

Lemma 15.2. Let P0 liP'

—

and P1

Poll < 1. Then the

such that

operator

T = Id+(P1 — Po)(2P0 — Id)

(15.2) belongs

denote two elements of

to EIIX and satisfies:

TP0=P1T.

(15.3) Proof. We have liT

— Id

ii

= il(P' — Po)(2P0 — Id)il

lIP1 — Poll ll2Po —

Id

Ii

ilPi—Poll 0 we can find a 1 t 2 with /2(5) E CL°° and deformation ft : Sk and 1 t 2. In fact, we can lift(s) — fi(x)Il < for all x E

choose a small 5 > 0 and a related "deformation retraction" of 51 onto 5! in such a way that

H0=Id H,(etO)=1 for—S 0 so that IIT'pII

for 'p E (kerT)1.

The first remarkable fact about the topology of the space of selfadjoint Fredholm operators is that it decomposes into three connected components which are distinguished by the essential spectrum. This will be proved in the following proposition. For a better understanding of this result, recall from ordinary Fredhoim theory: The space F, equipped with the operator norm, decomposes into Z connected components which are distinguished by the index. More generally, for any continuous family {Am}mEM of Fredhoim operators

parametrized by a compact topological space M, we can define a homotopy invariant, the index bundle index{Am}m€M E K(M),

where K(M) is the Grothendieck group of complex vector bundles over M. The Atiyah-Jänich theorem states that the index bundle defines a bijection [M, F] —' K(M), where [.., ..] denotes the homotopy

classes of mappings. Hence the homotopy type of the space F is completely identified: it is a classifying space for the functor K. It turns out that one can develop an analogous theory for the space F of self-adjoint Fredholm operators, not based on the index which vanishes on F, but on spectral invariants, namely the essential spectrum and the spectral flow. The most profound result is the following:

Proposition 16.2. Let F denote the space of self-adjoint Fredholm operators on a fixed complex separable Hubert space H. It has three

16. The Homotopy Groups of F

connected components F÷, 1..., and F.,. characterized by

T

is a positive (negative) operator for some compact operator k T— k

C

T

F. :

T not in T has both positive and negative essential spectrum.

Moreover, the components F+ and F_ are contractible. Proof. First check the equivalence of the two alternative descriptions given in the statement of the proposition: of the sets F+, F....., and If T — k is a positive operator for some compact operator k (one also says T is essentially positive), we have k compact

=

—

k)

T—k positive

C

(0, +oo).

Vice versa, if C (0, +oo), it follows from Proposition 16.1 that there exists e > 0, such that there are only finitely many discrete eigenvalues in the interval (—oo, e); hence T is a positive operator on the complementary space of the finite direct sum of these finitedimensional eigenspaces. The same arguments apply for F_ and F. so that also the spectral characterization of these spaces are established. To proceed further it is suitable to turn to the images of F,

and F. in the Calldn algebra C := B/K of the Hubert space H. Let Q denote the group of units of C and the subspace of seif-adjoint invertible elements in C (with respect to the natural Ce-algebra structure for C). Let p : B —p C denote the canonical projection. A widely

exploited fact is that F = p'(G) which is obvious, since the Fredholm operators on H can be characterized as those which are invertible modulo X. We also see that p maps F into g, and the mapping —, is also surjective: for g E and T E F with p(T) g g and T+2T E F. g By definition

specp(T) := (Al p(T) — Therefore the subsets ing subsets of

A

and F. are the inverse images of correspond-

under the projection p. Of course, here the

130

II. Analytical and Thpological Thols

speces5

—1

0

Fig. 16.1 The decomposition of

1

into its three connected components

point is that the

are defined via the essential spectrum and the g-subspaces only via the spectrum. These subspaces of Q are both open and closed and hence also its inverse images F± and Therefore the space F is not connected and decomposes into the union of these three mutually disjoint subsets. It turns out that each of these subsets is connected. This is obvious for Then (16.2)

does not lead out of

since by (16.1) spec

= (1—

=

(1

—

t)(specp(T)) + t t

T c Hence (16.2) provides a contraction of F÷ to the operator Id, cf. also Figure 16.1. Similarly F_ contracts to the operator — Id. We postpone the proof that is also connected. fl

16. The Homotopy Group8 of F

c.

p(T) Fig. 16.2 The retraction of

onto

: F.

Lemma 16.3. The natural map

131

is a homotopy equiv-

alence.

Proof. Recall the theorem of Bartle and Graves (see e.g. Bessaga & Pelczynski [1975; p. 86]) which states that each surjective, continuous, linear operator from one Frechet space into another Frechet space possesses a right (not necessarily linear) continuous inverse. Since the projection p: B —+ C is surjective, continuous, and linear, we get from the theorem of Bartle and Graves a section s in the bundle p: B C over C = B/K with total space B and fibre K, namely a continuous (in our case definitely not linear) right inverse s of p. Then s(g) +

g

(s(g)) 2

is a section (continuous right inverse) for the mapping

Modulo homotopy, the section we can retract .?. back onto := (1 —

is

also a right inverse of linearly by

t)T + ts1(p(T)),

0 t 1

since

II. Analytical and Thpologicai Tools

132

that all and in fact to belong to the same fibre p'(p(T)) of .1,. This is clear since psI = id, so sip(T) — T E K and therefore also T E K. See also Figure 16.2. for arbitrary T E

Note

[1

Thus the topology of the F-spaces has been replaced by the topol-

ogy of the c-spaces. However, the c-spaces are still rather large and their spectra are not easily grasped. For further topological investigations, it is convenient to replace these spaces by suitable deformation retracts, which only consist of single points, or for which at least the spectrum only consists of single points. We apply the standard algebra retraction (16.3)

:=

g

((1

—

t) id

o

t

i.

As a special case, we have that the group C of unitary elements of the Calkin algebra C is a deformation retract of (and hence homotopy equivalent with) C. Fortunately the c-spaces are stable under the retraction (16.3) because (16.4)

g*=g

g=gt.

Therefore (16.5)

S fl C is a deformation retract of S

for SE Denote the corresponding unitary retracts by Roman letters. Then

=gandg*g=idhenceg2=id}, (16.6)

Ô+={id},

Ô_={—id},

and

From Lemma 16.3 we get with (16.5) the following theorem:

Theorem 16.4. The space

is homotopy equivalent with the space

G*. Now we shall determine the homotopy groups of Ô..

16. The Homotopy Groups of Q *

133

16B. The Homotopy Groups of C. Let H be a complex separable infinite-dimensional Hubert space. consists of the seif-adjoint We have seen in (16.6) that the space involutions in the Calkin algebra C(H) = 8(H)/X(H) of H which are not equal to the identity:

Ô. = {j E C(H)

If = j, j2 = id, and j

±id}.

Thus it can be identified with a suitable Grassmannian

Gr(C(H)) := {q

(16.7)

E

C(H)

I

q2

=q =

0, id).

The isomorphism is given by

j

(16.8)

E

Gr(C(H)).

Consider the case of the corresponding Grassmannian in 8(H): (16.9)

Gr(H) := {P

If we fix a Po neous space

8(H) I P = P2 = and dim range P =00= dim range(Id —P)}.

Gr(H), then Gr(H) is isomorphic with the homoge-

(16.10) U(H)/U(Po(H))

—Po)H)

U(H)/U(H)

where 8(X) denotes the C-algebra of bounded operators on X for XE {H, P0(H), (Id—Po)(H)} and U(X) its unitary retract. The isomorphism is given by the map (16.11)

Rp0 : U(H)

U

UP0U'

E

Gr(H).

Repeating an argument from Chapter 15 (cf. especially the proof of Theorem 15.5 and Remark 15.6, see also Remark 24.5b below) we obtain the following theorem:

II. Anaiytical and Topological Tools

134

Theorem 16.5. The mapping R,'0 : U(H) —p Cr(H) of (16.11) defines a principal fibre bundle with fibre and structure group U(H) Note. By Kuiper's theorem, it follows that Gr(H) is the image of a contractible space and therefore itself contractible.

Now repeat this argument in the case of G.. The only essential difference is that now neither the total space, nor the structure group of the bundle is contractible. As before, let C denote the unitary group of C(H). Similar to how proved Lemma 16.3, see that C has the homotopy type of the space of Fredhoim operators and in particular (16.12)

Irk (C,30)

forkeven =

forkodd

0

for any base point E C. We fix jo (3m. As already noticed (in the proof of Lemma 15.2), P0 —(Id —P0) is an involution for any projection P0. Now the next elementary observation is proved:

Lemma 16.6. There exists P0 E Gr(H) such that p(Po—(Id —P0)) = jo, where p : B(H) — C(H) = B(H)/K(H) denotes the natural projection. Proof. Consider the following diagram which is not necessarily com-

mutative and contains a semi-continuous mapping P> (because of jumps of dimensions):

(16.13)

Gr(H) Here

p

P(T) denotes the spectral projection ofT onto [0,oo) and

is defined by (16.14)

:= p(2P — Id).

16. The Homotopy Groups of : —, is Begin with the fixed class jo E C. C g.. Since surjective, there exists T E.?. such that p(T) = jo• The operator T operator with = {±1}, and T is is a seif-adjoint essentially an involution:

(16.15)

:=

E

p'{id} = {Id+K}.

On the other hand 2 p2 (T) — Id is a true involution and

(T)) =

The following corollary completes the proof of Proposition 16.2.

Corollary 16.7. The spaces C. and F. are connected. Proof. First we show that C. is connected. Fix a P0 E Gr(H) and consider the following composition of maps

U(H)

Gr(H) —f-' &.

The first map is clearly continuous and surjective by Theorem 16.5.

The second map is continuous by construction and surjective by the

preceding lemma. Hence C. is the image of the connected (even contractible) space U(H) under a continuous map and therefore connected. Since, by Lemma 16.3, F. is homotopy equivalent with C., it is connected as well. U As seen below, the continuous image C. does not inherit the con-

tractibility from U(H). Similar to (16.10) above and Theorem 16.5 we now define a map r,0 : C —, C. on the unitary group C of the Calkin algebra by the formula (for fixed io) (16.16)

r,0(u) :=

We want

(1) to show that

a principal fibre bundle, (2) to derive its long exact homotopy sequence, (3) and to determine the homotopy groups of C.. We begin with the following result: : C —+ Ô. defines

II. Analytical and Topological Tools

136

Lemma 16.8. The map rj0 : C —'

is surjective.

Proof. Let j E a..

follows from Lemma 16.6 that j = p(P — Cr(H), and it follows from Theorem 16.5 that there exists a unitary operator U U(H) such that (Id —P)) for an appropriate P

UP0U' =

P, where P0 E Gr(H) was fixed such that p(P0 —

(Id —Po)) = jo• Set u := p(U). Then

j = p(P

(Id —P)) = p (U(P0

—

—

(Id —Po))U')

=

Now investigate the fibre

= {u E C I ujou1 = jo}.

Go :=

it is a subgroup of C. Moreover:

Proposition 16.9. We have Go Note.

C.

C

This result can also be formulated as follows: Go = p(Up0(H)),

(16.17)

where (16.18)

Up0(H) :

{U eU(H) UP0 — P0U

E AC(H)}

and P0 E Gr(H) with p(Po — (Id —P0)) = Then Proposition 16.9 states that any class u C0 can be expressed (in the Calkin algebra) as a class of direct sums of unitary operators on P0(H) and on (Id—Po)(H).

Proof. Define := ±jo). Assume that a Po E Gr(H) is given with jo = p(Po). Then for any class u Go, we have a decomposition (16.19)

u=u+u.... =u_.u+,

where

(16.20)

:=

+

The map iz (u+, u_) is an isomorphism of the group Co of unitary elements of the Calkin algebra onto

16. The Homotopy Groups of •

137

denotes the unitary part of the group p(.:F(P0H)) (or

where

—P0)(H))) resp.) which is a deformation retract of it.

U

To follow the line of arguments in proving Theorem 15.7, we now construct a local section of the bundle r30 : C Let j E with Iii — iou < 1. Then, following Lemma 15.2, the

id+(j — jo)(2jo

t(j) :

(16.21)

id)

—

is invertible in C(H) and we have

t(j)jo = jt(j),

(16.22)

therefore (16.23)

j = t(j)jot(j)'

hence

,

r30(t(j)) =j.

So t defines a local section. To be more precise, we have to take the unitary part of t(j) in order to get an element of C, since a priori only t E p(F(H)). This can always be done. Now following the proof of Theorem 15.7 we get the short exact sequence (16.24)

Z —. lr2k(ô.)

exploiting that lr2k(G

C)

Z

Z

and

lr2k(G)

Z.

Here as usual the homotopy groups are defined relatively to fixed base points. From (16.24) the main result of this chapter is obtained:

Theorem 16.10. Let T0 =

Po — (Id —Po) E and jo E be fixed Then the homotopy groups of the non-trivial component of the space of seif-adjoint Fredhoim operators and of the corresponding space of 8elf-adjoint non-trivial involutions of the Colkin algebra are given by

elements.

—

-.

fori

.

ir,(F.,To) = 7r(G.,30)

=

Z fort

even odd.

17. The Spectral Flow of Families of Self—Adjoint Operators

The spectral flow is investigated for families of seif-adjoint elliptic operators with non-vanishing symbol class over an oriented closed smooth manifold.

Our main tool for the treatment of global elliptic boundary problems for operators of Dirac type is the concept of the spectral flow, this peculiar blend of classical analysis and differential topology, which will be explained in detail in the present chapter.

17A. Continuity of Eigenvalues From a topological point of view, the most fundamental property of the spectrum of seif-adjoint operators is its continuity under continuous changes of the operator:

Lemma 17.1.

Let A be a seif-adjoint bounded linear operator in a Hubert space H and let a be a positive real number such that the inter-

section spec A fl (—a, a) consists only of a finite system of eigenvalues

(all the eigenvalues repeated according to their multiplicity). Then, for all seif-adjoint operators A' sufficiently close to A, the intersection spec A' fl (—a, a) consists of the same number of eigenvalues

and one has —

)41 hA

—

A'II for each j

E

{k,... , m}.

Proof. First decompose A into the difference A+ — A_ of two nonnegative operators. The operator A+ has the eigenvalues , and A_ has the eigenvalues —Ak,... , Since IIA+ — A'÷hI < — A'hI, it suffices to consider the case when A is non-negative.

17. The Spectral Flow

139

Recall that then the j-th eigenvalue A, can be characterized by the minimum-maximum principle (see e.g. Reed & Simon [1972] or Kato [1976; Section 1.6.10]) through

Ajsup{UA(vI,..,vj)Ivl,...

,v,€H},

where UA(v1,... ,v,) =

inf{(Aw,w) liwli = 1 I

and

WE £C(vl,

Now for each choice of vi,... ,v3 and for each > 0 w with liwli = 1, orthogonal to v1,... ,v3 such that (Aw,w) — UA(V1,... It

choose

a vector

,v,) <e.

follows that

UA'(Vl,...

,v,)

(A'w,w) = (Aw,w) +

((A'

UA(V1,...

,vj)+IIA—A'II+e;

- A)w,w)

hence

UAS'(Vl,... ,v,) UA(V1,... ,v,)+IIA—A'II.

In the same way we get

UA(v1,... which

,v,) UA'(Vl,... ,v,)+IIA—A'II,

proves the lemma.

Remarks 17.2.

(a) Recall that seif-adjoint elliptic operators of positive order over a closed manifold have a discrete spectrum {Aj},Ez of finite multiplicity. Moreover, there exists no essential spectrum, and the eigenfunctions span the whole L2E. We get the continuity of the eigenvalues immediately from Lemma 17.1. As usual, operators of order r can be reduced to operators of order 0 by the substitution

be a spectral decomposition of the More precisely, let {A,, Hubert space of square Lebesgue-integrable sections in a Hermitian vector bundle E generated by B, i.e., the {Aj}jEz are the eigenvalues

140

II. Analytical and Topological Tools

of B : WE —' L2E, where H" is the r-th Sobolev space, with the corresponding eigenfunctions {e, }jEZ spanning the whole L2E. Then

+ a spectral decomposition of L2E generated by B with all eigenvalues bounded by +1. Hence the continuity of the eigenvalues of B carries over to the eigenvalues of B. (b) The continuity of a finite system of eigenvalues can be obtained for all closed, not necessarily bounded or seif-adjoint operators as shown in Kato [1976, IV.3.5]. is

Now we could provide the definition of the spectral flow directly by exploiting the perturbation theory of the spectrum of closed, not necessarily bounded operators, as mentioned in the preceding remark. However, the topological meaning of the spectral flow will be more transparent, if instead, we recall the homotopy investigations from Chapter 16 and first work with seif-adjoint (bounded) Fredholin operators in Hubert space. Later we shall carry the arguments over to elliptic operators of positive order.

17B. The Spectral Flow on Loops in Now we define the spectral flow of a family {Bt}tE[o,1] C F.. We begin with operator families belonging to the subspace

andBhas only finitely many eigenvalues}.

We can argue in a similar way as in the proof of Proposition 15.4 (second step) in order to show that F°° is a deformation retract of However, we do not need the full argument, since we only want

to explain the notion of the spectral flow. Let {Bt}tEsl be a periodic family of operators of F°°. Then the spectrum of can be described in the following way.

Lemma 17.3. Let B: S' —,

F°° be a continuous map. Then the

graph of the spectrum of B is given by a finite monotone sequence of continuous functions j : I [—1, 1],

i.e.,

= {jo(t),jj(t),... ,jm(t)},

tEl.

17. The Spectral Flow

Definition 17.4. Let £ be the integer such that jk(I) = jk+t(O) for all k, where we eventually define

i-ei-t+i

1 and

...

= 1.

The integer £ is called the spectral flow of the family B.

Remarks 17.5. (a) The spectral flow is therefore just the difference between the number of eigenvalues, which change the sign from — to + as t goes from 0 to 1, and the number of eigenvalues, which change the sign from + to —. In Definition 17.4, we rearranged the eigenvalues to get a monotone sequence.

(b) Consider a family B: to a family with values in F

in .?,. Now we might deform B

this is not needed since the discussion in Section 17A showed that for any family B with values in F,, the graph of the spectrum of B near 0 can be parametrized by a finite set of continuous functions j' : I R. This yields once again a well-defined integer £ such that jk(1) = jk÷j(0). .

However,

Proposition 17.6. The spectral flow defines an isomorphism sf : iri(F,) —' Z.

Since 5,

connected, the choice of a base 2oint in F, is irrelevant to the definition of the homotopy group ir1 (F,). Note.

is

Proof. First the spectral flow is a homotopy invariant of the families: Let A = and B = be two periodic families parametrized by t E I which can be deformed into each other, i.e., there exists a two-parameter family {Ft,U}g,UEI such that for = , = each t, and for each u. Let C = be the boundary = of i.e. A49

forO<sl/4

—

F1,49..1

for 1/4 < s

—

B3_4.,

for 1/2 < s 0 and k 0 respectively, and v denotes the constant function equal to 1 (corresponding to k = 0).

Note that the reduction B

The preceding Example 17.8b raises two natural questions:

(1) Instead of the family = —id/dx + t} of self-adjoint elliptic operators over parametrized by t S', we may consider its suspension, namely the (non-seif-adjoint) elliptic operator 0 .8 A:=—-z—+t Ox Ot

17. The Spectral Flow

over

145

the torus T = S' x S' acting on the space

C°°(T;E) := {f E C°°(I x S') I f(1,x) = for all x E S'}. How are the index of the operator A and the spectral flow of the interrelated? In Theorem 17.13 below we shall explain why these two integers must coincide. family

(2) We noticed that the preceding examples (a) and (b) are interrelated. More generally, instead of the operator A —8/Ot +

or (equivalently) the family we may consider the spectral projections and P< belonging to B0, and the gauge transformation B1 = for a suitable morphism g. But how are the spectral flow of the family (or the index of A) and the index of the desuspension P> —g P< related to each other? In Theorem 17.17 it will be explained why these two integers also must coincide. As we shall show in Chapter 24—26, both questions are fundamental

for the understanding of the cutting and pasting of elliptic operators. Especially the second question seems to be central to understanding the geometry of the Grassmannian of all elliptic boundary problems and to solving single boundary problems. See also Remark 15.14 above, which shows that two pseudo-differential projections with the same principal symbol naturally define a family of seif-adjoint elliptic operators, and that the spectral flow of that family is the obstruction for the projections belonging to the same connected component in the Grassmannian.

17C. Spectral Flow and Index Let S' —p be a family of self-adjoint Fredhoim operators, } where the circle S' is parametrized by u E 10,11. By Proposition 17.6, such a family defines an element of the homotopy group ir1

two families define the same element if and only if they have the same spectral flow. We shall relate the spectral invariant with topological invariants: the indices of related single operators (of the suspension, Theorem 17.13, and of the desuspension, Theorem 17.17). To that purpose it is suitable to introduce an intermediate, typically differential invariant, the Chern character of the analytical index, which is a certain vector bundle canonically associated with the family We shall make all constructions as explicit as possible. and

II. Analytical and Topological Toots

146

The basic notations of K-theory may be found e.g. in Karoubi [1978), Gilkey [1984), or BooB & Bleecker [1985].

Recall that the elements of

K (S' x S1 /({O} x S1 u S' x {i})) are determined by the homotopy classes of the clutching mappings g: S' U(N) of the bundles, and that the generator is the class of the map g(u) = et". In this case the Chern character is the isomorphism

H'(S';Z)

ch : K1(S')

Z)

given by the formula (17.1)

22rs

I Js'

trg'dg=degg,

where g is assumed to be differential. Consider the diagram a-index

K'(S1)

To describe the horizontal map a-index which gives the identification, we recall the construction of Example 17.8a: For fixed compleboth having infinite-dimensional range and a mentary projections non-negative integer k, choose a subspace V, of range(P_) of dimension k. (For negative k one may correspondingly choose a subspace of dimension —k). Then any element of ir1(F.) is V, of defined by a family + 2uPk}

—

= k. Definition 17.9. With the preceding notations the analytical index with

denotes the homomorphism a-index :

—

given by := [hJ, where tary matrices defined by (17.2)

=

—

:=

K'(S') denotes the family of uni-

(Id on(Vk)1 e2" on Vk.

The preceding definition of the analytical index gives at once the identification of and

17. The

Proposition 17.10.

Let

Flow :

147

S' —' F be a continuous family.

Then

=

= k and represent the class of

Proof. Assume by the standard family

= 2irs j0

in iri(F.)

Then by (17.1) and (17.2)

= tr

1

t'

—/

Idvb = dim Vk I du = k. Jo

The remaining part of this chapter is less elementary since we shall

exploit the full strength of the index theorem including the index theorem for families. We proceed with the topological study of families of self-adjoint

elliptic operators in the realm of symbolic calculus. Recall that the homotopy class of an elliptic operator B over a closed manifold Y only

depends on the homotopy class of its principal symbol 0B. For any such symbol, the usual theory of elliptic symbols constructs a symbol class [oBJ, which is an element of K(TY), and which leads to the computation of the index of B by the topological index homomorphism K(TY) —' Z. If B is a self-adjoint elliptic operator over Y, then again its homotopy class only depends on the homotopy class of its principal symbol,

that is, B can be deformed into another seif-adjoint elliptic operator C through a smooth 1-parameter family of self-adjoint elliptic operators if and only if can be deformed into cc in the class of self-adjoint and (for each non-zero cotangent vector) invertible symbols. This is a standard exercise. However, the usual symbol class LaB] E K(TY) is vanishing. In Atiyah, Patodi & Singer [1976, §3], it was found that the characteristic element of a self-adjoint elliptic

operator lies in the group K'(TY): Definition 17.11. (a) Let B be a seif-adjoint elliptic operator acting on the C°°-sections of a Hermitian vector bundle E over a closed Riemannian C°°-manifold Y, and let

yEY,

and I,iI=1,

be

and Thpological Tools

II.

148

the principal symbol of B. Consider the family

for ir

1

and P< be the projections onto the —gPH. —v

This gives a graph of the spectrum of the family

as shown in

Figure 17.4, where P0, /.L1 parametrize the eigenvalues 2t —1 and 1— 2t with multiplicity m0 and m1; hence = m0—m1. As seen above,

we have

m0 = dim{v E P< H I

E P> H) = dim ker(P> —gP H I gv e P< H) = dimcoker(P —gP (B) restricted to range(p÷) is the identity. (b) For mathematical reasons and for the sake of applications, we define a subclass of the class of all elliptic boundary conditions which contains the original Atiyah-Patodi-Singer condition as well as compact perturbations of this condition. This subclass is distinguished We shall call such by requiring that R is a projection with r = boundary conditions generalized Atiyah-Patodi-Singer boundary conditions.

(c) We get a local elliptic condition when the range of r can be written as the lifting of the vector bundle V under the natural projection TY \ 0 —' Y. Then condition (1) of Definition 18.1 is automatically fulfilled. We shall discuss this in greater detail below (cf. Theorem 20.13 and Theorem 21.5). Note that on odd-dimensional manifolds, the decomposition of the spinors over the (even-dimensional) boundary defines a local elliptic boundary condition. We shall exploit this below in Chapter 21 to prove the cobordism theorem for Dirac operators. On even-dimensional manifolds we have a different situation, since for all Dirac operators there are topological obstructions which prevent the admission of local elliptic boundary conditions (see e.g. Boofi & l3leecker [1985, Section II.7.B] or Lawson & Michelsohn [1989, Section 11.6]). Nevertheless, systems of Dirac operators may admit lo-

cal elliptic boundary problems. This is always the case for systems associated with transmission problems, see especially Chapter 25 below.

(d) If, as in Remark 12.5b, we identify the space of initial values with the function space

= {w : R

(0, y;

=0

=

()

18. EBP's and Pseudo-Differential Projections

165

we can reformulate condition (2) of Definition 18.1 in the following way:

(2a) The equation r(y;

=

0

has no non-trivial solutions in

Thismeans

that

injective, and boundary conditions satisfying this condition are sometimes called injectively elliptic. is

(2b) For any 8

r(y, c(S), the equation r(y;

E

This means that

a solution in

=

8 has

:

ranger(y, () is surjective.

moreover, ranger(y, () = V,, then the pair (A, R) is sometimes called swjectively elliptic. Boundary conditions, which are both injectively and surjectively elliptic, we call local elliptic condiIf,

tions, whereas we require less for (global) elliptic boundary conditions, namely, that they only be injectively elliptic (2a) and that they satisfy (2b) and the technical condition (1).

What then is the meaning and strength of the concept of global elliptic boundary conditions? Such boundary problems present no serious analytical problems, but its index theory requires a different emphasis, namely somewhere between the broader class of injectively elliptic boundary problems (leading only to semi-Fredholm operators) and the smaller class of local elliptic conditions, where it suffices to

concentrate on the symbolic calculus with its approximative statements; in our situation of global elliptic boundary problems, as defined above, small changes of the boundary operator R without change

of its principal symbol r can lead to jumps of the index (see below Remark 22.25). Our main interests are the relations between the various expressions of a boundary problem, namely

• the full mapping pair (18.1)

(A,R):

given by the formula

(A,R)(u) := (Au,R(uly)); • the realization of A (18.2)

AR: domAR —4

as an unbounded operator in L2(X; domAR := {u

L2(X;S) with I

R(uIy)

IlL Applications

166

here R(uIy) =

0

means that u belongs to the kernel of the composite

continuous map

H1(X;S1)

L2(Y;V),

where W denotes the first Sobolev space of sections of the spinor bundle over X, '10 denotes the restriction to the boundary, and R0 is the well-defined continuous extension of R to H°(Y; = L2(Y; Note that '10 is even continuous H'(X) H'/2(Y) (Theorem 11.4 and Corollary 11.8); • and the boundary integral (18.3)

: H÷ (A) = range P÷

rangeR.

We have several questions:

(1) Under which restrictions on the operator R does the realization AR become a Fredhoim operator with smooth solutions and smooth solutions of the adjoint operator (AR)*? (2) Under which additional conditions for the operator R does the full mapping pair (A, R) become a Fredhoim operator with smooth solutions and smooth solutions of the corresponding adjoint problem? The question is whether the boundary conditions R, which provide a nice operator AR, are the same which give a nice operator (A, R). (3) Under which conditions does the boundary integral become a Fredhoim operator, and what about the stability of its index (which was called the virtual codimension i(R, P÷) of R in and discussed in Chapter 15 above)? (4) How can the general (global) and the conventional (local type) elliptic boundary problems be reformulated in the language of projections (pseudo-differential Grassmannians)? One of the reasons why we can answer these questions is that the projections, induced from the boundary condition R, are pseudodifferential operators, as we shall see now.

Definition 18.3. Let R: C°°(Y;

—, C°°(Y; V) be a pseudodifferential operator of order 0 such that the extension R(8)

: H8(Y; S1y)

H8(Y; V)

of R has a closed range for any real s. Then NR, resp. 'R shall denote the L2-orthogonal projections onto the kernel (Null space), resp. the range (Image) of R defined on resp. L2(Y; V).

18. EBP's and Pseudo-Differential Projections

167

When appropriate, we shall denote the restrictions of the resp. C°°(Y; V) by the same symbol. projections onto C°°(Y; Note.

Clearly we have

Lemma 18.4. With the preceding notation, the projection onto cokerR kerR can be written as HR• = Id 'R• The main purpose of this chapter is to prove the following result of Seeley [1968; Theorem IV.7 and Theorem VL6]:

Theorem 18.5. Let R be a (global) elliptic boundary condition in the sense of Definition 18.1. Then the L2-orthogonal projections HR, and HR. onto the kernel, the range, and the cokernel of R are pseudodifferential operators. The proof of Theorem 18.5 is divided into two parts. First we use condition (1) of Definition 18.1 to show that 0 is an isolated point of the spectrum of the seif-adjoint operator R*R (and of the principal

symbol of this operator as well). Then we apply the path integral representation from the proof of Proposition 14.2 to show that the orthogonal projection HR onto kerR is a pseudo-differential operator. We begin with some elementary functional analysis which is part of operator algebra folklore, but hard to find in written form in one place.

PropositIon 18.6. Let T: H1 —'

be a bounded operator acting between two Hubert spaces H1, H2. The following conditions are H2

equivalent:

is closed

(1)

range(T)

(2)

range(T) = (kerT*)i

(3) (4)

range(Tr) is closed

(5)

range(T*) is closed range(T*) =

(6)

range(T*T) is closed.

Proof. Since rangeT is dense in (1)

(2)

(3)

and

we (4)

clearly

(5)

have (6).

III. Applications

168

Now it only remains to prove the equivalence (3) (4). It follows from Proposition 9.4 that range TT is closed if and only if the self* adjoint operator I (ker T=ker

TT

has a bounded inverse, i.e. if and only if there exists a positive real number c1 such that I'r'T's V,V

(v,v)

for all v E (kerT*)

c1

or equivalently, (18.4)

ciIIvjI2 < (TTSV v) = IITvII2.

has a bounded inverse

The estimate (18.4) means that

which, once again by Proposition 9.4, is equivalent to closed range T*. U

be a bounded operator of closed Corollary 18.7. Let T: H1 range. Then 0 is an isolated point of the spectrum of the operator

T'T. Proof. It follows from Proposition 18.6.(6) that TT has a closed range and from Proposition 9.4 that

>6 for some e > 0.

U

Now we can prove Theorem 18.5: The results proved up to now

show that there exists 7> 0 such that the operator R*R — 'y is an invertible (elliptic) operator with kerR*R = and spec(R*R —'y) fl (—oo, 27) C {—'y}, where Va denotes the eigenspace of RR —7 corresponding to the real eigenvalue a.

Next we essentially repeat the argument given in the proof of Proposition 14.2. The projection HR onto kerR = ker RR is given by the path integral (18.5)

HR =

2irz

JA=ir

—

A)' dA.

To show that HR is a pseudo-differential operator we must compute the full symbol of HR. The only difficulty is to show that the principal This follows symbol of RR has no eigenvalues on the circle IAI = from the next proposition.

18. EBP's and Pseudo-Differentia' Projections

Proposition 18.8. Let T: C°°(Y; V)

169

C°°(Y; V) be a pseudo-

differential operator of order 0, acting on sections of a Hennitian vector bundle over a closed Riemannian manifold Y, and let 0T denote the principal symbol of T. Then the union (18.6)

spec(o'T)

spec ('T(y,()

U

(v,C)ET Y\0

of the symbol spectra, taken for all non-vanishing cotangent vectors of Y, is contained in the spectrum spec(T) of T.

Proof. Let Yo

Y, (e

Hence there is VO E

0, and let A E spec(or(yo, ()).

with (vo; vo) =

1

and Ày0 — UT(yo, C)vo = 0.

Extend v0 to a section v with (v; v) = 1 in a neighbourhood of Yo 0 for all y. and extend ( to a vector field e E C°°(Y; TY) with Let 0e,yo denote an approximation of the by smooth functions. From the expansion of the total symbol of T we obtain (18.7)

IIT(9e,yov)

0

—

for e

0.

By continuity we get A E spec(T).

The picture of the spectral correspondence between operator and principal symbol can be made even more clear.

Theorem 18.9. Let T be a self-adjoint pseudo-differential operator of order 0. Then spec(T) \spec(clT) consists of isolated points of finite multiplicity.

As for elliptic T, i.e. 0 spec(cYT), this generalizes the fundamental observation that its spectrum consists of isolated eigenNote.

values of finite multiplicity.

Proof. Let A0 E spec(T) \ spec(CT). We can assume that (18.8)

IAoI

>

I

yE Y and (E

Then the symbol A0 id —o-T(y, () is invertible and A0 Id —T is an elliptic pseudo-differential operator of order 0. Now either the Fredhoim operator Ao Id —T is invertible (which is impossible since A0 E spec(T)) or it has a non-trivial null space ker(Ao Id —T) of finite dimension, i.e. A0 is elgenvalue of T of finite multiplicity. We show that it is isolated. Let .M) denote the orthogonal projection onto

III. Applications

170

ker Ao Id —T. Then the operator A0 Id —T — M

is a seif-adjoint Fred-

hoim operator which is injective, hence surjective; and there must exist a whole real neighbourhood IA — Aol — 1/2 and t arbitrary)

III. Applications

186

Corollary 19.7. Let v E L2(X; St), Av E L2(X; S), and R7o(v) = 0. Then v belongs to H'(X; Si).

Remark 19.8. On the contrary, ( A does not in general have R'yo j \ a right-parametrix. Further details on left and right ellipticity and pararnetrices can be found in Grubb [1977], from which the proof of Theorem 19.5 is taken. Fortunately we only need Lemma 18.13 and Proposition 18.14 to show that the kernel and cokernel of the operators AR and

A

\ R'yoj

consist of smooth sections.

We shall close this sec)ion A

range of the operators

a discussion of the closedness of the

and AR =

That the respective

\ R'yo j ranges are closed is a consequence of the following elementary lemma. Lemma 19.9. Let T: H1

H2 denote a bounded operator between Hubert spaces H1 and Assume that T has a left-parametrix, i.e. there exists a bounded operator 5: H2 —, H1 such that

ST=IdH1-f-k, where k: H1 —p H1 is a compact operator. Then T has closed range.

Proof. According to Proposition 9.4, it must be shown that is invertible (i.e. the inverse is bounded). This is equivalent to the existence of a constant c> 0 such that (19.18)

IIfIIH,

cIITfIIH3

for any f E

Assume that such a constant does not exist. Then there exists a such that sequence {f, } c (ker (19.19)

= 1 and IITf,II
is generated by vectors of the form G(y)e, where e e is an eigenvector of 0i(B) (y, with negative real part eigenvalue. Recall from Corollary 12.6 that the closures of the infinite-dimensional spaces of Cauchy data and C'(H+(Aj) are orthogonal and complementary in L2(Y; Similarly, but more easily, we now obtain that the finite-dimensional vector spaces and are orthogonal and complementary in the space Moreover, range 00(IR. )(y, and range 0o(Id — TR• )(y, are orthogonal and complementary in by definition. Therefore, if 'R• is an elliptic boundary condition for its symbol provides an isomorphism from onto range oo(IR. )(y, and hence its complementary map (20.30)

C

—>

—i

III. Applications

196

must also be an isomorphism and so (20.28).

We can use the arguments of the preceding proposition to reprove the closedness of the operator 4 = which was obtained in Lemma 20.1. In fact, we have:

Corollary 20.5. Let 4 {u

(20.31)

Then 4

E

denote the operator

C°°(X;

I

the closure of 4(00)

with the domain

R(uly) = 0}.

as

an operator from a subset of

into L2(X;Sj.

Proof. A slight modification of the arguments, presented in the proof of Proposition 20.3, gives the following list of inclusions to be proved:

-

•

dom AQ c dom(AR

(1)

(ii)

dom 4 C C

where Q = G(Id —

dom4,

)G and 1R' an orthogonal projection with

—

— "IRS

Let T denote the operator 4

Then dom T C dom 4 which

implies (1), namely

domT j dom(4)* =

(20.32)

In order to prove (ii) we show for any v E domT that vly is welldefined in S1y) and Q(vly) = 0. We can repeat all the arguments of the second part of the proof of Proposition 20.3, where only C°°-test functions were involved. (i) and (ii) show that T* =

(20.33)

=

In order to prove (iii) we fix a u E dom 4 and apply once again Green's formula which yields the estimate

I(Av;u)I

(20.34)

for all v tion.

dom

since G*(vly) and uIy are orthogonal by defini-

Property of the Operator AR

20.

To prove (iv) we show that is well-defined in H4 (Y; and 'R (ulv) = 0 in a distributional sense for any u E We make the following modifications of the earlier proof: (20.35)

(IR.(ulv);f) = (uly;IR.f) = (CuIy;GIR.f) = (GuIy;wly) with WE C°°(X;S) and wIy = GIR.f =

=

(u;ATh,ow) —

shows the vanishing of IR' (uIy) for suitable C°°-functions with 1 close to Y and IIWIIL2 arbitrarily small. which in the (iii) and (iv) yield = so = +(oo) Hubert space means that AR is the closure of AR U which

.

The space of closed extensions of a Dirac operator A+

(or

of suitable

restrictions) does not only consist of the space 7Z. :=

I

R elliptic boundary problem)

of all realizations of elliptic boundary problems, but contains also two (see also Figure other prominent closed operators, and - --

20.2):

/

K. I

/

.0

..-1--

0

—— —

——

max

Fig. 20.2 The closed extensions of

III. Applications

198

Definition 20.6. Let tors in L2

which

the unbounded operaare determined by

and and

act like

denote

L2(X;S) and

:= {v E := {v E

E L2(X;S)}.

I

and

we define

Similarly

-y0v = 0},

are closed operaand Proposition 20.7. The operators into L2(X; Sj and in fact tors from subspaces of L2(X;

= {v E

I

= 0}.

Moreover

(A+ \*_A_ — max

an..d

(A+ \*_A_min max/ '

Note. We have C

C

for

all

E IZ

in the space of closed unbounded operators from L2(X;

into

L2(X; S—) (see Figure 20.2). is closed, let v, Proof. To prove that —i w in L2(X;S'), and l'ovn = 0. Then

(20.36)

v

in L2(X; St),

= (r+Ae+)A+vn;

= (Id hence, by Corollary 19.7,

—'

E H1 and

IIVnhIHl = II(Id —K+'yo)vflhIHl

is convergent in H' to a limit v' with 'y0v' = 0, but then it is convergent in V to the same = w. limit; hence v = v' and A+v = urn As in the proof of Lemma 20.1, we get that

To prove that is closed, we begin as above, but do not assume := (Id 7ovn = 0. Nevertheless, when we replace v,, by (20.36) remains valid and provides once again a limit v' E H' with A+vl

= n—.oo urn

—

= lirnA+vn = W.

Property of the Operator AR

20.

Moreover, K+ (i'OVn) = ker+(A+,O) and A+v = A+(v' + Similarly we prove =

v

v—

—

(v — v'))

199

v' E

= w.

Let v0, w0 E L2 and

w0) = (v; vo)

for any v E dom sequence w,

wO

We have to show v0 = Aw0. Choose a in L2, consisting of smooth sections, and get

= by Green's formula. C00° C dom hence dom is dense in L2. Therefore the sequence converges in L2. Let v' := urn := r+A_ hence = (Id is convergent in H' with := E H' and Aw' = = and we obtain w0) = lim(v; = (v; Aw'),

for any v E dom we get

hence v0 =

=

=

Aw' in L2. Since

is closed, and there is nothing left to prove.

Now we derive a formula for the index of the realization of an elliptic boundary value problem R: C°°(Y; C°°(Y; V) for a Dirac operator : C°°(X; —' C°°(X; S). Recall that the boundary integral —'

:

range(R)

is a Fredholm operator by Corollary 18.15. The following theorem provides a new proof of that fact and extends it to the realization.

Theorem 20.8. The operator =

:

dom4 —* L2(X;S)

is a &edholm operator with

= dim ker4 = =:

—

dim :

—'

ran ge(R))

III. Applications

200

Proof. Let T denote the operator

acting from

to

range(R). We have already shown that (1) is closed (by Proposition 19.11); (2) kerT (by Corollary 19.2); (3) and (At)' = (by Proposition 20.3); hence we only have to identify coker AR with coker T: (20.37)

= kerG(Id —IR.)G'P+(A)

cokerAR =

ker(Id 'R )(Id = =

: range(R) : range(R)

For the last equation we once more exploit the fact that the boundary

condition R (and hence R') has closed range by definition. That implies that R : range(R) —' is an isomorphism. Clearly the last space in (20.37) is isomorphic to coker T. Note.

U

In the following we shall return to the notation P÷ for

From Theorem 20.8 we get that the index of the Calderón boundary vanishes, since the corresponding boundary integral is problem just the identity. This can also be seen from Corollary 12.6 and Proposition 20.3 which give ) = A_. By definition of the Calderón projector, we have ker = ker = {0}. For the Atiyah-Patodi-

Singer boundary condition (= the spectral projection P> (At) onto correthe linear span of the eigensections of the tangential part of sponding to non-negative eigenvalues), the situation is quite different. First, we have

'

—

—A — P coincide if and only if P+ and P belong to the same connected of the Grassmannian this more closely in Chapters 21, 25, and 26.

We shall investigate

20. Fredholm Property of the Operator AR

201

Next we explain special features of the index of elliptic boundary problems of local type. Let A be a Dirac operator or a system of Dirac operators which admit a local elliptic condition (cf. Remark 18.2c). Then R : C°°(Y; C°°(Y; W) is a pseudo-differential operator of order 0 such that

—'

rop+(y,C) :

is an isomorphism. In particular the principal symbol of R is surjective.

S: V) C°°(Y; W) be a pseudo-differential operator of order 0 such that the principal symbol s(y, () : of S is surjective for any (y, Then S has finite-dimensional cokernel and in particular the range of S is closed. Lemma 20.9. Let

Proof. The operator

: C°°(Y; W) —' C°°(Y; W) is an elliptic operator since its principal symbol is an isomorphism. Therefore ker SS = ker is finite-dimensional. 0 We reformulate Corollary 19.10:

Proposition 20.10. For any real s>

'A' "

the

operator

—,

/

has a closed range. It remains to compute the cokernel of

We have the following

elementary lemma which sharpens Corollary 18.15, but is valid only for local elliptic boundary conditions:

Lemma 20.11. For local elliptic boundary conditions the operator RP÷ : is a Fredholm operator.

Proof. Once again we may consider everything on the level of Sobolev spaces. It is enough to show that dim ker RP÷ < oo and dim coker RP÷ < oo. We have already proved the first estimate. Moreover

coker RP+ = ker(RP+ )8 = ker P+ R

= and the operator RP÷R* is elliptic.

=

9

III. Applications

202

Theorem 20.12. For local elliptic boundary conditions one has

=

index

: H÷(A) —' C°°(Y; W)}.

H(A)

Fig. 20.3 The parametrization of coker(A,R) by

Proof. With regard to the results obtained earlier, it only remains to identify the cokernel of (A,R) with cokerRP÷(A) = (see Figure 20.3):

coker(A,R) = {(v,h) E

(As;v)

+ j(R(slv); Ii) dy = 0 for any 8 E C°°(X; and by Green's formula

0 = (As;v) +

= (s; A*v) + J(sIv; Rh — G'(vly))dy. Once again this implies Av = 0 and coker(A,R)

{(q,h) E with

= qr and Rh = q}

{(r,h) E with

=

= 0 and r = R*h}

20.

Property of the Operator AR

203

For later use we give the following reformulation of Theorem 20.12 which determines the relation between index AR and index(A, R) (notice that we suppress the subscripts + of the Calderón projector and the Cauchy data space):

Theorem 20.13. Under the preceding condition8 38

index(A, R) = index{RP(A) : H(A) —+ C°°(Y; W)} = index AR — dim(C°°(Y; W)/range(R)).

Note. Decompose the kernel of the operator P(A)R* : C°°(Y; V)

C°°(Y; E'y)

H(A)

into two finite-dimensional subspaces kerP(A)R = Wi

W2

with W1 := ker R* and W2 the orthogonal complement of Wi in kerp(A)R* (see Figure 20.4). Since ker(A, R) = kerAR and coker(A, R) coker(RP) = W1 W2 are already established, Theorem 20.13 shows that (20.39)

cokerAR

W2.

ker

(coker AR)IY

kcrl?

Fig. 20.4 The decomposition of' kerP(A)R and the isomorphism (coker AR)IY

204

III. Applications

Corollary 20.14. For local elliptic boundand conditions we have index AR = index(A, R), if and only if R is a surf ective operator.

Remark 20.15. We have seen in Chapter 15 above that a compact perturbation of R can change the connected component of the corresponding projection 'R• and we shall see in Section 22F below that in general the index of elliptic boundary problems is not a topological invariant (see Theorem 15.12 and Remark 22.25). However, when we

restrict ourselves to the class of local elliptic conditions, the index becomes a topological invariant and we have a corresponding AtiyahBott index formula (see Atiyah & Bott [19641 and also Hôrmander [1985; Section 20]). We do not discuss the Atiyah-Bott index formula here and refer the reader to the works of Atiyah, Bott, and Hörmander mentioned before.

21. Exchanges on the Boundary: Agranovië-Dynin Type Formulas and the Cobordism Theorem for Dirac Operators

We prove two variants of the cobordism theorem.

formula and the

In the preceding chapter we obtained the following index formula for the realization AR of an elliptic boundary condition R = R2 (cf. Theorem 20.8): (21.1)

index AR = index {RP(A) : H(A) —' rangeR},

where P(A) denotes the Calderón projector onto the Cauchy data space H(A) corresponding to the Dirac operator A. The obvious question is whether this integer is a topological invariant. Contrary to the case of local elliptic boundary problems, the answer is negative. The index is not a homotopy invariant of the principal symbols of the operators A and R. it is not difficult to deliver a specific example.

Example 21.1. Let X := D2 and take the Cauchy-Riemann operator

0 .0 0— — +2— — 0x

0y

jO .10— + —— rOço

close to the where we take the standard polar coordinates (r, boundary. In this special case the choice of the orientation of the transversal coordinate makes no real difference, but only a sign change

for the whole expression, since r is in both components here. The C) Calderón projector P(s) is the orthogonal projection of (orthogonal with respect to L2(S'; C)) onto the space £c{ek(ço) =

of the Cauchy data of all holomorphic functions on the disk D2. According to the results of Chapter 12, the projection P(0) is a pseudo-differential operator of order 0 with the principal symbol

=

11

(0

III. Applications

206

Actually, in this particular situation the Calderón projector incides with the spectral projection P(iO/ rôp). Now let us consider projections Pk given by the formula P,

co-

for k E Z.

:=

It follows from Theorem 20.8, see equation (21.1), that:

index&b = index PkP(9) = index PkPO =

(21.2) since

=

and

m max{0,k}}.

Therefore

dimkerPkPo:

k0:

k of the tangential part B of A). Then (21.6)

index AR2 — index AR1 = i(R2, R1).

Proof. By Theorem 20.8 we have (21.7)

= i(R3,'P(A)).

Then formula (21.6) is a consequence of Proposition 15.15.

III. Applications

208

The main purpose of this chapter, however, is to give a proof of the famous cobordism theorem for Dirac operators based on Proposition 21.3 (applied to an odd-dimensional case). Let Y be an odd-dimensional compact Riemannian manifold with

boundary Z. We assume that the metric of Y is a product near the boundary. Let S be a bundle of Ct(Y)-modules over Y with compatible connection. Let

B: denote the corresponding Dirac operator. We have a decomposition of B close to the boundary Z of the form (21.8)

where r is a unitary bundle automorphism with r2 = hence r defines a decomposition of SIz into the direct sum

and

of the subbundles of the ±i-eigenvalues of {rZ}ZEZ.

Theorem 21.5. (a) The operator B takes the following form close to Z with respect to the decomposition SIz 5+ S :

IL.))

(21.9)

(b) Let : S —' Then the operators (21.10)

denote the orthogonal projections of S onto defined by

:= {s E H'(Y;S) I Pj(sjy) = 0)

LB±:=B

are well-defined local elliptic boundary problems, and we have the following equations: (I)

(II) index

=

(III) contain the famous cobordism theorem for the index of operators of Dirac type. The cobordism theorem provided the main step in the original proof of the index theorem (see Palais (1965; Chapter XVII]) and has various generalizations.

21. Exchanges on the Boundary

Corollary 21.6. (Cobordism theorem). Let A÷ :

C°°(Z; S) denote the restriction of a Dirac operator on a Closed evendimensional manifold Z to the spinors of positive chirality. Then the index of vanishes if the couple (Z, is a "boundary". This means that there exists a manifold X with boundary Z and a bundle S of Clifford modules over X such that S restricted to Z is equal to

SEEDS. Proof of Theorem 21.5. (a) The special form of the decomposition of B follows from Chapter 7. (b) As in Theorem 7.3, we have A_ = Az... This shows that B is formally seif-adjoint. That the unbounded operators and 8_ are mutually adjoint follows immediately from the definition, see Remark 20.4c.

Equation (III) is an elementary consequence of Green's formula. We show that

ker(8÷) =

(21.11)

{O}

= ker(8_).

Let s be an element of the kernel of where we write slz = (s+, s_), and

i.e. Bs =

0

and s+ =

0,

denotes the projection P±s.

Then

0 = (Bs; s) =

(s;

Bs) —

= —

=

f

f

(r(y)s(y); s(y)) dvol(y)

s(y)) dvol(y) =

if

(s_ (y); s (y)) dvol(y)

II8_IIL2(SIz)

It follows that (21.12)

ker(8±) = {s Bs = 0 and sIz =

= 0} =

{O}

= 0 because of (I). Now we prove (II). We recall the Agranovië-Dynin formula (Proposition 21.3): and then index

index BR1 — index BR2 = index R12(B)R2,

where the boundary conditions Rj are given by projections. Hence we have (21.13)

index8_ —

index P_P(B)P+.

III. Applications

210

We show that the principal symbol of the 0-order pseudo-differential is equal (up to a factor to the principal symoperator when restricted to the bol of the first-order differential operator cotangent sphere bundle 5(Z). Let us denote by a the principal symbol of the (total) tangential operator A and let a± denote the principal symbols of the operators They satisfy the following equalities on S(Z): (21.14)

a_at = Ids+

= Ids-.

and

Hence the eigenvalues of

a(z,() =

(0

a_ 0

are all contained in the set {l, —l}. We have shown in Proposition 14.2 that the principal symbol p..,. of the Calderón projection P(B) is equal to the principal symbol of the (positive) spectral projection of is the orthogonal projection —' the operator A. So p,. (z, C) spanned by the eigenvectors of a(z, C) correonto the subspace of sponding to the positive eigenvalues. Here z Z and ( E T*(Z)z IS of length 1. Then we have the equality :

1

/

2

and hence

= P_

1(0 o\fi a_\(1 o\ O)=2a+.

flo

1

22. The Index Theorem for Atiyah-Patodi-Singer Problems

We present a proof of the index theorem for Atiyah-PatodiSinger boundary problems.

22A. Preliminary Remarks Let : C°°(X; C°°(X;S) be a Dirac operator on an even-dimensional smooth compact Riemannian manifold X with

boundary Y. Actually the odd-dimensional case is not different and we shall make the necessary comments after the proof of the index theorem. We assume that the Riemannian structure on X and the

Hermitian structure on S are products in a collar neighbourhood N [0,1] x Y of the boundary Y in X, that is, the metrics do not depend on the normal coordinate u when restricted to so that can be identified with Sly. Recall that then has the following form on N

Y

= G(OU + B),

(22.1)

where C : S+ — is a unitary bundle isomorphism, namely the Clifford multiplication by the inward normal vector, and B C°°(Y;

denotes the corresponding Dirac op-

C°°(Y;

erator on Y. Note that C and B do not depend on the normal coordinate on N. Moreover, the tangential operator B is seif-adjoint and elliptic of first order. Let {Ak; çok}kEz be a spectral decomposition of L2(Y; generated by B with Ak 0 for k 0. We know from Proposition 14.2 that the orthogonal projection P> := Po is a pseudo-differential operator and we have a wellonto defined Fredholm operator (22.2)

A

:

—'

L2(X;S).

It follows from the elementary index theory presented in Chapter 20 (see Theorem 20.8) that (22.3)

indexA =

III. Applications

212

where 1'..,. denotes the Calderón projector of A+ and

= index{P P..f : rangeP+

rangeP>}

the virtual codimension of in P+. Formula (22.3) does not involve the geometric data suitable for applications to concrete problems. Deeper insight is provided by the geometric Atiyah-Patodi-Singer index theorem for the operator A (Theorem 22.18 below). Following Atiyah, Patodi & Singer [1975],

we shall present a proof of this theorem. As in the heat equation proof of the Atiyah-Singer index theorem (accounted for e.g. in Cilkey [1984] and Berline, Cetzler & Vergne [1992]), the base of the Atiyah-

Patodi-Singer index theorem is the study of suitable heat operators and the related heat kernels and traces. A novel feature is of course to keep control of the boundary conditions. This is possible due to our assumption of the product structure in the collar neighbourhood of the boundary providing for explicit computations of the traces of the heat kernels involved.

We begin with some formulas for the adjoint boundary problem arid the related natural problems of second order:

Lemma 22.1. Let A denote the closed operator from L2(X;

A=

to L2(X; S—) which acts like {s E

I

P(sb') =

O}. Then

(a) The adjoint operator A* acts like A— and is determined by

domA = {s E H'(X;S) I P< G(sly) = O}, where P< := P. and P 0

=

—

=

—

—

= dim kerAA - dim kerAA Lemma 22.Id

=

.

. dim ker A — dim ker A

Thm. 20.8.

=

index A.

This proves

Lemma 22.2. The following equality holds for any t > 0 index A = tr

(22.6)

—

tr

Formula (22.6) is the base of the proof of the Atiyah-Patodi-Singer index theorem. The proof splits into three parts:

(1) We construct the heat kernels tors

and

where

and t

•

of the opera-

denotes the corresponding

boundary problem on the infinite cylinder [0, oo) x Y. We have also given the corresponding heat kernels ed and ed * for the operators AA and AA* on the closed double X of X (see Gilkey [1984], Roe [1988], or Berline, Cetzler & Vergne [1992] for the construction). (2) We use Duhamel's principle to construct the kernels and for the heat operators and — (3) We study to obtain the final index formula.

22B. Heat Kernels on the Cylinder We construct the kernels and of the operators and on the semi-infinite cylinder [0, oo) x Y, where acts like A and A is given by formula (22.1), but with u E [0, oo). For simplicity consider the normalized Dirac operator (22.7)

on the cylinder acting on sections of p*(S+ly) lifted to xY under the natural projection p). In this section the assumption G = Id is no restriction, since

AA=

22. The Index Theorem for Atlyab-Patodi-Singer Problems

215

for arbitrary unitary G. Clearly D is elliptic and its formal adjoint is :=

(22.8)

We shall show

Proposition 22.3. Let V and ir denote the closure of the operators defined by D and D with domains given by in L2(R.,. x Y; smooth functions subject to the boundary conditions (22.9)

P(fI{o}xy) =

0

P) denote the space of all C°°x Y; • let sections satisfying the boundary condition (22.9) (for u = 0) x Y;p*S+Iy;P); with subspace

• and let C8°(R+ x Y;

denote

the space of C°°-

sections with (compact) support totally contained in the interior of the cylinder R÷ x Y (i.e. vanishing for u C and for

u c). For later use notice that the inclusions (22.13)

x

C

Y;p*(S+Iy))

C

C

are dense.

We shall show that any equation Df = g with g

f'VOO

has a unique solution f E

X

V.

IY

x Y;p*S+Iy;P). More precisely:

Proposition 22.4. For any normalized Dirac operator D = over the cylinder R+ x Y there exists a linear operator

Q:

x Y;p*S+Jy)

—' C°°(R.,. x Y;p*S+Iy;P>)

such that (i) DQ9 = g for aug E

(ii) QDf = f for alif E

+B

x Y;p*S+Iy).

x Y;p*S+Iy;P).

(iii) Q extends to a continuous map

1

for all integers

s 1. Here

denotes the space of sections which are locally in the Sobolev space H8.

(iv) The kernel Q(u, y; v, z) of is a section in a suitable homoand y, z in Y) morphism bundle (with u, v running in which is C°° for u v.

22. The Index Theorem for Atlyab-Patodi-Singer Problems

217

here lives

R4xY

here lives Fig. 22.1 Separation of variables over

xY

Proc4 To solve (22.14) with

Df = g

/E

xY;

for g E

we make the usual separation of variables (products of sections from span x Y; pS4iy), see Figure and from L2(Y; 22.1) and expand / and gin terms of the previously fixed orthonormal of elgenfunctions of B (here and in the following basis we repeat A E spec(B) corresponding to its multiplicity):

f(u, y) =

g(u,y) =

and

IA(U)(PA(Y)

Then the partial differential equation (22.14) reduces to a family of

ordinary differential equations

AEspec(B),

(22.15)

half of them with initial conditions

forAO.

(22.16)

The textbook solution of (22.15) is JA(u) = fA(O)e

AU

+

j

dv.

With (22.16) and an appropriate choice of (22.17)

fx(O) := —

j

eAt? 9A(v)dv

for A < 0,

III. Applications

218

we get explicit solutions

2/ \ =

JAtU)

( fue)t(t)_u)g(v)dv 0

for A >0

J

forA-n then (22.41)

71B(25)

=

+ k=—n+1

+

22. The Index Theorem for Atiyah-Patodi-Singer Problems

229

for + 1). Recall that h := dim ker B is the multiplicity of the O-th eigen-

where ON(S) is holomorphic

and r is defined by r(s) :=

value, 7?B(S)

f°

for > 0 and extended to a meromorphic function on C with isolated simple poles at s = 0, —1, —2

Proof. (a) For t > 0, u 0, and y E Y one determines the kernel of x Y) x (R÷ x Y)

at the diagonal point (u, y; u, y) E by ftC(t;u, y)

{eA(t; u, u) —

= —

e,)(t; it, u)}

'ç-j( —e

e

®

—A2t

Id

+

}

+ Ae2"erfc

+

—

{ (22.42) e

—A2t

—A2t

—u2/tI

e —u2/t

+

IC(t)

I'PA(Y)12

=

j

(b) Note that, as t —'

j

}

Id

I'pA(y)12,

1. Integrating over R+ x Y, one

where, for convenience, sign(0)

(22.43)

}

+

=

obtains

'

u, y)dydu = >

Moreover, X(t) + in (22.43), )C(t) 0 exponentially as t —' oo. Also (22.43) shows that 1

1K(t)I

0

2

t

(22.30)


3/7} and {u < 4/7)}

22. The Index Theorem for Atiyah—Patodi-Singer Problems 91

K with

(22.47)

1 on supp

233

and

dist

Fort>O, x,x'EXwedefine (22.48)

Q (t; x, x') :=

soj (x)E3

(t; x, x')iJ.', (x'),

:= E2 := 6d, and the ii,, are considered as multiplication operators. Clearly Q is an approximate fundamental solution of the operator +4' A over X which satisfies the initial and boundary conditions, but not the equation over the whole of X. We define a in an analogous way. parametrix Q, of the operator where

Lemma 22.11. Under the preceding conditions let C(t; x, x') :=

+ ..4'A)Q(;

•,

the error term induced by the approximate fundamental solution Q. It vanishes for all x E X \ ([1/7,6/7] x Y) and also for d(x, x') < 1/7. The following inequality holds: denote

(22.49)

IC(t;x,x')I

for some constants c1,c2 > 0 as t —' Note.

0.

The concept underlying the whole range of arguments

between the preceding Lemma 22.11 and Theorem 22.14 below is referred to as Duhamel's principle. It goes back to Duhamel [1833], where the heat equation with time dependent boundary conditions is solved by taking the derivative of a superposition (convolution) of the solutions at successive times for the corresponding boundary conditions with frozen coefficients, see also Carsiaw & Jaeger [1988; Section 1.141. Similarly one speaks about Duhainel's principle when solving the inhomogeneous wave equation by replacing the force term on the right side by a succession of suitable initial impulses applied at various times to the solution of the homogeneous equation. The underlying idea is always the same, namely • write the error term between the true solution and the approximate solution at time t as an integral over the derivative of a convolution of the true and the approximate solution,

III. Applications

234

• iterate and apply power series analysis to get Levi's sum for the fundamental solution, • derive the needed estimates which ensure convergence and the vanishing of the error term for t —' 0, • and estimate the pole of the heat kernel for t —' 0 up to terms of magnitude (See Levi [1909), Minakshisundaram & Pleijel (1949), and McKean & Singer (1967].)

Proof. Since both e Q —* Id for t — 0, —

and Q satisfy the initial condition E —, Id and we have on the operator level

Q(t) = 6(t)Q(0)

= /

—

e(o)Q(t)

d uS

=

(22.50)

=

=

+

—

jt (e(s){_(A*A+ d(t—

f

s)) ds

))}Q(t — s)) ds

i(s) C(t — s) ds.

Hence on the kernel level we have the equality (22.51)

,

e(t; x, z') = Q(t; x, x') + J

J ox

x, z)C(t — s; z, x') dzds.

Next we derive a formula for the kernel of C(t). We fix an x' E X. Then we have (22.52)

—C(t;x,x')

+A*A) Q(t;x,x')

=

+A*A)

= 0

=

{

(x)6,(t; x, x')i,L',(x') + x for d(x, x')
2Ck(t;x,x'). To obtain the desired estimates of the heat kernels involved, we

need the following generalized triangle inequality.

Lemma 22.12. For any 0 < s ) denote the space of spinors s of positive chirality satisfying the boundary condition

P s(.,O)=O, where P> denotes the spectral projection of B corresponding to the eigenvalues 0. Then

C°°(X;S)

:

has

a finite

index given by

index

(22.68)

= where

h,

(i)

t —*

77B

are

J

h + 77B(0)

defined as follows:

is the constant term in 0) of the trace differences

e_t141(x)12 pEspec A+A

—

ao(x)

the asymptotic

—

AA+

expansion (as

242

III. Applications

denote the eigenvalues and eigenftinctions of where p, on the double of X, and p', are the corresponding objects for (ii) h := dim ker B = multiplicity of O-eigenvalue of B. (iii) 17a(8) sign AIAI8, where A runs over the eigenvalues of B. In (iii) the series converges absolutely for large, and then T7B(s) extends to a meromorphic function on the whole s-plane with a finite value at S = 0. Moreover, if the asymptotic expansion in (i) has no negative power oft, then 71B(s) is holomorphic for p(s) > — 22E. L2-Reformulatlon

To understand the Atiyah-Patodi-Singer index formula for a Dirac operator over a compact manifold X with boundary Y (and with + B) close to Y) in its correct analytical context, one has A= to reinterpret the action of the spectral projection P> defined by the tangential part B of which constitutes the Atiyah-Patodi-Singer boundary condition. Recall the analytical context of the Calderón projector namely its relation to the Poisson type integral which provides for all sections f over Y a global solution s of =0 with Roughly speaking, the action of the Calderón = projector is related to the inner extension problem of sections over the boundary to global solutions. The null space ker of the corresponding boundary problem, however, is uninteresting since by definition it consists only of the zero section. Now the remarkable fact is that the action of the spectral projection is opposite directed, namely toward the outer extension problem in such a way that can be identified with • the null space kerA (with A := the square-integrable solutions of = 0 on the complete non-compact elongation obtained from X by attaching the semi-infinite cylinder (—oo, 0] x Y to the boundary (see Figure 22.5); • the solution space kerA (with A* := a.) can be identified with the direct sum of all square-integrable solutions of A;s = 0 with a certain subspace of ker B which consists of the limiting values of all locally square-integrable solutions; • and that the dimensions of the two spaces of limiting values coming from = 0 and A,s = 0 add precisely to the

22. The Index Theorem for Atiyah-Patodi-Singer Problems

243

dimension of the null space of the tangential part B. (For the general case Muller [1988] and Melrose [1992, Chapter 7]).

xY

Fig. 22.5 The non-compact elongation We define:

Definition 22.19. Let X be a compact oriented Riemannian manifold with boundary Y, and let

C°°(X;S)

:

be a Dirac operator. Assume that the Riemannian structure on X and the Hermitian structure on S are products in a collar neighbourhood of Y. In particular, there we have A = with interior normal coordinate ti, unitary C, and self-adjoint B. (a) Since X is isometric to the cylinder Y x [0,1] near the boundary,

and since the Hermitian structure on S is product there, the noncompact Riemannian elongation

U (-oo,0]xY and related extensions of the bundles of Clifford modules to and of the Dirac operators to are well-defined and the unitary isomorphism C extends in the cylinder as well. (b) We define the space of L2-solutions of = 0 on := {s E

I

= 0}

and similarly (c) By an extended L2-solution of the equation = 0, we mean a solution which belongs to St,) such that, for large negative u, we have

s(u,y) =

8L2(U,y) + BIjm(y),

III. Applications

244

where

and 811m E ker B. Thus s has 8Iim as or limiting value as u —' —oo. We shall write

E

asymptotic

an

SE

A similar definition holds for extended L2-solutions $

+ 8ljm of

A;s = 0, but demanding G*SIim E ker B. (d) Let fl.1 C ker B denote the space of all limiting values of extended L2-solutions of =0, and define in a similar way C Gker B.

Proposition 22.20. As before, let A denote the closed operator from L2(X; to L2(X; Sj which acts like and is determined by domA = {s E H'(X : = 0}. Then we have natural I

isomorphisms ker A

(i) (ii)

and

L2

kerA*

fl.....

Due to the regularity of solutions of elliptic boundary

Note.

problems (Theorem 19.1) we have

kerA =

—'

:

G°°(X;S)}

and similarly kerA* = ker{A : C°°(X; S; P< G*)

Proof.

(i) Let

s

e ker A

C°°(X;

and expand it near the boundary in the

form

s(u,y) = is a spectral resolution of L2(Y; generated by B. Since = 0, we have ôs/Ou + Bs = 0, so that Since P>.(sly) = 0, we have fA(O) = 0 for A 0 fA(u) = and so where {A;

(22.69)

s(u, y) = >

fA(0)

A + index(A2)p2> = index(Ai UA2) — dim kerB.

Proof. This follows directly from Theorem 22.18 which yields

=

aj(x) —

(dim kerB + 1713(0))

and

= =

J a2(x) —

/ a2(z) —

(dim ker GBG' + ll_GBG-1 (0))

2

(dimkerB — '7B(O)).

We give the most general result of this type. Much deeper nonadditivity results can be obtained for spectral invariants like the analytic torsion and the ti-invariant (see Klimek & Wojciechowski [1992], Matteo & Meirose [1992], BooB-Bavnbek & Wojciechowski [1993], Bunke [1993], Luck [19931, and Muller (19931). The following result also generalizes the calculations about exchanges on the boundary for the Cauchy-Riemann operator given in Example 21.1.

Theorem 23.3. Let X1, X2 be two compact Riemannian manifolds with common boundary Y and let 4,, j = 1,2 be Dirac operators on X3 satisfying the consistency condition (22.1). Let a E R and let (A3)0 denote the operator We have (23.7)

index(Ai)0 + index(A2)_0 = index(Ai U A2) — dim V0

and (23.8)

index(Aj

+ index(A2)0 = index(Ai U A2) +

P!,

— dim V_0.

III. Applications

256

Remark 23.4.

Recall

that

denotes the weighted spectral projec-

tiori of the operator B, namely the orthogonal projection onto the subspace Va spanned by the eigensections of B corresponding to eigendenotes the corresponding weighted values in [a, oo), and that

spectral projection of the operator —CBG'. Moreover, Va denotes the a-eigenspace of B (if a E spec(B); otherwise V0 := {O}). By Moreover, for a b c we have definition we have = and by Proposition 15.15 =

+

(23.9)

Ph).

=

Finally, for a> b, we have two short exact sequences p1

p'

range

range

—'

V1,

{O},

b 0 such that specBfl(—2e,2e) C {0}

and

specBfl(—a—2e,—a+2e) C {—a}.

23. Some Remarks on the Index

257

Then clearly

=dimkerB

and

=

We also notice

index(Ai)o = index(Ai)o +

= index(Ai)o

for —2e < S 0. With (23.10) we obtain (23.11)

= I(C(Id — PLa +

= i(Id

= •lThl =

—

P_0), Id

—

Po))

1T)l rO,r>_0 — F_a/\ = 1*1>0 — fl \ .ini ni — FO, —

flu

'>—o+e

—

P0,

—

dim V_0

PLa) — dim V_a.

=

formula (Proposition 21.4)

We use the generalized and obtain (23.12)

+ Po)G')

G(Id —

and

index(Au)0 = index(A1)o +

index(A2)>0 = index(A2)>o +

We have = {index(Ai U A2) — dimkerB}

lfldeX(Ai)a +

+

+ +

{... }

+ i(PL, PLC) +

—

dim V_a

PLa) — dim

= index(Ai U A2) +

+ = index(Ai U A2) +

—

dim V_0

PLa) — dim V_6,

258

III. Applications

which proves (23.8). In the same way we prove (23.7):

index(Ai)0+ index(A2)_a = {index(Aj U A2) — dim ker B}

+ (Dla' Dl—e/\ + •(fl2

+ i(PL, = {.. . } + — dim V0 = {. } + =index(A1UA2)—dimV0.

' —

dim

Va

El

Theorem 23.3 shows that we can not expect additivity of the index for Atiyah-Patodi-Singer problems. In the following we discuss the important case of the signature operator. For this operator we give a natural boundary condition which provides the desired additivity. First we notice that the signature of X is not the index of a true Atiyah-Patodi Singer problem From Atiyah, Patodi & Singer [1975, Section 4, Theorem 4.14 and Equation (17.9)J we extract the

two following formulas:

signX index

= =

£(x)

£(x)

—

h

—

—

and 7lBev(O),

and hence the signature deficiency fonnula:

signX = indexAp> + h.

(23.13)

Here X is an oriented Riemannian manifold of dimension 4k with smooth boundary Y (and X is isometric to a product near Y), A IL. denotes the signature operator introduced in Chapter 7 earlier. The signature operator takes the form A = + B) close to Y. The tangential component B splits into B = Be" + Bodd with isomorphic to (23.14)

(0)

=

Hence (0)

and

h := dim ker Be" =

dim ker B.

Moreover, C(x) := Lk(pl(x),. .. where Lk denotes the k-th Hirzebruch L-polynomial and the p, denote the Pontrjagin forms of the Riemannian metric. We introduce the notation V

and

23. Some Remarks on the Index

259

so orth.

V>.

There exists a natural projection PE : to the Grassmannian Crp+ such that (23.15)

—,

range PE belonging

signX = index ApE,

or equivalently, by (23.1), such that (23.16)

index{PEP : rangeP>

= indexApE

—

indexAp> = h.

Clearly the projection PE onto any subspace E of range P> of codimension h will provide (23.15), since then PEP : rangeP> —* E is surjective and rangeP>/E. We use results of Section 22E to describe a choice of E which appears in this context in a natural way. There is actually a very natural subspace of range = V = V> + ker B of codimension h, namely the space V> + (23.17)

V> :

where

£c{cOA A > O}

and 71÷ denotes the space of limiting values of the extended solutions, as introduced in Definition 22.19d. Then the projection onto V> +11÷ will do the job of our desired alternative signature-amended AtiyahPatodi-Singer boundary condition PE. We recall from Section 22E the concept of a non-compact RiemannIan elongation of related extensions, especially of the signature operator A to and the concept of an extended solution (extended square-integrable harmonic form of even parity). As seen above, these concepts are meaningful in much more general situations. The following lemma, however, depends heavily on the internal symmetry of the

tangential part of the signature operator. The lemma is proved in Atiyah, Patodi & Singer [1975; pp 64—66 (before Theorem 4.14)] but not explicitly stated there:

III. Applications

260

Lemma 23.5. Let fl÷ denote the subspace of ker(B) which consists of the limiting values of extended L2-harmonic forms of even parity. It has dimension h = dim ker Be" = dim ker B. We fix the orthogonal projection ê ker B —. ker B of ker B onto N÷ and introduce the projection P> of the L2 differential forms of even parity over Y onto the subspace V> defined in (23.17). Now we define (23.18)

PE:=P>+e.

is only a perturbation of the pseudo-differential projection P by an operator of finite range, we obtain from equation (23.1) with Lemma 23.5 and equation (23.16) the following theorem: Since

Theorem 23.6. The projection FE defined in equation (23.18) beand defines a generalized Atiyahlongs to the Grassmannian Pat odi-Singer boundor,j value problem with

index ApE = sign X.

Corollary 23.7. We have (23.19)

L2-index(A)

=

J a(x)

—

IlBav(O),

denotes the non-compact elongation of X, the L2-index is defined as the difference of the dimensions of the spaces of 'extended solutions' of even and odd parity over the elongation (see Definition 22. 19d above), A denotes the signature operator of X, and Cr(x) denotes the index density of A which is naturally extended over the whole of the elongation. where

Remarks 23.8. (a) The preceding theorem permits a reformulation of the cutting and pasting invariance of the signature which we shall discuss in Chapter 25. Moreover, it gives a nice frame for the analytical explanation of the Novikov additivity of the signature (see e.g. Jänich [1968], Karras et al. [1973], and BooB & Rempel [1982]) and, when treating Riemannian manifolds with corners, an analytical explanation of the Wall non-additivity, see Wall [1969], Rees [1983], and Melrose [1992].

23. Some Remarks on the Index

261

(b) Theorem 23.6 and Corollary 23.7 do not hold for arbitrary Dirac operators. The reason is that in general h = h(A) h(A). However, one has the following formula for the L2-index (Corollary 22.21): (23.20)

indexAp> = L2-index(A)

—

and more generally

(23.21)

L2-index(A)

=J

a(s) —

—

(h(A)

—

h(A)).

See also Meirose [1981] and Roe [1988].

Anyway, we have no problems calculating the L2-index in the case of vanishing ker B.

Corollary 23.9.

Let A be a Dirac operator with non-singular tan-

gential part B. Then index Ap> = L2-index(A)

=

J a(s) —

Notice that we used the preceding corollary in Section 17D above.

One more reason for the choice of our alternative boundary condition is the odd-dimensional case. It is well-known that for dim X odd, Ap> is not a seif-adjoint operator whenever ker B {0}. See BooB & Wojciechowski [1989; Proposition 5]. However, one obtains that ApE is an elliptic seif-adjoint problem in this case.

24. Bojarski's Theorem. General Linear Conjugation Problems

We recover the index of a Dirac operator A over a closed par= c9X_ = fl titioned manifold M U X_ with X_ = Y from the Fredholm pair of Cauchy data spaces along Y. Similarly, the index of the linear conjugation (or trnnsmis= (s.,.jy) is given sion) problem = 0 in by twisting the Cauchy data spaces with 4. Related local elliptic boundary conditions for systems of Dirac operators are considered.

In this chapter we present some applications of the theory developed in the previous chapters and explain the geometric meaning of the spaces of Cauchy data and of the Calderón projector, i.e. objects which live on the submanifold Y but originate from global data. We deal with a closed Riemannian manifold M which is partitioned into two manifolds with joint boundary: (24.1)

C°°(M; F) be a Dirac operator on M. We assume the splitting form (9.3) in a bicollar neighbourhood of V (see

Let A : C°°(M; E)

Figure 24.1). Then on V and close to V we have the objects and relations introduced earlier in a slightly more restrictive situation, namely for the closed double M := K,. and A :=

Theorem 24.1. Let to the manifolds

A.,. U (A+)*.

denote the restrictions of the operator A with M, A as above. Then the following

equality holds:

(24.2) index A =

:

The theorem was conjectured by Bojarski in the middle of the type proof seventies. He envisaged it as the first step toward a of the Atiyah-Singer index theorem via a double induction argument: with respect to the number of handles in a Morse decomposition of

24. Bojarski's Theorem. General Linear Conjugation Problems

263

M

Fig. 24.1 The partitioned manifold M with bicollar neighbourhood M of Y

the manifold, and with respect to the dimension of the manifold. This will be discussed (to some extent) in the next chapter. Bojarski formulated his conjecture as the equality of the index of A and the index of the Fredholm pair of Cauchy data of the operator A along the submanifold Y. Let us explain his formulation. We begin with the concept of a Fredholm pair of subspaces.

Definition 24.2. Let H1, H2 be closed infinite-dimensional subspaces of a separable Hubert space H. We call (H1, H2) a pair of subspaces, if the following conditions are fulfilled: (24.3) (24.4)

a = dim(Hi fl H2)