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The Index of Elliptic Operators on Manifolds with Conical Points Boris Fedosov Bert-Wolfgang Schulze Nikolai Tarkhanov Institut fur Mathematik Universitat Potsdam Postfach 60 15 53 14415 Potsdam Germany August 29, 1997

Supported by the Deutsche Forschungsgemeinschaft.

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Abstract For general elliptic pseudodierential operators on manifolds with singular points, we prove an algebraic index formula. In this formula the symbolic contributions from the interior and from the singular points are explicitly singled out. For two-dimensional manifolds, the interior contribution is reduced to the Atiyah-Singer integral over the cosphere bundle while two additional terms arise. The rst of the two is one half of the `eta' invariant associated to the conormal symbol of the operator at singular points. The second term is also completely determined by the conormal symbol. The example of the Cauchy-Riemann operator on the complex plane shows that all the three terms may be non-zero.

AMS subject classication: primary: 58G10 secondary: 58G03. Key words and phrases: manifolds with singularities, pseudodif-

ferential operators, elliptic operators, index.

Contents

Contents

Introduction 1 Formal symbols 2 Pseudodierential operators 3 Algebraic index theorem 4 Two-dimensional case 5 Examples References

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5 8 12 15 23 31 35

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The Index of Elliptic Operators

Introduction

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Introduction Since Atiyah, Patodi and Singer APS75] investigated the index problem for

rst order elliptic dierential operators arising in Riemannian geometry on manifolds with cylindrical ends and introduced the famous `eta' invariant, there appeared a great number of papers dealing with index theorems for elliptic operators on manifolds with conical points. Among notable generalisations of their formula in the setting of Dirac operators are those to noncompact manifolds by Bruning and Seeley BS88], Bruning Bru90], Muller Mul87], Stern Ste89, Ste90] and Klimek and Wojciechowski KW93], to singular manifolds by Cheeger Che83, Che87], Melrose Mel93], Lesch Les97] and Hassell, Mazzeo and Melrose HMM96], to boundary value problems by Branson and Gilkey BG92], Douglas and Wojciechowski DW91], Muller Mul94], Grubb Gru92] and Grubb and Seeley GS95], to families by Bismut and Cheeger BC89, BC90] and Melrose and Piazza MP93], and also to de ne \higher" `eta' invariants by Lott Lot92], Getzler Get93] and Wu Wu93]. The problem of extending the Atiyah-Patodi-Singer index theorem to a general index theorem for elliptic pseudodierential operators on manifolds with conical singularities is of considerable interest. For a class of zero order pseudodierential operators on a closed manifold having a discontinuity of rst kind at isolated points, an index formula was proved by Plamenevskii and Rozenblum PR90, PR92]. Their approach is based on considering C -algebras generated by such operators it was further developed by Rozenblum Roz94]. In the framework of the b-calculus of Melrose Mel81] and Melrose and Mendoza MM83], an index formula for general elliptic operators was established by Piazza Pia93] who made use of the `zeta' function approach. His formula expresses the index of an operator acting on a Sobolev space of weight  2 , as the sum of an interior contribution given in terms of regularised `zeta' functions, and a boundary contribution generalising the `eta' invariant of Atiyah, Patodi and Singer. This second term measures the asymmetry of the spectrum of the indicial operator (or conormal symbol) with respect to a weight line ; in the complex plane. While the Atiyah-Patodi-Singer index theorem for a twisted Dirac operator can be obtained from the formula of Pia93], it is still an open problem as to whether, for a general elliptic b-pseudodierential operator A, the interior contribution depends only on the principal symbol of A. For an elliptic dierential operator on a manifold with a conical point whose conormal symbol is symmetric with respect to some weight line, Schulze, Sternin and Shatalov SSS97] proved a formula expressed the index as the sum of the ranks of spectral points of the conormal symbol and the integral from the Atiyah-Singer form over the smooth part of the manifold. The authors have

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The Index of Elliptic Operators

also contributed to this problem. But despite such an activity during the last 20 years the situation remains still unsatisfactory. The main diculty is that the algebra of pseudodierential operators on a manifold with conical singularities consists of components of dierent nature (interior and conormal) obeying some compatibility conditions, and it is not quite clear in which terms the index of elliptic operators should be expressed. In other words, one has to nd proper functionals on this algebra to express the index in a possibly simple way. Such an investigation was done by Melrose and Nistor MN96a] who computed the Hochschild and cyclic homology groups for the algebra of `cusp' pseudodierential operators on any manifold with boundary. These computations are closely related to the index problem since the index functional for this algebra can be interpreted as a Hochschild 1-cocycle. Their index formulas contain various extensions of the trace functional of the Wodzicki non-commutative residue type, thus resulting in a pseudodierential generalisation of the Atiyah-Patodi-Singer index theorem. In MN96a], they also computed K -theory invariants of algebras of pseudodierential operators on manifold with corners and proved an equivariant index theorem. Here we use another approach based on the algebra of formal symbols on a manifold with conical points blown up to a smooth manifold with boundary, M . This algebra is a particular case of the algebra of observables in deformation quantisation adopted to the special cotangent bundle structure of the symplectic manifold in question. It was previously used by the rst author Fed74] (see also Section 1 below for a short review) to approximate pseudodierential operators on a smooth manifold up to trace class operators. It turns out that the algebra of formal symbols is equally good to approximate the conormal components of pseudodierential operators on manifolds with conical points. This approximation leads to an index formula which we call algebraic, ind A = Tr (1 ; r  a)jN ; Tr (1 ; a  r)jN Z1   + 21i d Tr0 A;1 ( + i )A0c( + i ) ; Op@M (r( )  a0( )) jN ;1 c ;1 (0.1) (see Section 3). Here a and r are the formal symbols of the operator A and its parametrix R de ned up to the boundary,  stands for the product of formal symbols, Ac( ) and A;1 c ( ) are the conormal symbol of A and its inverse, Tr0 means the trace of a pseudodierential operator on the boundary of M , and N is the order of approximation, N  dim M . The subscript  signalls for a formal complex shift of the real argument, 1 (k) ( ) X a k a ( ) = ( ih ) (0.2) k=0 k !  de ning the weight line.

Introduction

7

It should be mentioned that more or less similar formulas were previously obtained in various particular cases, see e.g. PR90] or SSS97]. We have plucked up the courage to publish formula (0.1) because to our mind it seems new in such a form and such generality. The only assumptions we impose are that the symbols do not depend on the normal variable near the boundary (translation invariance in the terminology of MN96a]), and that Ac( ) and A;1 c ( ) are holomorphic in a strip around the weight line. As but one application of formula (0.1) we extend the result of Schulze, Sternin and Shatalov SSS97] to the case of pseudodierential operators. Namely, combining (0.1) with arguments of Section 4 ibid. thinking of M as half of a compact manifold without boundary by doubling across the boundary, we arrive at a nice index formula of the Atiyah-Singer type for operators satisfying the symmetry condition of SSS97] and possessing a meromorphic extension to the strip between the weight line and the symmetry axis ;0. More precisely, Z X X 1 rank z + rank z) sgn  + S M AS (A) ind A = ;( 2 z2spec A z2spec A =z=0

c

c

=z2(0 )

S M being the cosphere bundle of M and AS (A) being the Atiyah-Singer integrand manufactured from the principal interior symbol of A and dierential-geometric information in M by local operations. We use the machinery of cone pseudodierential operators, referring the reader to the book of Schulze Sch94]. The only dierence is that we do not use the Mellin transform applying the change of variables by r = e;t. Under this change the conical point becomes a `cylindrical end' and Mellin pseudodierential operators become Fourier pseudodierential operators on the weight line = =  . Such a modi cation has the advantage that we do not have to switch between Mellin and Fourier representations of pseudodierential operators. In particular, the compatibility condition for interior and conormal symbols looks much simpler for the Fourier representation. Our next goal is to simplify formula (0.1), in particular, to reduce its interior contribution to the Atiyah-Singer integral over the cosphere bundle. Unfortunately, we have succeeded only in the case dim M = 2. The reduced formula is Z 1 tr (a;1@a )3 ; 1 tr 0(a;1 @a ) ; 1 tr 1(@a a;1) ind A = 41 2 0 0 0 0 0 0 2 2 SM6   0 ;1 0 0 + 21i Tr A;1 (  + i ) A (  + i ) ; i ( A (  + i ) A (  + i )) c c c c  ! f a;1a1 ; i a;1 @a0 a;1 @a0  + 21i Tr 0 2 0 @ 0 @x (0.3) 0 and 1 being the curvature forms of the bundles E 0 and E 1 and tr meaning the matrix trace of matrix-valued functions.

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The Index of Elliptic Operators

The rst summand on the right-hand side of (0.3) is the usual AtiyahSinger integral. The next two summands contain the functionals Tr (regf (formal trace) introduced by Melrose Mel95]. In ularised trace) and Tr particular, the second summand is the `eta' invariant (or rather one half of it) of Ac( ) shifted formally to the real axis. On the other hand, the f is non-zero on functions a(x   ) on (T @M )  positively functional Tr homogeneous of degree ;1 = ; dim @M in (  ). For such functions it takes the form Z 1 f Tr a = 2i T @M tr a(x   )j =1 (0.4) =;1 ddx:

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The advantage of formula (0.3) is that it contains merely principal symbols of A, namely the principal interior symbol a0 and the conormal symbol Ac( ), which control the Fredholm property of A acting on the Sobolev space of weight  . The lower-order term a1(x   ) entering into the third summand of (0.3) is completely determined by the conormal symbol. Clearly, formula (0.3) can be extended to symbols which are not translation invariant near the boundary. We consider the example of the Cauchy-Riemann operator on the complex plane treating the point at in nity as a conical point. This is a particular case of the Riemann-Roch theorem in the interpretation of Melrose Mel93, 6.3]. This example shows that all the three terms in (0.3) may be dierent from zero. It is curious enough that they are half-integer (if non-zero). It would be interesting to extend formula (0.3) to higher dimensions, but even in the case of half-space n+ this seems to be a non-trivial homological problem.

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1 Formal symbols Recall an algebraic index formula on a smooth manifold, cf. Fed74]. Let M be a smooth manifold and E 0, E 1 vector bundles on M . A formal symbol a 2 Symb (M  E 0 E 1 ) is de ned by its local expressions and transitions rules. In a local chart U  M and for given trivialisations of the bundles E 0, E 1 over U the local expression is a formal series a = aU = a0(x ) + ha1(x ) + h2a2(x ) + : : :  where h is a formal parameter and ak (x ) are functions on U  n with values in the space of (r1  r0)-matrices of complex numbers. Here, n is the dimension of M and r0, r1 stand for the ranks of E 0 and E 1 respectively. These expansions form an associative algebra with respect to the product X (;ih)jj @ a @ b a  b = @ @x 2Z+ !

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n

Formal symbols

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usually called a star-product or the Leibniz product. For a change of coordinates y = f (x) in U \ V we de ne a transition rule by

aU (x )   i 0 ;1 0 V 0 ;1 = a (f (x) f (x))  exp h f (x)(f (x)+ f (x)(y ; x) ; f (y)) jy=x  (1.1) the symbol multiplication on the right-hand side being taken with respect to the variables y and  while x is xed. The vector (or rather the covector)

= f 0;1(x) in coordinates is de ned by the equality j (x) @f i = j @xi  and so (1.1) is a formal version of the change of variables for pseudodifferential operators. Note that (1.1) does not contain negative powers of h because of the second order zero in y ; x in the exponent. Changing frames of the bundles results in the following rule 1 aU = gUV aV  gV0 U :

(1.2)

For the leading term a0(x ) we have 1 aU0 = gUV f aV0 gV0 U 

where (f aV )(x ) = aV (f (x) f 0;1(x)). So a0(x ) varies like a function on T M with values in Hom ( E 0  E 1). The higher order terms vary in a more complicated way, in particular, 2f k @ 2 aV (f (x) ) aU1 = aV1 (f (x) ) + 2i k @x@ i@x (1.3) j @i @j 0 where = f 0;1(x). Choosing a torsion-free connection r on M , we may express (1.3) in another way by saying that 2 a0(x ) a1(x ) ; 2i k ;kij @ @ i@j behaves like a function on T M under change of variables, ;kij being connection coecients. Similarly, (x ) ;0 a1(x ) + i @a0@ i i behaves like a homomorphism of bundles, ;0i being connection coecients for the bundle E 0.

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The Index of Elliptic Operators

Given a function a(x ) on T M with values in Hom ( E 0  E 1), we can always construct a formal symbol a with the leading term a0(x ) = a(x ). To this end choose a covering (U ) by coordinate charts, local trivialisations of the bundles and a partition of unity ( ). For a chart U, we de ne the symbol  a =  (x)a(x ) to consist of the leading term only, for chosen trivialization. Other local expressions for the symbol  a are de ned by (1.1) and (1.2). Summing all the symbols  a, we obtain the desired symbol a. Clearly, if the local functions aUk (x ) are classical symbols of order m ; k, m 2 , on U  (for example, if they are homogeneous of order m ; k in  for large jj), then so are the functions aVk for any other local representation. The symbols with this property we denote by Symbm. The calculus of formal symbols is very similar to the calculus of complete symbols of classical pseudodierential operators. The only dierence is that for the formal symbols not only the asymptotic behaviour for large j j is relevant but the behaviour of their coecients for small j j as well. For example, formal symbols vanishing for large jj are meaningful objects unlike complete symbols. It is clear that for a classical complete symbol of order m we can construct a formal symbol whose local coecients ak (x ) coincide with the homogeneous components of degree m ; k of the complete symbol, for large j j. In this case we say that the corresponding pseudodierential operator A and the formal symbol a are compatible. For a pair (A a), A and a being compatible, we de ne a regularisation of A as follows. Let NX ;1 ajN = ak (x )

R R

k=0

be a partial sum for a with h = 1. Let further X Op (ajN ) := Op ( aU jN )~ 



(1.4)

be the pseudodierential operator on M de ned by a coordinate covering (U ), partition of unity ( ) and trivialisations of the bundles. The functions ~ are required to be identically equal to 1 on the supports of  but still satisfy supp ~  U . (In this way we obtain what will be referred to as `covering functions'.) As usual, Op means a standard pseudodierential operator on n de ned by the symbol  ajN . Of course, the operator (1.4) depends on the choices mentioned above. The operator regjN A := A ; Op (ajN ) is of order m ; N and thus is of trace class in each Sobolev space H s (M ) provided N ; m > n (we suppose M is compact). De ne a regularised trace of A by Tr regjN A := Tr (A ; Op (ajN ))

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Formal symbols

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A and a being compatible. The properties of the regularised trace are described in the following proposition.

Proposition 1.1 Let A be a classical pseudodierential operator of order m and let a 2 Symbm be a formal symbol compatible with A. Then: 1) the regularised trace Tr (A ; Op (ajN )) is independent of the choices of coordinate charts, trivialisations and partition of unity. 2) the regularised trace vanishes on commutators, more precisely, Tr regjN AB = Tr regjN BA if the regularisations of AB and BA are dened by the formal symbols a  b and b  a, respectively.

The proofs may be found in Fed74]. If a 2 Symbm with m < ;n, then the operator Op (ajN ) belongs to the trace class and its trace is correctly de ned by the symbol a and the truncation number N . In this case we abbreviate the notation of its trace to Tr ajN := Tr Op (ajN ): This number may be interpreted as an integral over T M , more precisely,  Z Z 1 tr ajN d dx (1.5) Tr ajN = (2)n M

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n

where the inner integral de nes a density on M not depending on the local representation of the formal symbol a. In (1.5) and subsequently, the notation tr stands for the matrix trace of the coecients ak (x ). We are interested in the application of the notion of formal symbol to the index theorem for elliptic operators. If A is an elliptic pseudodierential operator then there exists a parametrix R such that 1 ; RA and 1 ; AR are smoothing operators. If further, a and r are formal symbols compatible with the operators A and R the following algebraic index formula holds ind A = Tr (1 ; r  a)jN ; Tr (1 ; a  r)jN

(1.6)

for any N  n. This formula is an intermediate step between the analytical index formula ind A = Tr (1 ; RA) ; Tr (1 ; AR) (1.7) and the topological index formula due to Atiyah and Singer. For further references let us write down the Atiyah-Singer index formula for two-dimensional manifolds, Z 1 ind A = 42 S

1 tr (a;1@a)3 ; 1 tr 0 (a;1@a) ; 1 tr 1(@aa;1) (1.8) 2 2 M6

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The Index of Elliptic Operators

where S M is the cosphere bundle oriented as a boundary of the coball bundle B M with the orientation form !n = (di ^ dxi)n n! n! = d1 ^ dx1 ^ : : : ^ dn ^ dxn the covariant dierential @ of homomorphisms a : E 0 ! E 1 is de ned on sections u 2 C 1(M E 0) by (@a)u = @ 1(au) ; a@ 0u

@ 0, @ 1 being connections on E 0, E 1, and 0, 1 are the curvature forms of E 0 , E 1. We nish this section by a remark that the calculus of formal symbols is a particular case of a more general calculus, called deformation quantisation, on an arbitrary symplectic manifold. Here, the symplectic manifold in question is T M .

2 Pseudodierential operators From the point of view of analysis, a manifold with a conical point is in fact a smooth manifold with boundary (M @M ). A collar neighborhood of the boundary is dieomorphic to a cylinder @M  I , where I is the interval 0  r < 1. Typical dierential operators considered in the cone theory (see e.g. Schulze Sch94]) are of the form

A=r

;p

m X



@ ak (r) ;r @r k=0

!k

(2.1)

close to the boundary, with coecients ak (r) smooth up to r = 0, their values being dierential operators on @M of order m ; k. Usually one considers the weight factor in front of the sum with p = m but this is not necessary, we allow any p 2 . The analysis of such operators is based on the Mellin transform which evokes a technical diculty caused by switching between Fourier and Mellin representations of pseudodierential operators. To avoid it we pass from the collar neighbourhood to a cylindrical end by changing the variable r = e;t, 0 < t < 1. The operator (2.1) becomes

R

 !k m X @ : pt ;t e ak (e ) @t k=0

(2.2)

The exponential stabilisation of coecients as t ! 1 is inherited from the smoothness up to r = 0 in (2.1). The exponential weight factor ept may be included into basis sections of the bundle E 1 in the cylindrical charts.

Pseudodierential operators

13

This means that we pass from the frame e1(x0 t) : : : er1 (x0 t), r1 being the rank of E 1, to the frame

e01(x0 t) = e1(x0 t)ept : : :  e0r1 (x0 t) = er1 (x0 t)ept: In this new basis the operator (2.2) becomes 

!

k @ ak (e ) @t : k=0 m X

;t

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In the sequel by a conical point we mean a cylindrical end +  @M and exponential stabilisation of all the objects in question at t = +1. We assume that near the end everything has a cylindrical structure. So, vector bundles over the end are lifted from the boundary @M , local charts have the form U = +  U 0 with U 0  @M , thus resulting in the splitting of coordinates x = (t x0), x0 2 U 0 . Next, for a cylindrical chart Ui, we have

R

 (x) = 1 (t)0(x0) ~ (x) = ~1 (t)~0(x0)

R

where () is a partition of unity and (~ ) the set of covering functions. We emphasise that 1 2 C 1( +) is equal to 1 close to t = +1 and vanishes close to t = 0. Introduce the weighted Sobolev spaces H s (M E ) by requiring u 2 H s ( n) for the interior charts and e t u 2 H s( n) for the cylindrical charts, the norm being equal to

R

kuk2s =

R

X



k uk2s +

X

ke t j uk2s

R R j

where  runs over interior charts and j over cylindrical charts. For a C 1 function u(t x0) of compact support in +  n;1, the weighted Sobolev norm ke tuks may be written as Z

C

R ;1 n

Z

;

(1 + j s2, 1 > 2 (but not for s1 > s2, 1 = 2), Hilbert-Schmidt for s1 > s2 + n=2, 1 > 2, and of trace

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The Index of Elliptic Operators

class for s1 > s2 + n, 1 > 2 (see e.g. FS96]). The space H ;s; is dual for H s with respect to the usual paring given in local coordinates by (u v) =

Z

R

n

uv dx for u 2 H s  v 2 H ;s; :

We next describe pseudodierential operators acting between weighted Sobolev spaces. We con ne ourselves to operators which in the cylindrical charts are independent of t apart from a weight factor ept which we always omit including it into the frame of the bundle E 1 as was explained before. For the purposes of index theory this class is sucient. The cone pseudodierential operators are of the form

R

Au = 0 Aint ~0u + 1 A ~1u

R

(2.3)

where 1 2 C 1( +) is a non-negative function satisfying 1  0 near t = 0 and 1  1 near t = +1, and 0 = 1 ; 1 . The operator Aint is a classical pseudodierential operator of order m 2 in the interior of M . The operator A is de ned by a so-called conormal symbol Ac( ),  2 ; , which is a parametre-dependent pseudodierential operator of order m on @M . We assume that Ac( ) has an analytic extension to a strip j= ;  j < " around the weight line and on each horizontal line in this strip is a parametre-dependent pseudodierential operator on @M uniformly with respect to = in a smaller strip j= ;  j  "1, with "1 < ". Then, A acts 1 ( 1 on functions u(t) 2 Ccomp + C (@M E )) as

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A u (t) = := =

Opt (Ac( )) u (t) 1 Z eit A ( ) u^( ) d c 2 ; e; t Opt(Ac( + i )) e tu (t): 

(2.4)

Here we denote by Opt a standard pseudodierential operator with the shifted integration line = =  while the notation Opt = Op0t means that the integration line is the real axis. In both cases the symbols Ac are operator-valued functions. It is well known, cf. Schulze Sch94], that such operators are bounded, when mapping from H s to H s;m . The operators Aint and A in the cone theory are not independent. They satisfy a very important compatibility condition expressing the fact that A arises in practice as a result of rede nition of Aint. Roughly speaking, it 1 ( means that, for any 1 2 2 Ccomp +), the dierence

R

1 Aint 2 ; 1 A 2

(2.5)

is a smoothing operator on M and thus belongs to the trace class in the spaces H s . More precisely this condition can be formulated in terms of formal symbols.

Pseudodierential operators

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Let a be a formal symbol of Aint. We assume that in the cylindrical charts it takes the form

a(t  x0 0 )  a( x0 0) 2 Symbm(M  E 0  E 1): Being independent of t, the symbol a may be considered as a formal symbol on @M and we can construct a parametre-dependent operator Op@M (a( )jN ) as described in Section 1. De ne a formal imaginary shift of the parametre  by

a ( ) = a( + ih ) 1 X @ k a( ) (i )k := hk k1! @ k k=0 = e t a( )  e; t:

(2.6)

The compatibility condition in a more precise form means that, for any N , the dierence

R

RN ( ) = Ac( + i ) ; Op@M (a( + ih )jN )  2  is a classical parametre-dependent pseudodierential operator of order m ; N on @M bearing the following trace norm estimate, for m ; N < ;n + 1,

@ k R ( )k  C h im;N ;k+n;1 @ k N 1 where k  k1 stands for the trace norm of the operator in any space H s (@M ). For standard pseudodierential operators on n;1 whose symbols are homogeneous in ( 0) outside a ball these estimates are valid but in general we have to postulate them. Operator (2.5) may be written as k

R

1 (Aint ; Op (ajN )) 2 ; 1 e; t Opt(RN ( )) e t2 so, for compactly supported 1 and 2, both summands are trace class operators in H s , for each  . Denoting by C m the class of such operators (obeying the compatibility condition), we obtain what is called the cone algebra. Indeed, one can verify that the composition of A 2 C m and B 2 C l gives an operator BA 2 C l+m . In the case when the order m coincides with the exponent p of the weight factor ept in (2.2), this algebra was studied by the second author (see e.g. Sch94] and references therein). It is worth pointing out that the cone theory deals with Mellin pseudodierential operators near a conical point, but the results may be easily transported to our case by the change of variables r = e;t.

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The Index of Elliptic Operators

3 Algebraic index theorem In this section we consider elliptic operators of the type (2.3). Recall that ellipticity means that the principal homogeneous symbol a0(x ) is invertible for  6= 0 (interior ellipticity) and that the conormal symbol Ac( ) is an invertible pseudodierential operator on @M for any  2 ; (conormal ellipticity). The interior ellipticity implies that there exists a formal symbol r on M such that 1 ; r  a and 1 ; a  r vanish for jj large enough. We denote by Rint a pseudodierential operator on M compatible with r (see Section 1). The conormal ellipticity allows one to de ne an operator

R = Opt (A;1 c ( )): The operators 1 ; RintAint and 1 ; AintRint are smoothing in the interior of M (that is, being multiplied by a cut-o function  of compact support away from the boundary of M ), hence the compatibility condition for Rint and R follows from (2.5). Indeed,

R ; Rint = R (1 ; AintRint) + R (Aint ; A ) Rint and we see that both summands are smoothing in the interior. We introduce an operator

R = 0 Rint ~0 + 1 R ~1 2 C ;m

(3.1)

called a parametrix of A. Theorem 3.1 The operators 1 ; RA : H s ! H s  1 ; AR : H s;m ! H s;m are of trace class and the following index formula holds ind A = Tr 0(1 ; r  a)jN ; Tr 0 (1 ; a  r)jN Z1 1 0 + 2i Tr0 regjN ;1 A;1 c ( + i )Ac( + i ) d ;1

(3.2)

for any N  n. Let us explain the notations and give some comments. First of all, Tr0 denotes a trace on @M and regjN ;1 denotes a regularisation of the corre0 sponding A;1 c Ac (here the prime means the derivative with respect to  ) by means of its formal symbol as was explained in Section 1. It is important for this formal symbol (and thus the regularisation) to be de ned by the

Algebraic index theorem

17

R

interior formal symbols a and r as follows. In a cylindrical chart the symbols a = a( ) and r = r( ), being independent of t, may be considered as formal symbols on @M depending on the parametre  2 . Denote by a ( ) = a( + ih ) r ( ) = r( + ih ) the formally shifted symbols de ned by (2.6). Then we must take the symbol r ( )  a0 ( ) = (r( )  a0( )) 0 to regularise A;1 c Ac . In other words, 0 regjN ;1 A;1 c ( + i )Ac ( + i ) 0 = A;1 r( )  a0( )) jN ;1): (3.3) c ( + i )Ac( + i ) ; Op@M (( The cut-o function 0 entering into the rst two terms in (3.2) is inessential. Indeed, for two dierent choices of 0(t), we have $0 2 1 ( Ccomp +), so the dierence of the corresponding expressions is 1 Z 1 $ (t)dt Z 1 d (Tr0 (1 ; r( )  a( ))j ; Tr0 (1 ; a( )  r( ))j ) : 0 N N 2 0 ;1 But the expression in the parentheses vanishes because it is equal to the index of Ac( ) which is zero for Ac( ) is invertible. Proof. The starting point is formula (1.7). Using (2.3) and (3.1), we obtain 1 ; RA = 1 ; 0Rint~00Aint~0 ; 0Rint~01 A ~1 ;1 R ~1 0 Aint~0 ; 1 R ~1 1 A ~1 (3.4) and 1 ; AR = 1 ; 0Aint~00Rint~0 ; 1 A ~1 0Rint~0 (3.5) ;0 Aint~0 1 R ~1 ; 1 A ~1 1 R ~1 : The corresponding terms in (3.4) and (3.5) dier by the order of factors. We will transform these terms pairwise and write AB  C , BA  D if AB ; C and BA ; D are of trace class and their traces coincide. For example, AB  A0B 0 BA  B 0A0 (3.6) if both A ; A0 and B ; B 0 are of trace class since Tr (AB ; A0B 0) = Tr (A ; A0)B + Tr A0(B ; B 0) Tr (BA ; B 0A0) = Tr B (A ; A0) + Tr (B ; B 0)A0: As Op (0r  ~0  0a  ~0jN )  Op (0r  0ajN )

R

18

The Index of Elliptic Operators

the rst pair in (3.4) and (3.5) may be replaced by another one equivalent to it, namely Op (0r  0ajN ) Op (0a  0rjN ) (3.7) which is clear from Proposition 1.1. To transform the next pair we use the pseudolocality of pseudodierential operators. Since 01 has a compact support in +, we may replace 0 by 1 and 1 by 2, each of them having a compact support in +, such that 01 = 12. By ~1, ~2 we denote \covering" functions for 1, 2 still being of compact support in +. Let us justify, for example, the replacement of 1 by 2. We choose 2 so that (1 ; 2)~0  0. Then (1 ; 2)A ~1 0Rint~0 is a trace class operator by the pseudolocality property since 1 ; 2 and 0 have disjoint supports (details may be found in FS96]). The trace of this operator is 0 since Tr (1 ; 2)O~0 = Tr (1 ; 2)1 O~0 = Tr 1 O~0(1 ; 2) = 0: The corresponding term with the order interchanged is identically zero. Similarly we may replace step by step 0 by 1, ~0 by ~1 and ~1 by ~2. As a result we obtain an equivalent pair of the form 1Rint~12A ~2 2A ~21Rint~1: Now, using (3.6) and compatibility condition (2.5), we replace 2A ~2 by 2Aint~2 obtaining the pair 1Rint~12Aint~2 2Aint~21Rint~1: Since Op (1r  2ajN )  Op (0r  1ajN ) we may make use of Proposition 1.1 once again to replace the latter pair by Op (0 r  1ajN ) Op (1 a  0rjN ): (3.8) Similarly, the third pair in (3.4), (3.5) may be replaced by Op (1 r  0ajN ) Op (0a  1rjN ): (3.9) In the last pair the operators may be treated as pseudodierential operators on the half-axis t 2 + with operator-valued symbols. The cut-os ~1 may be omitted by pseudolocality. Denote temporarily O(t  ) = 1 (t)Ac( ) P (t  ) = 1 (t)A;1 c ( ):

R

R

R R

Algebraic index theorem

19

In these notations the last pair in (3.4), (3.5) is Opt (P (t  )) Opt (O(t  )) Opt (O(t  )) Opt (P (t  )): It is equivalent to the pair Opt (P (t  )  O(t  )jN ) Opt (O(t  )  P (t  )jN )

(3.10)

which is due to the theorem on the regularised trace of a product from FS96] similar to Proposition 1.1, n. 2). Here  means the symbol product of operator-valued symbols on the shifted real axis and the subscript N means truncation, that is

P  OjN =

NX ;1 k=0

(;i)k @ k P @ k O : k! @ k @tk

Gathering all the terms (3.7)-(3.10), we obtain 1 ; RA  1 ; Op (0r  0ajN ) ; Op (0r  1 ajN ) ;Op (1 r  0ajN ) ; Opt (1 A;1 c ( )  1 Ac ( )jN ) 1 ; AR  1 ; Op (0a  0rjN ) ; Op (1 a  0rjN ) ;Op (0 a  1 rjN ) ; Opt (1 Ac ( )  1 A;1 c ( )jN ): Further, ;1 1A;1 c ( )  1 Ac ( )jN = 1 Ac ( )  (1 ; 0 )Ac ( )jN = 1 ; 1A;1 c ( )  0Ac ( )jN

and similarly ;1 1Ac( )  1A;1 c ( )jN = 1 ; 1 Ac ( )  0 Ac ( )jN :

This results in 1 ; RA  Op (0(1 ; r  a)jN ) +Opt (1A;1 c ( )  0Ac ( )jN ) ; Op (1 r  0ajN ) (3.11) 1 ; AR  Op (0(1 ; a  r)jN ) +Opt (1Ac( )  0A;1 c ( )jN ) ; Op (1 a  0 rjN ): (3.12) The rst terms on the right-hand sides of (3.11) and (3.12) are trace class operators and the dierence of their traces gives precisely the interior terms in (3.2). In the rest terms all the symbols have compact supports in t 2 + since 1 0 has, so we may regard them as operators on the cylinder +  @M . To feel free with the exponential weight functions let us formulate the following lemma.

R R

20

R

The Index of Elliptic Operators

Lemma 3.2 Let a(t  ) be an operator-valued symbol with a compact

support on

+

. Then, for N large enough, the operator

ept Opt (a(t  )) e;pt ; Opt (epta(t  )  e;ptjN )

is of trace class and its trace vanishes.

Proof. This is a particular case of the theorem on the regularised trace

of a product. We take

A = Opt (epta(t  )) B = Opt (e;pt): Note that e;pt may be replaced by any function coinciding with e;pt on the support of a, which is due to pseudolocality. . Consider the last two terms on the right in (3.11). We will use the notations A~( ) = A( + i )  2  for the complex shift and

R

a (t  x0 0) = a(t  + ih x0 0 ) for the formal complex shift. By (2.4), ; t ;1 t Opt (1 A;1 c  0 AcjN ) = e Opt (1 A~c  0 A~c jN ) e

and

e t Op (1 r  0ajN ) e; t  Op (1 r  0a jN ) the last equality being a consequence of Lemma 3.2 and (2.6). Further, using the equality Op = Opt Op@M , we obtain nally for the last two terms in (3.11) the equivalent expression   ~ Opt 1A~;1   A j ; Op (  r  (  )   a  (  ) j ) (3.13) 0 c N @M 1 0 N c and similarly for the last two terms in (3.12)   j ; Op (  a  (  )   r  (  ) j ) : Opt 1 A~c  0A~;1 N @M 1 0 N c

(3.14)

The operators (3.13), (3.14) are of trace class due to the compatibility condition, so the conjugations by e; t do not aect their traces. The next transformation goes as follows. We carry the factor 0 through ~ Ac and a in (3.13), thus obtaining Opt fOp@M (1 (1 ; r  a )  0jN ) o ~ ;1 A~;1   A   ] j + Op (  r    a   ] j ) : c 0 N @M 1 0 N c

(3.15)

Algebraic index theorem

21

In (3.14) we move 1 through A~c and a as follows 1 A~c = A~c1 ; A~c 1] = A~c1 + A~c 0]: This gives n Opt A~c  1  0A~;1 c jN ; Op@M (a ( )  1  0r ( )jN ) o +A~c 0]  0A~;1 j ; Op ( a (  )   ]   r  (  ) j ) @M 0 0 N : (3.16) c N Now, expressions (3.15), (3.16) are trace class operators in H s0. Moreover, expressions in the curly brackets in (3.15), (3.16) are trace class operators on @M since each operator-valued symbol is regularised by subtracting a non-trace class part. We may change the order of operators in (3.16) changing simultaneously the order of symbols, thus arriving at Opt fOp@M (0(1 ; r  a )  1 jN ) o +0A~;1 c  A~c 0 ]jN ; Op@M (0 r ( )  a ( ) 0]jN ) : (3.17) Finally, taking traces of (3.15) and (3.17) and subtracting yields Z1 Z1   1 0 ~;1 ; dt d Tr A  A~c 0 ]jN ; Op@M (r ( )  a ( ) 0 ]jN ) : c 2 0 ;1 Now, 1 X (;i)k dk A~c dk 0  A~c 0] = k k k=1 k ! d dt 1 X (;i)k @ k a ( ) dk 0 : a ( ) 0] = @ k dtk k=1 k ! Integration over t yields zero unless k = 1. In the latter case Z 1 d 0 dt dt = ;1 0 and we obtain the desired result.

Remark 3.3 If a symbol a( ) decreases suciently fast then formal

complex shifts do not aect the value of the integral, that is Z

1

;1

a( )d =

Z

1 ;1

a( + ih )jN ;1 d:

Indeed, the terms of a(t + ih ) have the form (ih )k dk a( ) k! d k

22

The Index of Elliptic Operators

so the integration gives zero, for k large enough. This allows us to replace expression (3.3) by the formally shifted one. When shifting by ;ih , we obtain another expression for the integrand in (3.2) which is sometimes more convenient, ind A = Tr 0 (1 ; r  a)jN ; Tr 0 (1 ; a  r)jN Z1  0 Tr0 A;1 + 21i c ( + i ; ih )Ac( + i ; ih )jN ;1 ;1



; Op@M (r( )  a0( )jN ;1) d:

(3.18)

0 Remark 3.4 A special case of (3.18) arises when A;1 c ( )Ac ( ) is a mero-

morphic operator-valued function with nite-dimensional Laurent coecients at poles. In this case the integral in (3.18) may be reduced to the integral over the real axis (if there are no real poles) and a nite sum of residues in the poles between the real axis and the weight line. This is known as the Relative Index Theorem (see e.g. Melrose and Mendoza MM83]). Indeed, for a meromorphic function a(z) with a regular behaviour as 1 we have r0 = a;1 0 and, as we have seen, the integrand coincides with the Atiyah-Singer form which is now a constant multiple of tr (a0da0)3 since 0 = 1 = 0. Taking into account that a0 is a homogeneous function for r > 1, we obtain 

3

3 tr (a;1 = tr mr;1dr + a~;1 ~0 0 da0) 0 da  3  2 ;1 ;1 = tr a~;1 d a ~ + 3 mr dr ^ tr a ~ d a ~ 0 0 0 0  0:

RS

Here we have put a0 = rma~0, so that a~0 depends on two variables and thus (~a;1 ~0)3  0 while tr (~a;1 ~0)2  0 for any matrix-valued function a~0. 0 da 0 da The second integral may be extended to all of @M  2. We would like to represent it as a trace of a formal symbol on @M = 1 integrated over  2 . To this end we observe that d may be replaced by d ^ @=@ everywhere in (4.6) but the last term since other dierentials enter into the form ! = d ^ dx. The second integral in (4.6) becomes 1 Z 1 d 1 Z ftr (r a0 ) + tr (r a0 ; r0 a ) 0 0 1 0 0 1 2i ;1 2 S1R ! i @r0 @a00 ; @r00 @a0 ; tr 2 @x @ @ @x

R





! !

)

@ tr r @a0  ; @ (tr (r a0 )) ddx: + @ 0 0 0 @ @

(4.7)

The integrand may be represented as

s(x   ) := tr r( )  a0( )j1 @ i @ tr @r0 a0 + i @ tr @r0 a0 + i @ tr @r0 @a0 ; tr (r0 a1) ; @   2!@! @x 0 2 @x @ 0 2 @ @ @x @ tr r @a0  ; @ (tr (r a0 )): + @ 0 0 0 @ @ and we rewrite integral (4.7) as 1 Z 1 d Tr0 Op (s(x   )) @M 2i ;1 (recall that I has compact support). The index formula (3.18) becomes Z Z1 d ind A = AS (A) + 21i ;1 S M   0 0 Tr0 A;1 (  + i ; ih ) A (  + i ; ih ) j ; Op ( r  a  j ; s ) : 1 @M 1 c c (4.8)

30

The Index of Elliptic Operators The symbol



!



!

@ @r0 a0 + @ r a0  + i @r0 a0 r( )  a ( )j1 ; s( ) = ; 2i @x @ 0 @ 0 0 2 @x 0  !  ! ! @a i @r @ 0 0 @a0 ; @ r0 @  + 2 @ @x ; (r0a1) (4.9) may be regarded as a new regularisation of the operator 0 A;1 c ( + i ; ih )Ac( + i ; ih )j1  d  ;1 0 0 = A;1 c ( + i )Ac ( + i ) ; i d Ac ( + i )Ac( + i ) : This new regularisation is in a sense canonical since the behaviour of the symbols a and r inside the ball r < 1 does not aect the result. This property may be seen directly from (4.9) because all the terms involved are complete derivatives. In Remark 3.5, we have introduced another canonical regularisation given by the functional Tr. The two regularisations dier by 1 Tr Op (s ; r  a0j ) @M 1 2i and this expression may be computed easily using (4.9). The terms of the form (@=@)() or (@=@x)() are annihilated by Tr while Tr (@=@ )() can be computed by Proposition 6 of Mel95]. In the notation of Mel95], 1 Tr Op (s ; r  a0j ) @M 1 2i   ! ! @ i @r @a 1 0 0 f Op@M tr r0 = ; 2i Tr @  + 2 tr @ @x ; tr r0a1 0



!

Z

 =1 1 1 @r 0 @a0 = ; 42 1 2 tr @ @x + i tr r0a1

ddx: S R  =;1 For further references let us write down the index formula for n = 2 in detail, Z 1 tr (a;1@a )3 ; 1 tr a;1 1 ^ @a + a;1@a 0 1 ind A = 42 0 0 0 0 0 0 2 S M 6  ! 1 d ;1 0 ;1 0 + 2i Tr Ac ( + i )Ac( + i ) ; i d Ac ( + i )Ac( + i ) 



!

!

Z

 =1 1 1 ;1 @a0 ;1 @a0 ;1 ; 2 1 4 SR 2 tr a0 @ a0 @x + i tr a0 a1

 =;1 ddx: (4.10) Remark 4.2 Note that formula (4.10) being homotopy invariant implies that it extends to all symbols a(t  ) exponentially stabilising when t ! 1. In this case the boundary values in (4.10) are a( 1).

Examples

31

5 Examples

C

In this section we consider the Cauchy-Riemann operator on a complex plane treating the point at in nity as a conical point. Even for this simple operator the additional term in (4.10) may be non-trivial. Although all the bundles are trivial over , the trivialisations and connections may be chosen in dierent ways. Our goal is to observe how these choices aect the three terms in (4.10). So, we consider two cases: 1. the Cauchy-Riemann operator from functions to (0,1)-forms, @ : u 7! dz @u @ z  2. the Cauchy-Riemann operator from functions to functions, @ : u 7! @u : @ z @ z Near z = 1 we introduce new coordinates  = t + i' t 2 + ' 2 (mod 2) setting z = e . Case 1. We take the global frame 1 for E 0 and dz, d as local frames in U0 = fjz j < 2g and U1 = f 0g for E 1 . In these frames we have @ +i @  A = @x @y a0 = i( + i ) in U0 and @ +i @  A = @t @' a0 = i( + i) ! @ Ac( ) = i  + @' in U1 . Thus,  !;1 @ ;1 0 Ac ( )Ac( ) =  + @' : Case 2. We take again the global frame 1 for E 0 and 1, e;t as local frames in U0 and U1 for E 1  = E 0. Again we have @ +i @  A = @x @y a0 = i( + i )

C

R R

32

The Index of Elliptic Operators

in U0, but

!



@ +i @  A = @t @' i' a0 = ie ( + i) ! @ Ac( ) = iei'  + @' ei'

in U1 . Then, just as in Case 1, !



;1 @ ;1 0 (5.1) Ac ( )Ac( ) =  + @'  and so in both cases the -invariant term is the same. The functions eik', k 2 , form an orthonormal basis of eigenfunctions of (5.1) with eigenvalues

Z

Z

Z

k ( ) = ( + ik);1

provided that  6= ;ik for k 2 . The conormal ellipticity holds for weight lines ; with  62 . 0 To compute Tr A;1 c Ac consider   0 (  ) A (  ) = ; Tr0 dd A;1 c c

1 X

k=;1

( + ik);2

=  d coth : d The rst integration gives  coth  . Next, if  < 0, then 



0 Tr A;1 c ( + i )Ac ( + i ) = Tlim !1

ZT ;T

 coth ( + i ) d

Z T +i

= Tlim coth z dz !1 ; T +i = ;2in ; i + 2i where n is the number of poles of coth z in the strip =z 2 (0  ). This may be seen by integrating over the contour which is the boundary of the rectangle with vertices ;T , T , ;T + i , T + i and with removed small disc centered at the origin. The principal value of the integral over the interval (;T T ) vanishes since coth z is an odd function. The second term under Tr functional gives  ! Z1 d d ;1 0 Tr ;i d (Ac ( + i )Ac( + i )) = ;i  coth ( + i ) d ;1 d = ;2i:

Examples

33 



Thus, the whole -invariant term is equal to ; n + 21 , for  < 0. In general, this term is equal to ; n + 12 sgn  . 3 Consider now the interior term. The form tr (a;1 0 @a0) vanishes identically since a;1 0 @a is a scalar 1-form. Thus, the interior term vanishes in Case 2 since 0 = 1 = 0 in this case. In Case 1 the interior term is 1 Z d ( + i ) 1 ; 2 (5.2) 8 S M  + i

where the orientation is de ned by the form d ^ dx ^ d ^ dy on T M . We

rst change the orientation form to d ^ d ^ dx ^ dy and then compute the integral of the rst factor in (5.2) obtaining

i Z 1 4 M with the orientation form dx ^ dy = dt ^ d'. Introduce connection forms ;10, ;11 in U0, U1 for the bundle E 1 = %01, i.e., de ne covariant dierentials of the frames dz in U0 and d in U1 . In U0 we set @ (dz) := (d0) dz + d (1 e )d = d0 + d(1 e )e; dz so that ;10 = d0 + d(1 e ) e;

= 1 d: Similarly, ;11 = ;0 d. It is a simple matter to see that these forms satisfy the usual transition rule for connection coecients. Then 1

whence

= = = =

d01 d11 d1 ^ d ;i d1 ^ d'

i Z 1 = 1: 4 M 2 Finally, let us compute the additional term in (4.10). The lower order symbol a1 vanishes in both cases. Moreover, in Case 1 the whole additional term is zero, since a0 does not depend on x.

34

The Index of Elliptic Operators For Case 2 we have Z Z 1  1  1

 =1 Z 2 Z1 1 1 1 d

; 2 1 ; dd' = d'

2 4 S ;1 2  + i  =;1 4 0 ;1 1 +  2 = 12 : So, we obtain

ind @ = ind @@z   1 1 = 2 ; n + 2 sgn :

For  < 0 one can easily recognise the classical Liouville theorem for entire functions. Indeed, the inclusion u 2 H s with  < 0 implies that juj  Ce; t as t ! 1, or juj  C jz j; . If u 2 ker @, then, by the Liouville theorem, it is a polynomial of degree not greater than ; ], and dim ker @ = ; ] + 1. The cokernel in this case is empty (as is always the case for dierential operators and  < 0).

References

35

References Ang93] N. Anghel, An abstract index theorem on non-compact Riemannian manifolds, Houston J. Math. 19 (1993), 223{237. APS75] M. F. Atiyah, V. K. Patodi, and I. M. Singer, Spectral asymmetry and Riemannian geometry. I, Math. Proc. Camb. Phil. Soc. 77 (1975), 43{69. BC89] M. Bismut and J. Cheeger, Eta-invariants and their adiabatic limits, J.A.M.S. 2 (1989), 33{70. BC90] M. Bismut and J. Cheeger, Families index for manifolds with boundary, superconnections and cones, Invent. Math. 89 (1990), 91{151. BG92] Th. P. Branson and P. B. Gilkey, Residues of the eta function for an operator of Dirac type, J. Funct. Anal. 108 (1992), no. 1, 47{87. Bru90] J. Bruning, L2-index theorems on certain complete manifolds, J. Dierential Geom. 32 (1990), no. 2, 491{532. BS88] J. Bruning and R. Seeley, An index theorem for rst order regular singular operators, Amer. J. Math. 110 (1988), no. 4, 659{714. Che83] J. Cheeger, Spectral geometry of singular Riemannian spaces, J. Dierential Geom. 18 (1983), 175{221. Che87] J. Cheeger, Eta invariants, the adiabatic approximation and conical singularities, J. Dierential Geom. 26 (1987), 175{221. Cho85] A. W. Chou, The Dirac operator on spaces with conical singularities and positive scalar curvatures, Trans. Amer. Math. Soc. 289 (1985), no. 1, 1{40. DF83] H. Donnelly and C. Feerman, L2-cohomology and index theorem for the Bergman metric, Ann. of Math. 118 (1983), no. 3, 593{ 618. DW91] R. G. Douglas and K. P. Wojciechowski, Adiabatic limits of the eta-invariants. The odd-dimensional Atiyah-Patodi-Singer problem, Comm. Math. Phys. 142 (1991), 139{168. Fed74] B. V. Fedosov, Analytic formulas for the index of elliptic operators, Trans Moscow Math. Soc. 30 (1974), 159{241.

36 FS96]

FST96] Get93] GL83] Gru92] GS95] HMM96] KW93] Les97] Lot92] Mel81] Mel93] Mel95] MM83]

The Index of Elliptic Operators B. V. Fedosov and B.-W. Schulze, On the index of elliptic operators on a cone, Schrodinger Operators, Markov Semigroups, Wavelet Analysis, Operator Algebras, Advances in Partial Dierential Equations, vol. 3, Akademie-Verlag, Berlin, 1996, pp. 347{ 372. B. V. Fedosov, B.-W. Schulze, and N. N. Tarkhanov, On the Index of Elliptic Operators on a Wedge, Preprint MPI/96-143, Max-Planck-Inst. fur Math., Bonn, 1996, 46 pp. E. Getzler, Cyclic homology and the Atiyah-Patodi-Singer index theorem, Contemp. Math. 148 (1993), 19{45. M. Gromov and H. B. Lawson, Positive scalar curvature and the Dirac operator on complete Riemannian manifolds, Publ. Math. I.H.E.S. 58 (1983), 83{196. G. Grubb, Heat operator trace expansions and index for general Atiyah-Patodi-Singer boundary problems, Comm. Part. Di. Equ. 17 (1992), 2031{2077. G. Grubb and R. Seeley, Weakly parametric pseudodierential operators and Atiyah-Patodi-Singer boundary problems, Invent. Math. 121 (1995), 481{529. A. Hassel, R. Mazzeo, and R. B. Melrose, A signature formula for manifolds with corners of codimension two, Preprint, MIT, Boston, September 1996 (to appear in Topology). S. Klimek and K. Wojciechowski, -invariants on manifolds with cylindrical ends, Dierential Geom. Appl. 3 (1993), 191{201. M. Lesch, Operators of Fuchs Type, Conical Singularities, and Asymptotic Methods, Teubner-Texte zur Mathematik, vol. 136, B.G. Teubner Verlagsgesellschaft, Stuttgart-Leipzig, 1997. J. Lott, Higher eta-invariants, K-Theory 6 (1992), 191{233. R. B. Melrose, Transformation of boundary problems, Acta Math. 147 (1981), no. 3{4, 149{236. R. B. Melrose, The Atiyah-Patodi-Singer Index Theorem, A K Peters, Wellesley, Mass, 1993. R. B. Melrose, The eta invariant and families of pseudodierential operators, Math. Res. Letters 2 (1995), no. 5, 541{561. R. B. Melrose and G. A. Mendoza, Elliptic Operators of Totally Characteristic Type, Preprint MSRI 047-83, MIT, Berkeley, 1983.

References

37

MN96a] R. B. Melrose and V. Nistor, Homology of Pseudodierential Operators. I: Manifolds with Boundary, Penn State Preprint, May 1996.

R

MN96b] R. B. Melrose and V. Nistor, The k-Equivariant Index Theorem and C -Algebras of b-Pseudodierential Operators, Penn State Preprint, November 1996 (to appear in GAFA). MP93]

R. B. Melrose and P. Piazza, Families of Dirac Operators, Boundaries and the b-Calculus, Preprint, MIT, Boston, 1993.

Mul87] W. Muller, Manifolds with cusps of rank one, spectral theory and L2-index theorem, Lect. Notes Math., vol. 1244, Springer-Verlag, Berlin, 1987. Mul94] W. Muller, Eta invariants and manifolds with boundary, J. Di. Geom. 40 (1994), 311{377. Pia93]

P. Piazza, On the index of elliptic operators on manifolds with boundary, J. Funct. Anal. 117 (1993), 308{359.

PR90]

B. A. Plamenevskii and G. V. Rozenblum, On the index of pseudodierential operators with isolated singularities in symbols, Algebra i Analiz 2 (1990), no. 5, 165{188 (Russian).

PR92]

B. A. Plamenevskii and G. V. Rozenblum, Pseudodierential operators with discontinuous symbols: K -theory and index formulas, Funkts. Analiz 26 (1992), 266{275 (Russian).

Roz94] G. Rozenblum, Index Formulae for Pseudodierential Operators with Discontinuous Symbols, Preprint 32, Kobenhavns Univ., Copenhagen, December 1994. Roz96] G. Rozenblum, The Index of Cone Mellin Operators, Preprint, Goteborg Univ., Goteborg, April 1996. Sch94]

B.-W. Schulze, Pseudo-Dierential Boundary Value Problems, Conical Singularities, and Asymptotics, Akademie-Verlag, Berlin, 1994.

SSS97] B.-W. Schulze, B. Sternin, and V. Shatalov, On the Index of Dierential Operators on Manifolds with Conical Singularities, Preprint 97/10, Univ. Potsdam, Potsdam, May 1997. ST97]

B.-W. Schulze and N. N. Tarkhanov, The index of elliptic operators on manifolds with cusps, Dierential Equations, Asymptotic Analysis, and Mathematical Physics, Math. Research, vol. 100, Akademie-Verlag, Berlin, 1997, pp. 344{359.

38 Ste89] Ste90] Wu93]

The Index of Elliptic Operators M. Stern, L2-index theorems on locally symmetric spaces, Invent. Math. 96 (1989), 231{282. M. Stern, Eta invariants and Hermitian locally symmetric spaces, J. Dierential Geom. 31 (1990), 771{789. F. Wu, The Chern character and the eta-invariant of the Dirac operator, K-Theory 7 (1993), 145{174.