Dixmier s trace for boundary value problems

manuscripta math. 96, 23 28 (998) c Springer-Verlag 998 Ryszard Nest Elmar Schrohe Dixmier s trace for boundary value problems Received: 3 January 998 Abstract. Let X be a smooth manifold with boundary of dimension n >. The operators of order n and type zero in Boutet de Monvel s calculus form a subset of Dixmier s trace ideal L, (H ) for the Hilbert space H = L 2 (X, E) L 2 ( X, F ) of L 2 sections in vector bundles E over X, F over X. We show that, on these operators, Dixmier s trace can be computed in terms of the same expressions that determine the noncommutative residue. In particular it is independent of the averaging procedure. However, the noncommutative residue and Dixmier s trace are not multiples of each other unless the boundary is empty. As a corollary we show how to compute Dixmier s trace for parametrices or inverses of classical elliptic boundary value problems of the form Pu = f; Tu= with an elliptic differential operator P of order n in the interior and a trace operator T. In this particular situation, Dixmier s trace and the noncommutative residue do coincide up to a factor. Introduction In 988, A. Connes established the analog of the Yang-Mills action within the framework of noncommutative geometry [2]. The remarkable theorem motivating his construction is the fact that, for a pseudodifferential operator of order n, acting on a vector bundle over an n-dimensional manifold, Dixmier s trace coincides with Wodzicki s noncommutative residue. In particular, while Dixmier s trace in general depends on the choice of an averaging procedure, this is not the case on pseudodifferential operators of order n, where it can be computed by integration of the principal symbol. Fedosov, Golse, Leichtnam, and Schrohe [6] showed that there is a trace on Boutet de Monvel s algebra on a manifold with boundary which is unique up to multiples and extends Wodzicki s noncommutative residue. It is easy to see that there is a subalgebra of Boutet de Monvel s algebra for which Dixmier s trace makes sense. It is a natural question whether it coincides with the noncommutative residue up to a factor. We show here R. Nest: Department of Mathematics, University of Copenhagen, DK-2 Copenhagen, Denmark. e-mail: rnest@math.ku.dk E. Schrohe: Institut für Mathematik, Universität Potsdam, D-445 Potsdam, Germany. e-mail: schrohe@mpg-ana.uni-potsdam.de Mathematics Subject Classification (99): 35S5, 58G2, 47B

24 R. Nest, E. Schrohe that this is not the case. Still, Dixmier s trace can be determined in a similar fashion; it is given by a local expression, and it is independent of the averaging procedure. We start with a review of Dixmier s trace and the noncommutative residue. There follows a short description of Boutet de Monvel s calculus. In Section 2 we prove that on B n, (X), i.e., on the operators of order n and type zero, Dixmier s trace makes sense and derive a formula for it (Theorem 2.6). We next show how these results can be applied to find Dixmier s trace for the inverse operator (or the parametrix) to a classical boundary value problem, using a technique of Grubb and Geymonat [8]. Indeed, suppose we are given a boundary problem on X of the form Pu = f in X, T u = on X with a differential operator P of order n and a trace operator T of order n. It induces an unbounded operator on L 2 (X, E). Whenever this operator has an inverse or a parametrix in a sense specified below, we show that Dixmier s trace for the parametrix is given by (2π) n n X S tr E p n (x, ξ) σ ξ dx, where σ ξ is the surface measure on the sphere S ={ξ=}over x X and p n is the principal symbol of P. In particular, the expression is independent of the boundary condition and the averaging procedure involved in the definition of Dixmier s trace. It coincides with (2π) n /n times the noncommutative residue in the form of Sect..4, below.. Review: Wodzicki s noncommutative residue and Dixmier s trace.. The ideal L, (H ) Let H be a Hilbert space and K(H ) the ideal of compact operators. By L, (H ) we denote the space of all K K(H ) such that the ordered sequence {µ,µ 2,...}of the singular values satisfies sup N log N N µ k <. k= A construction of Dixmier allows us to endow the space L, (H ) with a family of traces {Tr ω } ω. The subscript ω indicates the choice of an averaging procedure: For positive K, the sequence a N = (log N) N µ k is bounded, and ω is used to associate to it a limit, Tr ω (K), which coincides with the

Dixmier s trace for boundary value problems 25 usual limit lim a N whenever {a N } converges. The construction turns out to be additive on positive K s and therefore extends by linearity to the full ideal. The Tr ω are all traces, and they all vanish on the ideal L (H ) of trace class operators. For details see Dixmier [4] or Connes [3]..2. Wodzicki s noncommutative residue and Connes theorem In the rest of this paper let M be a closed compact manifold of dimension n. We fix an arbitrary Riemannian metric on M, so that we have the notion of L 2 (M). By (M) denote the algebra of all classical pseudodifferential operators on M. We first assume dim M>. Given P (M) we let res P = p n (x, ξ)σ ξ dx. () M S Here p n is the component of order n of the complete symbol of P over a local coordinate chart, and σ ξ is the (n )-form n σ ξ = ( ) j+ ξ j dξ... dξ j... dξ n. The hat indicates that dξ j is removed. It is well-known that σ ξ S is the surface measure on the sphere S ={ξ=}. The fact that () is independent of the local representation and furnishes a trace on (M) is non-trivial. For details see Wodzicki [2,3] or [6]. In general, P might be acting on sections of a finite-dimensional vector bundle E over X. We then replace p n (x, ξ) by tr E p n (x, ξ) with a trace tr E on Hom(E). There is a more intrinsic way to write this: Let ρ be the radial vector field and ω the symplectic form on the cosphere bundle S X. Then res P = ( ) n(n )/2 (p ρ ω n /n!). (2) Here ( ) denotes the term of degree zero in the asymptotic expansion of homogeneous forms, and is the contraction of vector fields with forms. If M is one-dimensional, then a simpler consideration shows that there exists only one trace on the classical pseudodifferential operators (up to multiples), given by res P = tr E {p (x, ) + p (x, )}dx. M We would get two traces, if we restricted ourselves to orientation preserving diffeomorphisms. S X

26 R. Nest, E. Schrohe Theorem. (Connes 988). Let P (M) have order n. Then P : L 2 (M, E) L 2 (M, E) defines an element of L, (L 2 (M, E)), and independent of ω. For a proof see [2]. Tr ω (P ) = res P, (2π) n n.3. Boutet de Monvel s calculus For detailed introductions see Boutet de Monvel [], Rempel-Schulze [9], or Grubb [7]. A short version can be found in []. In the following, X is an n-dimensional manifold with boundary X embedded in M. We assume that dim X>. Definition.2. (a) We write (R ± ) for the rapidly decreasing functions on R ±, i.e., (R) R±. (b) By e ± we denote the operator of extension (by zero) of functions on R ± to functions on R, byr + the restriction operator on functions or distributions from R to R +. We use the same notation for extension by zero from X to M and restriction from M to X. (c) We let H + ={(e + u) : u (R + )}; H ={(e u) : u (R )}. The hat denotes the Fourier transform. (d) H s (X) is the Sobolev space of order s on X, i.e. the restrictions of distributions in H s (M) to X, while H s ( X) is the Sobolev space at the boundary. We consider matrices of operators acting on sections of vector bundles E,E 2 over X and F,F 2 over X. An operator of order m Z and type zero is a matrix ( ) P+ + GK A = : T S C (X, E ) C (X, E 2 ). C ( X, F ) C ( X, F 2 ) In view of the fact that we shall deal with endomorphisms we assume E = E 2 = E, F = F 2 = F. P is a classical pseudodifferential operator of order m on M. The subscript + indicates that we consider P + = r + Pe +. The operator P is supposed to have the transmission property; this means that the homogeneous component p j of order j in the asymptotic expansion

Dixmier s trace for boundary value problems 27 p k= p m k of the complete symbol p = p(x,x n,ξ,ξ n )of P in local coordinates near the boundary satisfies k x n α ξ p j(x,,, +) = ( ) j α k x n α ξ p j(x,,, ). The operator G is a classical singular Green operator of order m and type zero, T is a classical trace operator of order m + and type zero, K is a classical potential operator of order m, and S is a classical pseudodifferential operator of order m +. We shall write B m, (X) for the collection of all operators of order m and type zero. In order to avoid superfluous notation, we shall no longer write classical. All operators, however, are assumed to be classical. There is no need to go into the details of the definition of G, T, and K, for it is sufficient to know the following facts: (a) The sum in the upper left corner is direct up to regularizing pseudodifferential operators. B m, (X) isafréchet space, and the composition of operators (with the present choice of orders of the entries) yields a continuous map B m, (X) B m, (X) B m+m +, (X), provided m. (b) For all s> /2, the elements of B m, (X) extend to bounded operators H s (X) H s ( X) H s m (X) H s m ( X). The topology on B m, (X) is stronger than the operator topology. (c) If K is a potential operator of order m and T is a trace operator of order m and type zero, then KT is a Green operator of order m + m and type zero. The composition TK yields a pseudodifferential operator on the boundary of order m+m. Moreover, the linear span of the compositions KT forms a dense set in the space of Green operators of order m + m. (d) Let A B m, (X) and A B m, (X) have entries P + + G and P + + G in the upper left corner. Then the composition (P + + G)(P + + G ) is of the form (P P ) + + L(P, P ) + P + G + GP + + GG. Here PP is the usual composition of pseudodifferential operators, L(P, P ) = P + P + (P P ) +, the leftover term, is a singular Green operator of order m + m and type zero (assuming m ), the same is true for the remaining three terms. For these statements to hold it is essential that P and P have the transmission property. In particular: If either P or P is zero, then also the pseudodifferential part of AA is zero.

28 R. Nest, E. Schrohe (e) A singular Green operator coincides with a regularizing pseudodifferential operator in the interior of X; i.e. ϕgϕ is regularizing for ϕ C (int X). For a smooth function ϕ supported in a single coordinate neighborhood diffeomorphic to R n + we have ϕgϕ = op G g for a Green symbol g. Here g is a smooth function on R n R n R R with g(x,ξ, ξ ξ n, ξ η n ) S m cl (R n R n ) ˆ π H + ˆ π H. It has an asymptotic expansion g k= g m k into terms g j which are positively homogeneous of degree j in (ξ,ξ n,η n ). Note that the highest degree of homogeneity for a singular Green operator of order m is m. The action op G g of g is defined by (op G g)u(x,x n ) =(2π) n 2 e ixξ g(x,ξ,ξ n,η n )(e + u) (ξ,η n )dη n dξ. (f) The intersection over the spaces of singular Green operators of type zero as the order varies over Z is called the space of regularizing singular Green operators of type zero. They can be characterized as integral operators with smooth kernel sections on X X.IfP is a regularizing pseudodifferential operator on M, then P + also has a smooth kernel on X X; conversely, given an operator with a kernel of this kind, it coincides with P + for some regularizing pseudodifferential operator on M. Remark.3. In general, the operators in Boutet de Monvel s calculus may have a type d N. The composition AB of an operator A with all entries of order m and type d with an operator B with all entries of order m and type d yields an operator of order m + m and type max{m + d,d }. Operators of order m and type d, however, have the mapping property in Sect..3(b) only for s>d /2. Boundedness on L 2 therefore requires the type to be zero. This is the reason why we focus on these elements..4. The noncommutative residue for manifolds with boundary It was shown in [6] that there is up to multiples a unique continuous trace on Boutet de Monvel s algebra. In the notation of Sect..3 it is given by resa = tr E p n (x, ξ)σ ξ dx X S { } 2π tr E g n (x,ξ,ξ n,ξ n )dξ n + tr F s n+ (x,ξ ) σ ξ dx. X S This residue extends Wodzicki s: The formulas coincide for X =.

Dixmier s trace for boundary value problems 29 Remark.4. There is no trace on the algebra of all classical pseudodifferential on X endowed with the Leibniz product. This was already noticed by Wodzicki; a proof can be found in [6, Theorem 3.2]. In particular, the expression τ(p) = tr E p n (x, ξ)σ ξ dx X S does not define a trace if we admit pseudodifferential operators of all orders. In Boutet de Monvel s calculus, the quantization p P + gives rise to the leftover terms as explained in Sect..3(d), and it is via their contribution that the formula in Sect..4 works. The situation changes if we focus on operators of order n: On pseudodifferential operators of order n, the functional τ indeed defines a trace in the sense that τ([p,q]) = whenever P is pseudodifferential of order n and Q is of order zero. On B n, (X) the contribution of the term G vanishes: Since G is of order n, the highest degree of homogeneity in the asymptotic expansion is n. We therefore have at least two linear independent traces on B n, (X), namely the trace τ and the noncommutative residue on the lower left corner. In fact there are many others. 2. Dixmier s trace for boundary value problems Proposition 2.. A bounded operator on L 2 (X) with range in H n (X) is an element of L, (L 2 (X)); if its range even is contained in H n+ (X) then it is trace class. Proof. Let A be bounded on L 2 (X) with range in H n (X). It is well-known that there is an invertible pseudodifferential operator R of order n on M such that R + : L 2 (X) H n (X) is an isomorphism with inverse equal to (R ) +, for a proof see e.g. [7, Theorem 3.2.4]. We then may write A = R + (R ) + A. The composition (R ) + A yields a bounded operator on L 2 (X), since (R ) + : H n (X) L 2 (X) is bounded. On the other hand the singular values of R + on L 2 (X) can be estimated in terms of the singular values of R on L 2 (M) and the norms of the operators e + : L 2 (X) L 2 (M) and r + : L 2 (M) L 2 (X). Since R is of order n, it is an element of L, (L 2 (M)) according to Theorem.. So we get the first assertion. The second statement is proven similarly, noting that operators of order n are trace class. We write write H = L 2 (X, E) L 2 ( X, F ). The following lemma is now obvious from Proposition 2. and Sect..3(b):

2 R. Nest, E. Schrohe Lemma 2.2. B n, (X) L, (H ), and B n, (X) L (H ). In particular, Dixmier s trace applies to elements in B n, (X); it vanishes on B n, (X). Obviously, no multiple of the noncommutative residue in Sect..4 can coincide with Dixmier s trace (unless X = ): The contributions by P and S have opposite signs, while Dixmier s trace is a positive functional. We shall now determine Dixmier s trace on B n, (X). We start by considering the subspace where the pseudodifferential part vanishes. Moreover, we shall use the following simple observation. Lemma 2.3. Let L K(L 2 (X, E)). Then, apart from zero, the spectrum of L) in L(L 2 (X, E)) including multiplicities coincides with the spectrum of in L(H ) and the spectrum of X L X in L(L 2 (M, E)). ( L Here X denotes the characteristic function of X in M, and we identify L 2 -functions on X with their extensions by zero. ) Lemma 2.4. Let B n, (X) have zero pseudodifferential part. Then ( G T K S ( ) GK Tr ω = T S res S (2π) n (n ) with Wodzicki s residue on X. There is no contribution from G, T,orK. Proof. Tr ω is a trace on L, (H ), therefore Tr ω ([B,C]) = whenever B B n, (X) and C L(H ), cf. [3, IV.2, Proposition 3]. In particular, ([( ) ( )]) (( )) = Tr ω, = Tr I T ω ; T ([( ) ( )]) (( )) K K = Tr ω, = Tr I ω. Dixmier s trace therefore vanishes on the off-diagonal entries. Next we deduce from (an analog of) Lemma 2.2 and Theorem. that (( )) res S Tr ω = Tr S ω (S) = (2π) n (n ). Finally let K be a potential operator of order zero and T a trace operator of order n and type zero. Then K : L 2 ( X, F ) L 2 (X, E), K : H n ( X, F ) H n (X, E), and T : L 2 (X, E) H n ( X, F )

Dixmier s trace for boundary value problems 2 are bounded according to Sect..3(b). Moreover, ) B n, (X). Thus ( T ( ) K L(H ), and ([( ) ( )]) (( )) K K = Tr ω, T = Tr T ω T K. By Sect..3(c), K T is a singular Green operator of order n and type zero, and T K is a pseudodifferential operator of order n on the boundary. T K therefore is of trace class (dim X = n!), and we conclude that Tr ω ( K T ) =. The span of the operators K T of this form is dense in the space of singular Green operators. Hence Dixmier s trace vanishes on the upper left corner, since it is continuous on L, (H ), and the topology on the singular Green operators is stronger than the operator topology. By linearity the proof is complete. Corollary 2.5. In order to determine Dixmier s trace on B n, (X) we are therefore reduced to studying the contribution from the pseudodifferential part P + in the upper left corner. By Lemma 2.3 we may first reduce to the case where H = L 2 (X, E), so that we have to find Tr ω (P + ) in L(L 2 (X, E)). Again by 2.3 this amounts to computing Tr ω ( X P X ) in L(L 2 (M, E)). In view of the fact that Tr ω is a trace, Tr ω ( X P X ) = Tr ω ( X P). Note that the fact that P satisfies the transmission property is no longer of importance. Proposition 2.6. Let P be a pseudodifferential operator of order n, acting on L 2 (M, E). Then Tr ω ( X P) = tr (2π) n E p n (x,ξ)σ ξ dx. n X S Here we employ the notation of Sect..2 Proof. First assume additionally that P is positive. Choose sequences of smooth functions {ϕ k }, {ψ k } with ϕ k X ψ k and ϕ k,ψ k X pointwise. Since ϕ k,ψ k are smooth, ϕ k P and ψ k P are the pseudodifferential operators with symbols ϕ k p and ψ k p, respectively. For them, Dixmier s trace is given by Theorem.. In view of the fact that Tr ω is a positive functional, we have Tr ω ( X P ϕ k P) = Tr ω (( X ϕ k )P ) = Tr ω ( X ϕ k P X ϕ k ).

22 R. Nest, E. Schrohe Applying a similar identity for ψ k, we deduce that tr (2π) n E ϕ k (x)p n (x, ξ)σ ξ dx n M S = Tr ω (ϕ k P) Tr ω ( X P) Tr ω (ψ k P) = tr (2π) n E ψ k (x)p n (x,ξ)σ ξ dx. n M S The difference ϕ k ψ k tends to zero pointwise; it is uniformly. Lebesgue s theorem on dominated convergence therefore gives the above formula. Now for the general case: Since P may be written P = 2 (P + P ) i 2 (ip ip ), and since P ± P is pseudodifferential of order n, we may assume that P is selfadjoint. Let be the Laplace-Beltrami operator associated with the Riemannian metric. Define T = (I ) n/4. This is a positive selfadjoint pseudodifferential operator of order n/2. It is invertible, and T is of order n/2. The operator T PT hence is of order zero and therefore bounded; it also is selfadjoint. For large t R, we conclude that T PT + ti is positive. Hence also P + tt 2 = T(T PT +ti)t is positive. We can therefore write P as the difference of the positive operators P +tt 2 and tt 2. In view of the additivity of the integral the proof is complete. Altogether we now have established the following result. Theorem 2.7. For an operator ( ) P+ + GK A = B n, (X) T S acting on H = L 2 (X, E) L 2 ( X, F ), n = dim X>2, we have Tr ω (A) = tr (2π) n E p n (x,ξ)σ ξ dx n X S + tr (2π) n F s n+ (x,ξ )σ ξ dx. (n ) X S Here, S is the sphere {ξ = } in T M over a point x X, σ ξ is the form introduced in Sect..2, while S is the corresponding sphere {ξ = } in T X over a point x in X. For dim X = 2 the second summand becomes tr F (s (x, ) + s (x, )) dx. 2π X

Dixmier s trace for boundary value problems 23 3. Classical boundary value problems In the more traditional literature one adopts slightly different way of looking at boundary value problems. We shall now show how our results apply to this situation. 3.. The framework Let P : C (X, E) C (X, E) be a differential operator of order m>, acting on sections of a vector bundle E over X. Note that for differential P we may omit the subscript +, since the way of extending distributions is irrelevant. Furthermore let T : C (X, E) C (Y, F ) be a trace operator. It is customary to assume that F = F... F m, where the dimension of each F j might be zero (in a standard application, m would be even and half of the F j would be zero). Correspondingly one writes T = (T,...,T m ), asking that T k be a differential boundary operator of order m k <minvolving at most k normal derivatives. In local coordinates near the boundary, T k can be written in the form T k = jα (x )Dx α γ j. j<k, j+ α m k a [k] Here, γ j = γ j x n is the operator of evaluation of the j-fold normal derivative at the boundary. Of course, T k can be neglected when the dimension of F k is zero. One usually wants to solve either the inhomogeneous problem Pu = f on X; Tu=g on X (3) for given f and g, or else the homogeneous problem Pu = f on X; Tu= on X (4) for given f. For (3) one considers the operator ( ) L 2 (X, E) P : H m (X, E) T m k= H m mk /2 ( X, F k ). (5)

24 R. Nest, E. Schrohe In order to treat problem (4), one studies the realization P T which is defined as the unbounded operator P T on L 2 (X, E) with domain D(P T ) ={u H m (X, E) : Tu=} (6) and acting like P. In both cases one is interested in the question whether the corresponding operator has the Fredholm property. The link between the two approaches is given by the following lemma. Lemma 3.. Let H, H and H 2 be Hilbert spaces over C, and let P : H H as well as T : H H 2 be linear operators. Then the following are equivalent: (i) ( ) P : H T H H 2 is a Fredholm operator. (ii) P T = P ker T :kert H is a Fredholm operator, and im T is finite codimensional in H 2. We omit the proof, since the point of main interest, namely the construction of a Fredholm inverse for P T starting from a Fredholm inverse for ( P) T will be clarified in the proof of Theorem 3.2. Note that in standard cases one will often have T surjective so that the second condition in (ii) is easily fulfilled. 3.2. Ellipticity and parametrices to the inhomogeneous problem It is well-known that the operator in (5) is a Fredholm operator if and only if the following two conditions are fulfilled: (i) For all (x, ξ) S M X the principal symbol p m of P is invertible as an endomorphism of E, and (ii) for all (x,ξ ) S X the principal boundary symbol is invertible. ( pm (x,,ξ ),D n ) : (R t m (x,ξ +,E),D n ) (R +,E) F (x,ξ ) Here t m is the operator-valued principal symbol of the boundary operator T. In local coordinates near the boundary, it is the vector with entries j<k, j+ α =m k a [k] jα (x )ξ α γ j. We may replace each T k by T k = m m k /2 T k, where = ( X ) /2 is order-reducing along X. In view of the fact that the powers of are

Dixmier s trace for boundary value problems 25 invertible, this affects neither (6) nor the ellipticity of ( P) T, but it allows us to use the space L 2 ( X, F ) instead of m k= H m mk /2 ( X, F k ) on the right hand side of (5). In addition, the resulting boundary operator T is a trace operator of order m in the sense of Boutet de Monvel (there is a difference of /2 between the order of T in Boutet de Monvel s calculus and its order as a boundary operator). The ellipticity implies that there is a parametrix to A = ( ) P T in Boutet de Monvel s calculus. It is of the form B = (Q + + G K); the pseudodifferential part Q is a parametrix to P, while G is a singular Green operator of order m and type zero and K is a potential operator of order m. Being a parametrix here means that AB I = S and BA I = S 2 are regularizing operators; their types are and m, respectively. As a consequence the operator S is an integral operator with a smooth kernel section. Theorem 3.2. Let ( P T ) be the above elliptic boundary value problem and B = (Q + + G K) its parametrix. Then there is a regularizing singular Green operator G of type zero, i.e. an integral operator with smooth kernel on X X, such that R = Q + + G + G has the following properties: (i) R maps L 2 (X, E) to D(P T ) and (ii) RP T I and P T R I are finite rank operators whose range consists of smooth functions. Proof. This follows from a method developed by Grubb and Geymonat [8], cf. also [7, Section.4]. We supply a proof for the convenience of the reader, using the notation introduced above. Note that the only point here is to modify the operator Q + + G, i.e., the first entry in the parametrix B, so that it maps into D(P T ) = ker T = ker T. Elliptic regularity implies that N = ker A consists of smooth functions in H m (X, E) and that there is a finite number of functions in C (X, E) C ( X, F ) spanning the orthogonal complement C to im A in L 2 (X, E) L 2 ( X, F ). We denote by C the operator acting like the inverse of A from C to N while mapping C to zero. Then CA = I pr N and AC = I pr C, where here and in the following pr denotes the orthogonal projection onto the space in the subscript. Both pr N and pr C may be written as integral operators with smooth kernel; they are therefore regularizing elements of type zero. We have C B = C(AB S ) B = (I pr N )B CS B = pr N B C(I pr C )S. Let us show that the right hand side is a regularizing operator of type zero in Boutet de Monvel s calculus: Obviously, pr N B is regularizing of type zero, so

26 R. Nest, E. Schrohe we only have to consider the second operator on the right hand side. Having a smooth integral kernel, S extends to a continuous operator S : H σ (X, E) H σ ( X, F ) C (X, E) C ( X, F ) for arbitrary σ>. Here, H σ (X, E) denotes the closure of C (int X, E) in H σ (M, E). As a consequence, (I pr C )S has the same property. Moreover, this operator maps into the range of A. Since C acts like an inverse to A on the range, we deduce from elliptic regularity that also C(I pr C )S has the above mapping property. Hence it is an integral operator with a smooth kernel section. Therefore C = (Q + + G + G K + K ) with a suitable regularizing singular Green operator G of type zero and a suitable regularizing potential operator K, while Q, G, and K are as above. Write pr C asa2 2matrix of operators with respect to the direct sum L 2 (X, E) L 2 ( X, F ) and denote by S ij its entries in row i and column j. The identity AC = I pr C implies that P(Q + +G+G ) = I S, and (7) T(Q + +G+G )= S 2. (8) We let R = (Q + + G + G )pr kers2. According to (8) we have TR =. We have pr C = pr C, hence S 2 = S2, and both are finite rank operators with values in smooth functions. Therefore, pr ker S2 = I pr ims = I pr 2 ims2 differs from the identity only by a regularizing singular Green operator of type zero, and R is the desired operator. Corollary 3.3. Suppose ( P) T is elliptic, and R is another parametrix for P T, i.e. an operator of order m and type zero in Boutet de Monvel s calculus satisfying P R = I + S, RP = I + S 2, and T R = for regularizing singular Green operators S and S 2 of type and m, respectively. Then R coincides with the operator R in Theorem 3.2 up to a regularizing singular Green operator of type, since R = RPR RPG RS = (I + S 2 )R RPG RS = R RPG + S 2 R RS. Here, S is the upper left entry in the operator matrix S above, and G is as in Theorem 3.2. In particular, any such parametrix for P T differs from the first entry Q + + G in an arbitrary parametrix (Q + + GK)to ( P T ) only by a regularizing singular Green operator of type zero. Since these are trace class, we get the following result.

Dixmier s trace for boundary value problems 27 Corollary 3.4. In case m = n and ( P) T is elliptic, Theorem 2.7 applies and shows that we may compute Dixmier s trace for an arbitrary parametrix R to the operator P T in the sense of Corollary 3.3: We have Tr ω R = tr (2π) n E p n (x, ξ) σ ξ dx. (9) n X S The expression is the same for all parametrices and independent of the choice of the boundary condition. Moreover, it coincides with the noncommutative residue for R. For certain classes of parameter-elliptic boundary value problems identity (9) also follows from the eigenvalue asymptotics established first by H. Weyl [] for the Dirichlet problem in the plane and extended subsequently by numerous authors. Remark 3.5. Note, however, that Theorem 2.7 covers much more general situations, in particular, non-elliptic operators. Also the construction of Theorem 3.2 relies only on the existence of a parametrix to ( ) P T in Boutet de Monvel s calculus. We thus get the result of Corollary 3.3 for all elliptic boundary value problems ( P +G) T of order n in Boutet de Monvel s calculus (i.e., without requiring parameter ellipticity and with additional singular Green terms as well as more general trace operators); formula (9) continues to hold. Acknowledgements. The second author would like to thank T. Krainer and B.-W. Schulze for valuable discussions on the subject and G. Grubb for her comments on a preliminary version of the paper. References [] L. Boutet de Monvel: Boundary problems for pseudo-differential operators. Acta Math. 26, 5 (97) [2] A. Connes: The action functional in non-commutative geometry. Comm. Math. Phys. 7, 673 683 (988) [3] A. Connes: Noncommutative Geometry. New York, London, Tokyo: Academic Press, 994 [4] J. Dixmier: Existence de traces non normales. C. R. Acad. Sci. Paris, Sér. A 262, 7 8 (966) [5] B.V. Fedosov, F. Golse, E. Leichtnam, and E. Schrohe: Le résidu non commutatif pour les variétés à bord. C. R. Acad. Sci. Paris, Sér. I Math., 32, 669 674 (995) [6] B.V. Fedosov, F. Golse, E. Leichtnam, and E. Schrohe: The noncommutative residue for manifolds with boundary. J. Funct. Anal., 42, 3 (996) [7] G. Grubb: Functional Calculus for Boundary Value Problems. Second edition. Number 65 in Progress in Mathematics. Boston, Basel: Birkhäuser, 996 [8] G. Grubb and G. Geymonat: The essential spectrum of elliptic symstems of mixed order. Math. Ann. 227, 246 276 (977) [9] S. Rempel and B.-W. Schulze: Index Theory of Elliptic Boundary Problems. Berlin: Akademie-Verlag, 982

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