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The index of generalised Dirac-Schr\"odinger operators

We study the relation between spectral flow and index theory within the framework of (unbounded) KK-theory. In particular, we consider a generalised notion of 'Dirac-Schr\"odinger operators', consisting of a self-adjoint elliptic first-order differential operator D with a skew-adjoint 'potential' given by a (suitable) family of unbounded operators on an auxiliary Hilbert module. We show that such Dirac-Schr\"odinger operators are Fredholm, and we prove a relative index theorem for these operators (which allows cutting and pasting of the underlying manifolds). Furthermore, we show that the index of a Dirac-Schr\"odinger operator represents the pairing (Kasparov product) of the K-theory class of the potential with the K-homology class of D. We prove this result without assuming that the potential is differentiable; instead, we assume that the 'variation' of the potential is sufficiently small near infinity. In the special case of the real line, we recover the well-known equality of the index with the spectral flow of the potential.


Introduction
A 'classical' Dirac-Schrödinger operator (also called a Callias-type operator) is an operator of the form D − iV , where D is a Dirac-type operator, and the 'potential' V is a self-adjoint endomorphism on some auxiliary vector bundle (of finite rank). Under suitable assumptions on the potential V , one can then prove that D − iV is Fredholm (see, for instance, [Cal78,Ang90,BM92,Ang93,Råd94,Bun95]).
In this paper we aim to prove a similar statement in the case where the auxiliary vector bundle is of infinite rank, and the 'potential' consists of a family of unbounded operators. Since any locally trivial Hilbert bundle with (separable) infinite-dimensional fibres is in fact globally trivial (see [Dix82,Theorem 10.8.8]), we will restrict our attention to the case of a globally trivial Hilbert bundle with infinite-dimensional fibre H. (In fact, instead of H we will more generally consider a Hilbert module over an arbitrary σ-unital C * -algebra, but in this introduction we limit our attention to the simpler case of a Hilbert space.) One motivation for studying the case of a potential acting on infinite-dimensional fibres comes from the notion of spectral flow. Namely, there is a well-known 'index = spectral flow' equality, which states that the spectral flow of a (suitably continuous) family of unbounded self-adjoint operators {S(x)} x∈[0,1] (with invertible endpoints) is equal to the index of ∂ x +S(·) (see e.g. [RS95,Wah07,AW11]). Here, we note that the operator −i∂ x − iS(·) is a special case of a Dirac-Schrödinger operator, where the underlying manifold is just the real line. The operator S(·) is a regular self-adjoint Fredholm operator on C 0 (R, H), which defines a class [S(·)] in the odd K-theory group KK 1 (C, C 0 (R)) ≃ K 1 (C 0 (R)). Under the Bott periodicity isomorphism β : K 1 (C 0 (R)) ≃ − → K 0 (C) = Z, the class [S(·)] corresponds precisely to the spectral flow of {S(x)} x∈[0,1] . By observing that the Bott periodicity isomorphism is implemented as the Kasparov product with the generator [−i∂ x ] of the K-homology group KK 1 (C 0 (R), C), we see that we have the equality Our main goal in this paper is to generalise this equality, replacing R by an arbitrary manifold M , and −i∂ x by some first-order differential operator on M . Such a generalisation has already been obtained by Kaad and Lesch [KL13,§8], under the assumption that the family of operators {S(x)} x∈M is differentiable (in a suitable sense). However, on the real line, such a differentiability assumption is not necessary to obtain the above-mentioned 'index = spectral flow' equality [AW11, Theorem 2.1]. Our aim is therefore to make a link between these two approaches, by studying Dirac-Schrödinger operators on higher-dimensional manifolds (as in [KL13]), without assuming that the family of operators is differentiable (as in [Wah07,AW11]).
More precisely, we will consider the following setup. Let M be a connected Riemannian manifold, and let D be a self-adjoint elliptic first-order differential operator on M . Let {S(x)} x∈M be a family of selfadjoint operators with compact resolvents and with common domain W on a Hilbert space H such that S : M → B(W, H) is norm-continuous, and S(x) is uniformly invertible outside a compact subset K ⊂ M . We then consider the 'generalised Dirac-Schrödinger operator' D S := D − iS(·).
The family {S(x)} x∈M on H defines a regular self-adjoint Fredholm operator S(·) on the Hilbert C 0 (M )module C 0 (M, H), which defines a class [S(·)] in the odd K-theory group KK 1 (C, C 0 (M )) ≃ K 1 (C 0 (M )). Our main theorem is complementary to the results of [KL13,§8]; although we do not need to assume any differentiability of S(·), we do need stronger assumptions on the 'variation' of S(·) near infinity. Furthermore, our theorem generalises the aforementioned 'index = spectral flow' equality of [AW11] (which is obtained in the special case M = R).
The This product operator is precisely given by the standard formula for the construction of the unbounded Kasparov product [Mes14,KL13,BMS16,MR16]. Unfortunately, we cannot directly apply these results on the construction of the unbounded Kasparov product, since we do not assume any differentiability for the potential S(·). However, it is interesting to note that in our scenario, despite this lack of differentiability, the formula (1.1) for the unbounded Kasparov product nevertheless remains correct. Let us provide a brief outline of this paper. In Section 2, we start by recalling the main facts regarding (unbounded) KK-theory and the Kasparov product. Next, we discuss in detail how an unbounded Fredholm operator defines a class in K-theory, following the approach of [Wah07]. We also provide a purely KKtheoretic description of the notion of spectral flow.
In Section 3 we define our notion of generalised Dirac-Schrödinger operators, and we prove that they are self-adjoint. In Section 4, we show that these Dirac-Schrödinger operators are also Fredholm. Furthermore, we prove a relative index theorem, generalising Bunke's K-theoretic relative index theorem [Bun95]. As a useful consequence, we show that the index of a Dirac-Schrödinger operator is unaffected, if one replaces a suitable open subset of the manifold by a cylindrical end.
In Section 5 we prove our main theorem, stating that the index of the Dirac-Schrödinger operator D S is equal to the Kasparov product of [S(·)] with [D]. The proof consists, roughly speaking, of the following three steps. First, using the same methods as in [KL13], we can prove the theorem for a differentiable family.
Second, on a manifold with cylindrical ends, we can always find a homotopy which replaces our continuous family {S(x)} x∈M by a differentiable family. Third, using the cut-and-paste invariance of the index, we may replace any manifold by a manifold with cylindrical ends.

Index and spectral flow in unbounded KK-theory
Let B be a σ-unital Z 2 -graded C * -algebra. Recall that a Z 2 -graded Hilbert B-module E is a vector space equipped with a Z 2 -graded right action E × B → E and with a B-valued inner product ·|· : E × E → B, such that E is complete in the corresponding norm. The endomorphisms End B (E) are the adjointable linear operators E → E. For any ψ, η ∈ E, the rank-one operators θ ψ,η are defined by θ ψ,η ξ := ψ η|ξ for ξ ∈ E. The compact endomorphisms End 0 B (E) are given by the closure of the space of finite linear combinations of rank-one operators. For two Hilbert B-modules E 1 and E 2 , the adjointable linear operators E 1 → E 2 are denoted by Hom B (E 1 , E 2 ). To avoid confusion, we sometimes denote the norm on Hom B (E 1 , E 2 ) as · E1→E2 . For a detailed introduction to Hilbert modules and Z 2 -gradings, we refer to [Bla98,Lan95]. A densely defined operator S on E is called semi-regular if the adjoint S * is densely defined. A semiregular operator S is called regular if S is closed and 1 + S * S has dense range. A densely defined, closed, symmetric operator S is regular and self-adjoint if and only if the operators S±i are surjective [Lan95, Lemma 9.8]. We say that a semi-regular operator S is essentially regular self-adjoint if its closure S is regular selfadjoint. If B = C, then a Hilbert C-module is just a Hilbert space H, and we write B(H) = End C (H) and K(H) = End 0 C (H). In this case, any closed operator on H is regular. Regular operators on a Hilbert B-module have similar properties as closed operators on Hilbert spaces. In particular, there is a continuous functional calculus for regular self-adjoint operators [Wor91,Kus97,Kuc02].
Given a densely defined, symmetric operator S on E, we can equip Dom S with the graph inner product ψ|ψ S = (S ±i)ψ|(S ±i)ψ = ψ|ψ + Sψ|Sψ . The graph norm of S is then defined as ψ S := ψ|ψ S 1 2 . We prove a few basic lemmas which will be used in this paper.
Lemma 2.1. Let T be a regular self-adjoint operator on a Hilbert B-module E, and let S be an operator such that Dom S ⊃ Dom T . For some a ≥ 0, we have S(T ± i) −1 ≤ a if and only if Sψ ≤ a ψ T for any ψ ∈ Dom T .
The following lemma is a consequence of the closed graph theorem. A proof of this statement for Hilbert spaces can be found for instance in [Sch12,Lemma 8.4].
Lemma 2.2. Let S be a closed operator on a Hilbert B-module E, and let T be a closable operator such that Dom S ⊂ Dom T . Then T is relatively bounded by S.
Proof. We consider Dom S as a Hilbert module equipped with the graph norm of S, and we denote by T the closure of T . We will show that T | Dom S : Dom S → E is closed. Consider a sequence ψ n ∈ Dom S which converges to ψ ∈ Dom S (with respect to the graph norm of S) such that T ψ n converges in E. Since T is closable (and ψ n → ψ in E), we know that T ψ n converges to T ψ. But ψ ∈ Dom S ⊂ Dom T , so T ψ = T ψ. Hence T | Dom S : Dom S → E is a closed everywhere defined operator. The closed graph theorem then implies that T | Dom S is bounded.
Lemma 2.3. Let S and T be closed symmetric operators on a Hilbert B-module E, such that Dom S ∩Dom T is dense in E. Let Dom(S + T ) denote the closure of Dom S ∩ Dom T in the graph norm of S + T . If Dom(S + T ) ⊂ Dom S, then also Dom(S + T ) ⊂ Dom T , and therefore Dom(S + T ) ⊂ Dom S ∩ Dom T .
Proof. Since Dom(S + T ) ⊂ Dom S, we know from Lemma 2.2 that there exists C > 0 such that Sψ ≤ C ψ S+T for all ψ ∈ Dom(S + T ). Let ψ ∈ Dom(S + T ), and consider a sequence ψ n ∈ Dom S ∩ Dom T which converges to ψ in the graph norm of S + T . Then we also have Sψ n → Sψ, because S(ψ n − ψ) ≤ C ψ n − ψ S+T . Hence T ψ n = (S + T )ψ n − Sψ n also converges, which means that ψ ∈ Dom T .
Finally, we mention the Kato-Rellich theorem for regular self-adjoint operators on Hilbert modules.
Theorem 2.4 ([KL12, Theorem 4.5]). Let T be a regular self-adjoint operator on a Hilbert B-module E, and let S be a symmetric operator such that Dom S ⊃ Dom T . Suppose there exist a ∈ (0, 1) and b ∈ [0, ∞) such that Sψ ≤ a T ψ + b ψ for any ψ ∈ Dom T . Then T + S is regular and self-adjoint on the domain Dom(T + S) = Dom T .
We remark in particular that, if S(T ± i) −1 < 1, then we obtain the inequality Sψ ≤ a T ψ + b ψ with a = b = S(T ± i) −1 , so the Kato-Rellich theorem applies.

Kasparov modules
We consider two Z 2 -graded C * -algebras A and B. Kasparov [Kas80] defined the abelian group KK(A, B) as a set of homotopy equivalence classes of certain Kasparov A-B-modules. Below we briefly recall the main definitions. More details can be found in e.g. [Bla98].
An odd Kasparov A-B-module (A, π E B , F ) is defined in the same way as above, except that A, B, and E are assumed to be trivially graded, and F is not required to be odd. Two Kasparov modules (A, π1 E 1B , F 1 ) and (A, π2 E 2B , F 2 ) are called unitarily equivalent if there exists an even unitary in Hom B (E 1 , E 2 ) intertwining the π j and F j (for j = 1, 2). A homotopy between (A, π0 E 0B , F 0 ) and (A, π1 E 1B , F 1 ) is given by a Kasparov The ( In this article we will mostly focus on the unbounded representatives of KK-elements introduced by Baaj and Julg [BJ83]. Definition 2.6 ( [BJ83]). An (even) unbounded Kasparov A-B-module (A, π E B , D) is given by a Z 2 -graded, countably generated, right Hilbert B-module E, a Z 2 -graded * -homomorphism π : A → End B (E), a separable dense * -subalgebra A ⊂ A, and a regular self-adjoint odd operator D : Dom D ⊂ E → E such that 1) we have the inclusion π(A) · Dom D ⊂ Dom D, and [D, π(a)] ± is bounded on Dom D for each a ∈ A; 2) the resolvent of D is locally compact, i.e. π(a)(D ± i) −1 is compact for each a ∈ A.
If no confusion arises, we will usually write (A, E B , D) instead of (A, π E B , D) and a instead of π(a). If B = C and A is trivially graded, we will write E = H and refer to (A, H, D) as a spectral triple over A (see [Con94]).

Given an unbounded Kasparov
An odd unbounded Kasparov A-B-module (A, π E B , D) is defined in the same way as above, except that A, B, and E are assumed to be trivially graded, and D is not required to be odd. We can then consider the 'even double' given by where e denotes the generator of

The unbounded Kasparov product
There exists a bilinear associative pairing KK(A, B) × KK(B, C) → KK(A, C), called the Kasparov product [Kas80]. The following theorem by Kucerovsky provides sufficient conditions for when an unbounded Kasparov module represents the Kasparov product.
Let φ : B → C be a C * -algebra homomorphism, and let [φ] ∈ KK 0 (B, C) be the corresponding class represented by the (even) unbounded Kasparov B-C-module (B, φ C C , 0). The homomorphism φ induces maps φ * : KK • (A, B) → KK • (A, C) and φ * : for any x ∈ KK • (A, B) and y ∈ KK • (C, A). The following lemma will be useful to us later. Theorem 2.7. The second and third conditions are obviously satisfied. For the first condition, we compute for ψ, ξ ∈ A · Dom D and c 1 , c 2 ∈ C the commutator and we see that this (graded) commutator is bounded for each ψ ∈ A · Dom D. This shows the first condition of Kucerovsky's theorem, and we conclude that [ D ⊗ 1] represents the Kasparov product of [ D] with [φ].

Symmetric elliptic operators
Let D be a symmetric elliptic first-order differential operator on a (possibly Z 2 -graded) vector bundle

Fredholm operators
Let B be a (trivially graded) C * -algebra, and let E be a (possibly Z 2 -graded) Hilbert B-module. In Section 2.3, we will consider a (suitably continuous) family of regular self-adjoint operators {D(x)} x∈X on E parametrised by a locally compact Hausdorff space X, such that we obtain a regular self-adjoint operator D(·) on the Hilbert C 0 (X, B)-module C 0 (X, E). We would like to associate to D(·) a class in KK 1 (C, C 0 (X, B)), without assuming that D(·) has compact resolvent (i.e. D(·) does not define an unbounded Kasparov module). Instead we only assume that D(·) is Fredholm, following the approach of [Wah07,§2]. In this subsection, we recall the notion of Fredholm operators on Hilbert modules, and we describe how a Fredholm operator determines a class in KK-theory.
Definition 2.10 ([Joa03, Definition 2.1]). Let D be a regular operator on a Hilbert B-module E. A right parametrix for D is an adjointable endomorphism Q R ∈ End B (E) such that DQ R is closable, DQ R is adjointable, and DQ R − 1 is compact. Similarly, Q L ∈ End B (E) is a left parametrix if Q L D is closable, Q L D is adjointable, and Q L D − 1 is compact. We say Q is a parametrix for D if Q is both a left and a right parametrix. The operator D is called Fredholm if there exists a parametrix.
Given a regular operator D on a Hilbert B-module E, the following statements are equivalent: . Definition 2.12 ([Wah07, Definition 2.3]). A normalising function for a regular self-adjoint operator D on a Hilbert B-module E is an odd non-decreasing smooth function χ : R → R with χ(0) = 0, χ ′ (0) > 0, lim x→∞ χ(x) = 1, and χ(D) 2 − 1 ∈ End 0 B (E). From Proposition 2.11, it follows that a regular self-adjoint Fredholm operator on a Hilbert module admits many normalising functions.
Definition 2.13. Let D 0 and D 1 be (even or odd) regular self-adjoint Fredholm operators on Z 2 -graded Hilbert B-modules E 0 and E 1 , respectively. A homotopy between (E 0 , D 0 ) and (E 1 , D 1 ) is an (even or odd) regular self-adjoint Fredholm operator D on a Z 2 -graded Hilbert C([0, 1], B)-module E such that ev j ( E, D) ≃ (E j , D j ) (for j = 0, 1). Here ≃ denotes unitary equivalence, and ev t ( E, D) := ( E⊗ ρt B, D⊗ 1) (with ρ t as in Definition 2.5). We say that D 0 and D 1 are homotopic if there exists a homotopy between (E 0 , D 0 ) and (E 1 , D 1 ). Proof. Let χ be any normalising function for D. By assumption, χ(D) 2 − 1 is a compact endomorphism. Since χ(D) is also self-adjoint, we see that (E B , χ(D)) is a (bounded) Kasparov C-B-module.
We consider the following standard isomorphisms of In the even (Z 2 -graded) case, consider an odd Fredholm operator ). We will see in Section 2.3 how the class [D] ∈ KK 1 (C, B) of an unbounded regular self-adjoint Fredholm operator is related to the spectral flow.

Spectral flow and the Kasparov product
Let X be a locally compact, paracompact space, and consider a family of operators {D(x)} x∈X on a Hilbert B-module E. We define the operator D(·) on the Hilbert C 0 (X, B)-module C 0 (X, E) by In order for D(·) to be a densely defined operator on C 0 (X, E), we of course need to assume that the family {D(x)} x∈X is suitably continuous. The following lemma characterises the continuity required for regular self-adjoint operators.
The notion of spectral flow for a path of self-adjoint operators (typically parametrised by the unit interval) was first defined by Atiyah and Lusztig, and it appeared in the work of Atiyah, Patodi, and Singer [APS76,§7]. Heuristically, the spectral flow of a path of self-adjoint Fredholm operators counts the net number of eigenvalues which pass through zero. An analytic definition of spectral flow was given by Phillips in [Phi96]. For a path of Fredholm operators on a Hilbert module, the most general definition of spectral flow (to the author's knowledge) is given in [Wah07]. 1 Definition 2.16 ([Wah07, Definition 3.9]). Let X be a locally compact, paracompact space, and consider a regular self-adjoint operator D(·) = {D(x)} x∈X on the Hilbert C 0 (X, B)-module C 0 (X, E). We say there exist locally trivialising families for D(·) if for each x ∈ X there exist a precompact open neighbourhood O x of x and a family of bounded operators Lemma 2.17. Let X be a locally compact, paracompact space. Let D(·) = {D(x)} x∈X be a regular self-adjoint operator on the Hilbert C 0 (X, B)-module C 0 (X, E) for which locally trivialising families exist. Suppose that there exists a compact set K ⊂ X such that D(·)| X\K is Fredholm. Then D(·) is Fredholm.
Proof. By assumption, for each x ∈ X there exist a precompact open neighbourhood O x of x and a trivialising operator For each n we have a trivialising family {A n (x)} x∈Un as above, and we write Q n := (D(·)| Un + A n (·)) −1 . Let U 0 := X\K, and let Q 0 be a parametrix for D(·)| X\K . Let χ n be a partition of unity for the finite open cover {U n } N n=0 of X.
Thus, if there exist locally trivialising families for D(·), and D(·) is 'Fredholm near infinity', then D(·) is Fredholm and therefore defines a class [D( This suggests the following generalisation of the notion of spectral flow. Definition 2.18. Let X be a locally compact, paracompact space, and let D(·) be a regular self-adjoint Fredholm operator on C 0 (X, E B ). Then we define the KK-theoretic spectral flow of D(·) as In the special case X = R, and using the Bott periodicity isomorphism β : , we can view the KK-theoretic spectral flow of D(·) as an element in the even K-theory of B. In this case, the KKtheoretic spectral flow agrees with the usual notion of spectral flow under the Bott periodicity isomorphism. This was already shown by Wahl [Wah07] for the spectral flow of a family on the unit interval.
We point out that the spectral flow is also well-defined on the whole real line R, as long as the family of operators is uniformly invertible outside a bounded interval [a, b] ⊂ R (i.e. sup x∈R\[a,b] D(x) −1 < ∞). Indeed, in this case we can simply define the spectral flow by restricting to this bounded interval: [a,b] . This is well-defined, because the invertible part of the family does not contribute to the spectral flow.
Proposition 2.19. Let D(·) be a regular self-adjoint operator on the Hilbert C 0 (R, B)-module C 0 (R, E). Suppose that D(x) is uniformly invertible outside a bounded interval [a, b] ⊂ R, and that there exist locally trivialising families for {D(x)} x∈R . Then D(·) is Fredholm, and we have the equality Proof. From Lemma 2.17 we know that D(·) is Fredholm, so the KK-theoretic spectral flow is well-defined. We know that D(x) is uniformly invertible outside a bounded interval [a, b] ⊂ R. Without loss of generality, we may assume that [a, b] = [0, 1]. Consider then the homotopy Since D(·) is regular self-adjoint, we know from Lemma 2.15 that the resolvents (D(x) ± i) −1 are strongly continuous. By construction, the resolvents (D t (x)±i) −1 are then strongly continuous as well (jointly in t and x). Hence we know (again from Lemma 2.15) that D • (·) is a regular self-adjoint operator on C 0 ([0, 1]× R, E). Let Q be a parametrix for D(·). Then D • (·) is also Fredholm, because using a partition of unity we can 'patch together' the parametrix Q Hence D • (·) indeed gives a homotopy between D 0 (·) and D 1 (·) = D(·), and we have reduced the problem to the case where D(·) is constant outside a bounded interval. For this case, it has been shown in [Wah07,Proposition 4 We would like to describe the spectral flow in terms of the Kasparov product. Consider the standard spectral triple ( Proposition 2.20. Let D(·) be a regular self-adjoint Fredholm operator on C 0 (R, E). Then Proof. For the first statement, we observe that the Bott periodicity isomorphism β : 3 Dirac-Schrödinger operators

Families of unbounded operators
Let X be a locally compact, paracompact, topological space, let B be a (trivially graded) σ-unital C * -algebra, and let E be a countably generated Hilbert B-module. Let {S(x)} x∈X be a family of regular self-adjoint operators on E satisfying the following assumptions.
(a1) The domain W := Dom S(x) is independent of x ∈ X, and the inclusion W ֒→ E is compact (where W is viewed as a Hilbert B-module equipped with the graph norm of S(x 0 ), for some x 0 ∈ X). (a2) The map S : X → Hom B (W, E) is norm-continuous. (a3) There exists a compact subset K ⊂ X such that S(x) is uniformly invertible on X\K.
Here we say that S(x) is uniformly invertible on X\K if S(x) is invertible for all x ∈ X\K and we have a uniform bound sup x∈X\K S(x) −1 < ∞. Furthermore, we say that the graph norms of S(x) are uniformly equivalent (to the norm of W ) if there exist constants C 1 , C 2 > 0 such that Proof. Let x ∈ X, and consider W x := Dom S(x) as a Hilbert B-module equipped with the graph norm.
Since W x = W = Dom S(x 0 ), we know from Lemma 2.2 that the graph norm of S(x) is equivalent to the norm of W . Therefore we know that S is also continuous as a map X → Hom B (W x , E). In particular, (S(y) − S(x))(S(x) − i) −1 depends continuously on y. The statement then follows from the inequality Lemma 3.2. The family {S(x)} x∈X gives an essentially regular self-adjoint operator S(·) on the Hilbert for all ψ in the initial domain C c (X, W ). Furthermore, if the graph norms of {S(x)} x∈X are uniformly equivalent, then the closure of S(·) is regular self-adjoint on the domain Dom S(·) = C 0 (X, W ).
Proof. By Lemma 3.1, the resolvents (S(x)±i) −1 are norm-continuous, so we already know from Lemma 2.15 that (the closure of) S(·) is regular self-adjoint, with the domain given in Eq. (2.2). We need to check that Since ψ ∈ C c (X, W ) is continuous, and S(x) depends norm-continuously on x as a map W → E, the above inequality shows that S(x)ψ(x) ∈ E also depends norm-continuously on x.
Finally, if the graph norms of S(x) are uniformly equivalent, it follows that the graph norm of S(·) is equivalent to the supremum-norm on C c (X, W ), so that Dom S(·) = C 0 (X, W ).
Proof. We know from Lemma 3.1 that the family ( . Proposition 3.4. The operator S(·) on the Hilbert C 0 (X, B)-module C 0 (X, E) is regular self-adjoint and Fredholm, and hence it defines a class [S(·)] ∈ KK 1 (C, C 0 (X, B)).
Proof. We have seen in Lemma 3.2 that S(·) is regular self-adjoint. From Lemma 3.3 we know that S(·) has locally compact resolvents.
Lemma 3.5. Let H be a separable Hilbert space, and consider a family of self-adjoint operators {S(x)} x∈X on H, satisfying assumptions (a1)-(a3). Then there exist locally trivialising families for S(·).
Proof. Consider a point x 0 ∈ X. Since S(x 0 ) has compact resolvents, the spectral projection Remark 3.6. In the special case X = R, the above lemma shows that the spectral flow of {S(x)} x∈R is well-defined. From Proposition 2.19 we then know that the class [S(·)] ∈ KK 1 (C, C 0 (R)) ≃ Z is given by the spectral flow of {S(x)} x∈R . However, it is not clear if the statement of the above lemma remains valid for families of operators on Hilbert modules (rather than Hilbert spaces).
In Section 5, we will compute a Kasparov product with the class [S(·)] ∈ KK 1 (C, C 0 (X, B)). However, the existing literature on the unbounded Kasparov product deals only with unbounded Kasparov modules. Although the Fredholm operator S(·) represents a class in KK 1 (C, C 0 (X, B)), in general it does not define an unbounded Kasparov C-C 0 (X, B)-module, because it might not have compact resolvents. Therefore we show next that we can replace S(·) by an operator which does have compact resolvents (following [KL13,§8]), so that we can make use of existing results on the Kasparov product of unbounded Kasparov modules. The operators (S(x) ± if (x)) −1 are compact and uniformly bounded, and by Lemma 3.1 they are continuous in x. Therefore (f −1 S(·) ± i) −1 = f (S(·) ± if ) −1 lies in C 0 (X, End 0 B (E)), so f −1 S(·) has compact resolvents and hence it defines an odd unbounded Kasparov module.
Proof. First, we note that the restricted family {S(x)} x∈U also satisfies assumptions (a1)-(a3), so that [S(·)| U ] is well-defined. Let f ∈ C 0 (U ) be a strictly positive function such that f (x) = 1 for all x ∈ K. By Lemma 3.7, we know that f −1 S(·)| U defines an unbounded Kasparov module and . We need to show that the latter module is homotopic to (C, C 0 (X, E), S(·)).

The product operator
From here on, we make the following standing assumptions.
Standing Assumptions. Let B be a (trivially graded) σ-unital C * -algebra, and let E be a (trivially graded) countably generated Hilbert B-module. Let M be a connected Riemannian manifold, and let D be an essentially self-adjoint elliptic first-order differential operator on a hermitian vector bundle F → M . Let {S(x)} x∈M be a family of regular self-adjoint operators on E satisfying the following assumptions.
Lemma 3.9. The graph norms of S(x) are uniformly equivalent.
Proof. For any x, y ∈ M , since Dom S(x) = Dom S(y), we know from Lemma 2.2 that the graph norms of S(x) and S(y) are equivalent. Using compactness of K and continuity of S(x), it follows that the graph norms of S(x) are uniformly equivalent for all x ∈ K. For x ∈ M \K, we know from assumption (A4) that x ∈ V j for some j. Then Using a Neumann series argument, we know that S(x j )S(x) −1 ≤ 1 1−aj . Therefore These inequalities show that for x ∈ M \K the graph norms of S(x) are uniformly equivalent to the graph norm of S(x 1 ).
The operator 1 ⊗ D is not well-defined on L 2 (M, E ⊗ F). Instead, using the canonical isomorphism Alternatively, we can extend the exterior derivative on C 1 0 (M ) to an operator Denoting by σ the principal symbol of D, we can define an operator 1 We note that, despite our use of the term 'Dirac-Schrödinger' operator, we do not assume that the operator D is of Dirac-type (although a Dirac-type operator is of course a typical example). We introduce some further notation. For any x ∈ M , we write S x (·) for the operator corresponding to the constant family S x (y) := S(x) (for all y ∈ M ). We can then consider the product operator D x S := S x (·) × D defined as above. Lastly, we define on L 2 (M, E ⊗ F) ⊕2 the operators Remark 3.11. The product operator D S is given by the standard formula for the (odd) unbounded Kasparov product of S(·) with D [Mes14, KL13,BMS16,MR16]. In these references, the proof that this formula indeed represents the Kasparov product, relies on the condition that the commutator [D, S(·)] is well-behaved. However, since we only assume that S(·) is continuous (but not necessarily differentiable), we have no direct control on this commutator. In Section 5 we will show that the operator D S nevertheless represents the Kasparov product.

Regularity and self-adjointness
Our first task is to prove that the product operator D S := S(·) × D is regular self-adjoint. Our proof is a generalisation of the methods used in [AW11, §2] for the case M = [0, 1] ⊂ R. We start with the following well-known lemma (for more general statements, see e.g. [Mes14, Theorem 6.1.8] or [KL12, Theorem 7.10]).
We will make use of the following generalisation of Tietze's extension theorem.  , and a continuous map S : A → L whose image is contained in the ball of radius a around S(x). By Theorem 3.14 there then exists a continuous extension S U : M → Hom s B (W x , E) whose image is contained in the same ball, i.e. such that sup y∈M (S U (y) − S(x))(S(x) ± i) −1 ≤ a. It follows from the Kato-Rellich theorem that S U (y) is regular self-adjoint (on the domain W x ) for all y ∈ M . This proves the first statement.
The second statement is proven similarly, by equipping W x with the equivalent norm ψ Wx := S(x)ψ (in this case, we note that the Kato-Rellich theorem still applies, because S(x)(S(x) ± i) −1 ≤ 1). Furthermore, in this case, using the inequality (S U (y) − S(x))S(x) −1 ≤ a, a Neumann series argument shows that S U (y) −1 ≤ 1 1−a S(x) −1 (for any y ∈ M ).
Proposition 3.16. The product operator D S := S(·) × D is regular self-adjoint on Dom D S = Dom S(·) ∩ Dom D.
Proof. By assumption (A4), there exists a finite open cover {V j } l j=1 of M \K with points x j ∈ V j and positive numbers a j < 1 such that S(x) − S(x j ) S(x j ) −1 ≤ a j for all x ∈ V j . For x ∈ K, we know by continuity that there exists a precompact open neighbourhood U x of x such that sup y∈Ux (S(y) − S(x))(S(x) ± i) −1 ≤ 1 2 . Since K is compact, there exist finitely many points x l+k ∈ K such that the open sets V l+k := U x l+k cover K. Setting a l+k := 1 2 , we therefore have a finite open cover {V j } of M and numbers a j < 1 such that By Lemma 3.15, there exists for each j a family of self-adjoint operators {S Vj (x)} x∈M such that S Vj (x) = S(x) for all x ∈ V j , and sup x∈M (S Vj (x) − S(x j ))(S(x j ) ± i) −1 ≤ a j . By Lemma 3.13, the corresponding product operators D For any We can pick λ sufficiently large, such that the norm of K ± (λ) := j [ D, χ j ] D Vj S ± iλ −1 χ j is less than one. Then 1 + K ± (λ) is invertible, and R ± (λ)(1 + K ± (λ)) −1 is a right inverse of D S ± iλ. Similarly, we can also obtain a left inverse, which proves that D S is regular self-adjoint on the domain Dom D S = Dom S(·) ∩ Dom D.
4 Index theory

The Fredholm property
In this section, we will show that the operator D S is Fredholm, and therefore Index(D − iS(·)) is well-defined. The Fredholm property will be a consequence of the local compactness of the resolvents and the 'invertibility near infinity'.
Proof. The statement is (essentially) a special case of [KL13, Theorem 6.7]. However, since we do not assume any differentiability for the family {S(x)} x∈M , we need to check that the proof of [KL13, Theorem 6.7] still holds. For clarity, we provide the details of the proof (rewritten to suit our context). There exists a frame {e j } ∞ j=1 for E B , i.e. a sequence {e j } ∞ j=1 such that for each ξ ∈ E B the series ∞ j=1 e j e j |ξ converges to ξ (see [FL02]). Given an approximate unit u m for the algebra C 0 (M ), we then obtain an approximate unit θ m := For φ ∈ C 0 (M ) we know from Lemma 3.3 that φ(S(·) ± i) −1 ∈ C 0 (M, End 0 B (E)) is a compact endomorphism on C 0 (M, E). Since θ m is an approximate unit for the algebra of compact endomorphisms C 0 (M, End 0 B (E)), it follows that θ m φ • ι S converges in norm to φ • ι S .
Recall from Proposition 3.16 that Dom D S = Dom S(·) ∩ Dom D. Now consider the following commuting diagram of continuous inclusions.
Dom D ιD ' ' P P P P P P P P P P P P P Dom D S D 9 9 r r r r r r r r r r Hence φ • ι is compact, and therefore φ( D S ± i) −1 is compact. Finally, if ( S(·) ± i) −1 is in fact compact, then we can repeat the argument with φ = 1 to conclude that also ( D S ± i) −1 is compact.
Lemma 4.2. Suppose that S(x) is uniformly invertible for all x ∈ M , and there exist x 0 ∈ M and a < 1 S ψ . It follows that S x0 (·)( D x0 S ) −1 ≤ 1, and we obtain the inequality (4.1) A Neumann series argument then shows that D S is also invertible, and that its inverse is given by We then calculate that The operators [ D, χ j ] are smooth and compactly supported, and therefore bounded.
is also bounded (for j = 0), it follows from Proposition 4.1 that D S Q − 1 is compact. Hence Q is a right parametrix for D S . A similar calculation shows that Q is also a left parametrix, and therefore D S is Fredholm.
hence Index D−iS(·) = 0. Such Γ exists, for instance, if D is of Dirac-type and M is even-dimensional. 2) Consider the classical case of a Dirac-Schrödinger operator D −iV (also called a Callias-type operator), where D is a Dirac-type operator, and the potential V is a self-adjoint endomorphism on some auxiliary vector bundle (of finite rank ). Under suitable assumptions on the potential V , one can then prove that D − iλV is Fredholm for sufficiently large λ ∈ (0, ∞) (see, for instance, [BM92,Ang93]). In our case, S(·) plays the role of the potential. As observed in Lemma 4.4, D − iλS(·) is Fredholm for any λ ∈ (0, ∞). However, we stress here that this does not provide a generalisation of the classical result. Indeed, our theorem applies to a 'potential' S(·) acting on the Hilbert module C 0 (M, E).
Restricting to the case of finite-rank bundles, this means our theorem only applies to trivial bundles.
In the case of non-trivial bundles, there are examples in which the Fredholm property of D − iλV fails to hold for some λ ∈ (0, ∞) (see, e.g., [BM92,§4]). 3) In the classical case of a Dirac-Schrödinger operator D − iV , the index vanishes whenever the manifold is compact. Indeed, in this case V is bounded and D has compact resolvents. Therefore D − iV is a relatively compact perturbation of D (which is self-adjoint), so that Index(D − iV ) = Index(D) = 0. In our setup, the Hilbert B-module E is in general not finitely generated projective (i.e., in the case B = C, the Hilbert space E = H is in general not finite-dimensional). Therefore, although D has compact resolvents on L 2 (M, F), the operator D ⊗ 1 on L 2 (M, E ⊗ F) in general does not have compact resolvents. Hence the index may be non-zero on compact manifolds. Consider for instance the simple case of a circle M = S 1 (with D = −i∂ x ). Then the index of ∂ x + S(·) is equal to the spectral flow of {S(x)} x∈S 1 (as we will discuss at the start of Section 5), and this spectral flow could certainly be non-zero.

A relative index theorem
Before we describe the relative index theorem, we will first prove an auxiliary lemma, stating that the class of an unbounded Fredholm operator vanishes if there exists a suitable symmetry. A similar statement is given in [Bun95, Lemma 1.15] in terms of bounded Kasparov modules. Proof. Since e[D, e] ± is bounded and symmetric, we know that D t := D − t 2 e[D, e] ± is regular self-adjoint for each t ∈ [0, 1]. Furthermore, since [D, e] ± is relatively compact, we also know that D t is Fredholm, and any parametrix for D is also a parametrix for D t . Then the operator D • on C([0, 1], E) is also regular self-adjoint and Fredholm, so it provides a homotopy between D 0 = D and D 1 = 1 2 (D − eDe). Since e is an odd unitary, we can identify E 0 with E 1 using e, so that E ≃ E 0 ⊕ E 0 . Then e, D, and D 1 take the form The class of D 1 is given by Index((D 1 ) + ) ∈ K 0 (B). Since (D 1 ) For j = 1, 2, let F j → M j , D j , and S j (·) be as in the Standing Assumptions. We assume that the operators {S j (x)} x∈M j act on the same Hilbert B-module E. Suppose we have partitions where N j are smooth compact hypersurfaces. Let C j be precompact open tubular neighbourhoods of N j , and assume that there exists an isometry φ : C 1 → C 2 (with φ(N 1 ) = N 2 ) covered by an isomorphism Φ : for all x ∈ C 1 . We will identify N 1 and N 2 via φ, and we simply write N . Define two new Riemannian manifolds Moreover, we glue the bundles using Φ to obtain hermitian vector bundles F 3 → M 3 and F 4 → M 4 . For j = 3, 4, we then obtain corresponding operators D j and S j (·) satisfying the Standing Assumptions. For j = 1, . . . , 4, we consider the product operators D j S := S j (·) × D j . By Theorem 4.3, these operators are regular self-adjoint and Fredholm. Under the standard isomorphism KK 0 (C, B) ≃ K 0 (B), the class , and hence Index(D 1 − iS 1 (·)) + Index(D 2 − iS 2 (·)) = Index(D 3 − iS 3 (·)) + Index(D 4 − iS 4 (·)) ∈ K 0 (B).
Proof. We roughly follow Bunke's proof of the K-theoretic relative index theorem [Bun95, Theorem 1.14], except that we work with unbounded operators. For j = 1, . . . , 4, we write E j := L 2 (M j , E ⊗ F j ) ⊕2 . Each E j is Z 2 -graded, with the grading operator Γ j given by 1 ⊕ (−1). We write ( E j ) op = E j for the same Hilbert module with the opposite grading (Γ j ) op = (−1) ⊕ 1. We define the Hilbert module E := . For j = 1, 2, we choose smooth functions χ U j and χ V j such that Note that L 2 (U 1 ∪ C 1 , E ⊗ F 1 ), which is a subspace of L 2 (M 1 , E ⊗ F 1 ), can also be viewed as a subspace of for this identification. Similarly, we have identifications β 14 : We define Consider on E the operator Then X = −X * and X 2 = −1. We calculate Since all entries are smooth compactly supported endomorphisms, [ D S , X] is bounded. Furthermore, from The following useful consequence of the relative index theorem allows us to replace the manifold M by a manifold with cylindrical ends. Let C ′ = φ(C) be the collar neighbourhood of ∂U in M ′ . We equip M 2 with a complete Riemannian metric which agrees with the metric of M ′ on C ′ ∪ ∂U × (0, ∞) , and which is of product form on (−∞, 1] × ∂U . We extend the vector bundle F ′ | C ′ ∪(∂U×(0,∞)) to a bundle F 2 → M 2 , and we pick an operator D 2 (satisfying the Standing Assumptions) such that D 2 | C ′ ∪(∂U×(0,∞)) = D ′ | C ′ ∪(∂U×(0,∞)) (for instance, we can take D 2 to be of product form on (−∞, 1] × ∂U ). We define a family {S 2 (x)} x∈M 2 by S 2 (y, r) := S(y) for all y ∈ ∂U and r ∈ R. Then F 2 → M 2 , D 2 , and S 2 (·) satisfy the Standing Assumptions. By cutting and pasting along ∂U

Incomplete manifolds and manifolds with boundary
From Proposition 4.8 we know that we can cut and paste certain open ends of the manifold M without changing the index of D − iS(·). More precisely, suppose that S(·) is uniformly invertible outside a compact set K, and let U be a precompact open neighbourhood of K with smooth boundary ∂U , such that M \U is a finite disjoint union of open subsets on which the variation of S(·) is 'sufficiently small' (as in assumption (A4)). Then in particular the index of D − iS(·) is completely determined by its restriction D − iS(·)| U to the subset U , even though D − iS(·)| U may not be Fredholm. This observation allows us to obtain a well-defined index also more generally, starting with a possibly non-self-adjoint operator D.
Consider the following setup, in which we drop the self-adjointness of D from the Standing Assumptions. Let (M, g) be a Riemannian manifold. Let D be a symmetric elliptic first-order differential operator on a hermitian vector bundle F → M . Let {S(x)} x∈M be a family of regular self-adjoint operators on a Hilbert B-module E satisfying the assumptions (A1)-(A3). We fix a compact subset K of M such that S(·) is uniformly invertible outside of K, and a precompact open neighbourhood U of K with smooth compact boundary ∂U = j N j which consists of finitely many connected components {N j }. Assume that there exist points x j ∈ N j and positive numbers a j < 1 such that Consider the manifold M ′ := U ∪ ∂U (∂U × [0, ∞)). Equip M ′ with a Riemannian metric which is of product form on ∂U × [1, ∞), and which agrees with g| U on U . Let F ′ → M ′ be a hermitian vector bundle which agrees with F| U on U . Let D ′ be a symmetric elliptic first-order differential operator of bounded propagation speed on F ′ → M ′ , which agrees with D| U on U . Consider the family {S ′ (x)} x∈M ′ given by Then the family {S ′ (x)} x∈M ′ satisfies assumptions (A1)-(A4) (for assumption (A4), we can take the open subsets V ′ j := N j × (0, ∞)). Hence we obtain a regular self-adjoint Fredholm operator D ′ S := S ′ (·) × D ′ , and therefore we have a well-defined class A similar procedure as above also allows us to deal with a Dirac-Schrödinger operator on a compact manifold with boundary. Indeed, viewing U as a compact manifold with boundary ∂U and interior U , we obtain a class [ D S ] cyl ∈ K 0 (B) by attaching a cylindrical end (as described above). In this case, it would be interesting to see if this class [ D S ] cyl is equal to the index of a Fredholm operator on U , obtained as an extension of (D − iS(·))| U by imposing suitable boundary conditions on ∂U . In the case of a classical Dirac-Schrödinger operator (in which case the potential acts on a finite-rank bundle), it has been shown in [Råd94] that this is indeed the case (using a natural choice of boundary conditions on M ). It is left as an open problem whether or not this result can be generalised to our setup.

The Kasparov product
Consider a Dirac-Schrödinger operator D S = D − iS(·) on a Riemannian manifold M . By Proposition 3.4, we know that the operator S(·) on the Hilbert C 0 (M, B)-module C 0 (M, E) is Fredholm, and therefore we obtain a well-defined class [S(·)] ∈ KK 1 (C, C 0 (M, B)) as defined in Proposition 2.14. Furthermore, since D is an essentially self-adjoint first-order differential operator, and since the ellipticity of D ensures that D also has locally compact resolvents [ Proof. By Lemma 3.5, there exist locally trivialising families for {S(x)} x∈R , so that the spectral flow is well-defined. We may assume (without loss of generality) that a = 0 and b = 1. Consider the family Using the relative index theorem (Theorem 4.7), we will show that we also have the equality Index ∂ x + S(·) = Index ∂ x + S ′ (·) . Indeed, consider the partitioned manifolds and S 2 (x) = S(0). After cutting and pasting, we obtain the potentials By Lemma 4.2, the operators D 2 S and D 3 S are invertible, so that the relative index theorem implies that Index(∂ x + S(·)) = Index(∂ x + S 4 (·)). This means that we can replace the potential on (−∞, 0) by a constant potential. Similarly, we can also cut and paste (1, ∞), and we conclude that Index ∂ x + S(·) = Index ∂ x +S ′ (·) . Finally, it has been shown in [AW11, Theorem 2.1] that the spectral flow sf {S ′ (x)} x∈[0,1] is equal to Index ∂ x + S ′ (·) , which completes the proof.
Proof. We have the equalities where the first equality is from Theorem 5.1, and the second equality is from Proposition 2.20.
In the remainder of this section we aim to prove the equality on a manifold M of arbitrary dimension, where the operators S(·) and D satisfy the Standing Assumptions. This equality can be interpreted as a generalisation of the 'index = spectral flow' theorem to higherdimensional manifolds. We point out that a similar equality has already been obtained in [KL13] under the assumption that the family {S(x)} x∈M is suitably differentiable. First, we will adapt the methods of [KL13,§8]  Subsequently, we will show that the equality remains valid without assuming any differentiability.

A differentiable family
In this subsection, following [KL13,§8], we consider the special case in which the family {S(x)} x∈M is not only continuous but in fact differentiable. While in [KL13,§8] only families of operators on a Hilbert space were considered, we will show that the results of [KL13,§8] remain valid for families of operators acting on a Hilbert module over a C * -algebra B.
. Let M be a smooth manifold, and let E 1 and E 2 be countably generated Hilbert B-modules. A map S(·) : M → Hom B (E 1 , E 2 ), x → S(x), is said to have a uniformly bounded weak derivative if the map is weakly differentiable (i.e. the map x → S(x)ξ, η is differentiable for each ξ ∈ E 1 and η ∈ E 2 ), the weak derivative dS(x) : is bounded for all x ∈ M , and the supremum sup x∈M dS(x) is finite.
Definition 5.4 (see [KL12,Assumption 7.1]). Let S and T be regular self-adjoint operators on a Hilbert B-module E. We will say that [S, T ](S − iµ) −1 is well-defined and bounded if 1) there exists a submodule E ⊂ Dom T which is a core for T ; 2) for each ξ ∈ E and for all µ ∈ R\{0} we have the inclusions 3) the map [S, T ](S − iµ) −1 : E → E extends to a bounded operator in End B (E) for all µ ∈ R\{0}.
Assumption 5.5. Let B be a (trivially graded) σ-unital C * -algebra, and let E be a countably generated Hilbert B-module. Let M be a connected complete Riemannian manifold, and let D be a symmetric elliptic first-order differential operator with bounded propagation speed on a hermitian vector bundle F → M . Let {S(x)} x∈M be a family of regular self-adjoint operators on E satisfying assumptions (A1)-(A3).
Let σ be the principal symbol of D. The boundedness of the propagation speed of D means that sup ξ∈T * M, ξ =1 σ(ξ) is bounded. We point out that the completeness of M and the bounded propagation speed of D ensure that D is essentially self-adjoint [HR00, Proposition 10.2.11].
Lemma 5.6. Let D and S(·) be as in Assumption 5.5. Suppose that the graph norms of S(x) are uniformly equivalent, and that S : M → Hom B (W, E) has a uniformly bounded weak derivative. Let φ ∈ C 1 (M ) be any differentiable function such that φ(x) ≥ 1 for all x ∈ M , and dφ is bounded. Then D, φS(·) φS(·) ± i −1 is well-defined and bounded. Moreover, we have the following pointwise norm-estimates: where c > 0 is chosen such that [−c, c] does not intersect the spectrum of S(x) for any x ∈ M \K.
Proof. The operator D, S(·) S(·)±i −1 is well-defined and bounded, which follows from the same argument as in the proofs of [KL13, Lemma 8.5 & Theorem 8.6] (in fact, in [KL13] only the case B = C was considered, but the same argument also works for an arbitrary C * -algebra B). We have Hence for any x ∈ M we obtain We can estimate Thus for any x ∈ M we obtain which proves the first inequality. For x ∈ M \K, we can use the invertibility of S(x) to improve this inequality. Indeed, we can estimate Hence for x ∈ M \K we obtain which proves the second inequality. In particular, since this second inequality does not depend on the size of φ(x), we conclude that D, φS(·) φS(·)±i Lemma 5.7. Let D and S(·) be as in Assumption 5.5. Suppose that D, S(·) S(·) ± i −1 is well-defined and bounded. Then there exists λ 0 ≥ 1 such that the operator D λS = D − iλS(·) is Fredholm, for any λ ≥ λ 0 .
The second condition requires the domain inclusion Dom D ′ S ⊂ Dom D. From Lemma 5.6 we know that D, S ′ (·) S ′ (·) ± i −1 is well-defined and bounded. It then follows from [KL12, Theorem 7.10] that D ′ S is regular self-adjoint on the domain Dom S ′ (·) ∩ Dom D. In particular, the second condition holds.

Manifolds with cylindrical ends
We now consider the general case without assuming any differentiability for the family {S(x)} x∈M . We will show that S(·) is nevertheless homotopic to an operator obtained from a differentiable family. For this purpose, we cite (a special case of) Wockel's generalisation of Steenrod's approximation theorem. , E) satisfies the assumptions (A1)-(A4) (note that our assumptions on V j and x j imply that H t 1 (x j ) is independent of t, which ensures that (A4) remains satisfied). Hence we may assume (without loss of generality) that S(x, r) = S(x, r ′ ) for any x ∈ ∂U and r, r ′ ∈ [0, ∞) (i.e., S is constant in the 'axial direction' of the cylindrical ends).
Let Hom s B (W, E) denote the (real) Banach space of symmetric operators S on E with Dom S = W (equipped with the operator norm of maps W → E). We consider the globally trivial bundle M × Since h t (y) is a symmetric perturbation of S(y) such that (h t (y) − S(y))(S(y) ± i) −1 < c < 1, it follows from the Kato-Rellich theorem that h t (y) is regular self-adjoint with Dom h t (y) = Dom S(y) = W for any y ∈ M . Furthermore, a Neumann series argument shows that h t (y) is invertible.
Proposition 5.12. Let M be a manifold with cylindrical ends, satisfying the assumptions of Lemma 5.11. Suppose furthermore that M is complete, and that D has bounded propagation speed. Then the class [ D S ] ∈ KK 0 (C, B) is the (internal) Kasparov product of [S(·)] ∈ KK 1 (C, C 0 (M, B)) with [D] ∈ KK 1 (C 0 (M ), C).