Spectral Flow Argument Localizing an Odd Index Pairing

arXiv:1802.04518v2 [math-ph] 28 May 2018 oabuddoeao mr rdtoal,oerqie commutat requires one traditionally, (more C operator bounded a to ai nltclpoete fteseta o sgvni 1,1,5 nature. 11, analytic [10, functional in given purely as of flow hence spectral is the It of properties analytical basic netbe(ia)operator (Dirac) invertible ipeadt u at ipybatflagmn onciga odd an connecting argument note both short avoids beautiful this It simply of aim taste localizer. the our is spectral It to [3]. and flow simple spectral to connection its and nasprbeHletspace Le Hilbert 6]. separable [4, theory a index on non-commutative of framework general and uz pee.Frteseilcs faoedmninlsystem one-dimensional a of pr case The special with cases. well-defined the K many For is in signature spheres. numerically this fuzzy index the that called the showing matrix access odd estimates dimensional suitable analytic to finite a possible a pairing with of it index torus signature makes dimensional odd the odd the as an that calculated over proved be space authors Hilbert the the [9], on paper result recent the a In of Statement 1 ∗ tertcpr ftepofcnb elcdb pcrlflwar flow spectral a by replaced be can proof the of part -theoretic agbacontaining -algebra pcrlflwagmn oaiiga d ne pairing index odd an localizing argument flow Spectral e sbgnb ttn h anrsl.I xed h anstateme main the extends It result. main the stating by begin us Let 2 e a ecluae stesgaueo nt dimensional finite a flo of spectral signature a the by shown as localizer. is calculated it be Here can index an dex index. is this Noether to with Associated operator invertible. resol given compact the with with operator tor Dirac so-called unbounded an by eatetMteai,Friedrich-Alexander-Universit¨ Mathematik, Department nodFehl ouefragvnivril prtro Hi a on operator invertible given a for module Fredholm odd An 1 eateto ahmtc n ttsis nvriyo Ne of University Statistics, and Mathematics of Department er .Loring A. Terry A ohv hspoet) soitdto Associated property). this have to D ihcmatrsletsc httecmuao [ commutator the that such resolvent compact with H nubuddodFehl ouefor module Fredholm odd unbounded An . K ter swl sthe as well as -theory 1 n emn Schulz-Baldes Hermann and Abstract 1 tErlangen-Nürnberg, Germany at η arn ntrso Fredholm a of terms in pairing ivrat n nta ssonly uses instead and -invariant, D etadbuddcommuta- bounded and vent sas-aldHryprojection Hardy so-called a is arxcle h spectral the called matrix S21:1K6 46L80 19K56, MSC2010: ruethwti in- this how argument w twsas hw o the how shown also was it , uetuigthe using gument t ,adrcle nappendix. in recalled and ], r o es usti a in subset dense a for ors A br pc sspecified is space lbert K eio USA Mexico, w opoieamr direct, more a provide to fa netbeoperator invertible an of ea netbeoperator invertible an be tertcagmnson arguments -theoretic pcrllclzr This localizer. spectral to 9 otenatural the to [9] of nt rdommdl can module Fredholm ne arn othe to pairing index 2 o n[]combined [9] in oof A saselfadjoint a is ,D A, η -invariant extends ] Π= χ(D > 0) where χ denotes the indicator function. Then it is well-known, e.g. [4] or p. 462 in [6], that [Π, A] is compact and the Toeplitz operator T = Π A Π+(1 Π) , (1) − is a bounded Fredholm operator on . Its associated Noether index is denoted by Ind(T ). H The operator T and its index are called the index pairing of (the K1-class of) A with (the odd Fredholm module specified by) D. In this situation, the spectral localizer is defined as the operator κ D A L = ∗ , κ A κ D − acting on where κ > 0 is a tuning parameter to be specified later on. It measures the size w.r.t. HA, ⊕ which H is supposed to satisfy A 1. We will consider finite volume restrictions of the spectral localizer. For given ρ> 0, letk usk ≥ set = Ran χ(D2 ρ2) . (2) Hρ ≤ As D has compact resolvent, each ρ is finite-dimensional. The surjective partial isometry H ∗ from to will be denoted by π . For any operator T on , we then set T = π T π which H Hρ ρ H ρ ρ ρ is an operator on ρ. This corresponds to restricting T with Dirichlet boundary conditions. Let us also use 1 H= π π∗ for the identity on . We also write π and 1 for the surjective ρ ρ ρ Hρ ρ ρ partial isometry and identity on ρ ρ. With these notations, the finite volume spectral localizer on is H ⊕ H Hρ ⊕ Hρ ∗ κ Dρ Aρ Lκ,ρ = πρ Lκ πρ = ∗ . (3) A κ Dρ ρ − The following was proved in [9] for the special case of = ℓ2(Zd, CN ) with d odd and D built from the components of the position operator on ℓ2(ZdH, CN ), see the example below. Theorem Let g = A−1 −1 > 0 be the gap of an invertible bounded operator A on a separable Hilbert space. Let Dk specifyk an odd Fredholm module for A. Set g3 κ0 = . (4) 12 A [D, A] k kk k Suppose that κ and ρ are such that 2 g 2 g < ρ . (5) κ0 ≤ κ

Then the matrix Lκ,ρ satisﬁes the bound g2 (L )2 1 . (6) κ,ρ ≥ 4 ρ

In particular, Lκ,ρ is invertible and thus has a well-defined signature Sig(Lκ,ρ). Then 1 Ind Π A Π+(1 Π) = Sig(L ) . (7) − 2 κ,ρ 2 The main hypothesis is that the commutator [D, A] is bounded, that is, A is differentiable in the non-commutative sense. Then one can choose sufficiently small κ and large ρ such that both inequalities in (5) hold. The identity (7) then allows to determine the index pairing numerically in many interesting situations, see [8, 9] which is based on the example below.

Let us note that κ = 0 forces ρ = . Indeed, L0 has vanishing signature for any finite ρ. ∞ ,ρ Thus Lκ,ρ is not invertible for some κ< 2g/ρ. Actually the spectral flow for κ [0, 2g/ρ] Lκ,ρ is equal to the half-signature on the r.h.s. of (7). However, our proof of the∈ Theorem7→ is not based on this spectral flow, but rather starts out with Phillips’ Theorem [11] expressing the index pairing on the l.h.s. of (7) in terms of a suitable spectral flow, see Section 3 below. Another comment on the Theorem concerns the choice of the finite dimensional Hilbert space . The proof below shows that any other finite subspace of may be used without Hρ H altering the signature, provided that it contains ρ for some ρ satisfying (5). For the example below this means that one can choose finite volumesH to be balls or cubes or triangles, etc. Example: Let d be odd. The Hilbert space is ℓ2(Zd) over a d-dimensional lattice Zd, extended N 2 d N by a fiber C . Thus = ℓ (Z , C ). On this Hilbert space act the d components X1,...,X of H d the selfadjoint commuting position operators defined by Xj n = nj n where n =(n1,...,nd) Zd and n ℓ2(Zd) is the Dirac Bra-Ket notation for the| uniti vector| i localized at n. Suppose∈ | i ∈ furthermore that γ1,...,γd is a self-adjoint irreducible representation of the Clifford algebra Cd d−1 on CN . This determines N =2 2 in terms of d. From this data, the selfadjoint Dirac operator is d D = X γ + 0 0 γ1 . j ⊗ j | ih | ⊗ j=1 X The last summand is added to ensure that D is invertible. Clearly, D has a compact resolvent. It defines a Fredholm module for A if these operators have a bounded commutator

[D, A] < . k k ∞ Now for d = 1, Π = χ(D > 0) is the classical Hardy projection and, if furthermore A is periodic, T is the classical Toeplitz operator (strictly speaking extended by 1 Π to the full Hilbert space). Its index is then known to be the winding number of A which according− to (7) can hence be calculated readily from the spectral localizer. For higher odd dimension d, the index pairing plays an important role in the theory of topological insulators, e.g. [8, 7, 12]. ⋄ Acknowledgments: The authors thank the Simons Foundation (CGM 419432), the NSF (DMS 1700102) and the DFG (SCHU 1358/3-4) for ﬁnancial support.

3 2 The proof of the constancy of the signature

The ﬁrst step of the proof, namely that (6) follows from (5), is preparatory. It will also be shown in this section that the signature is the same for all pairs κ, ρ satisfying (5). While this is already contained in [9], the argument given here is simpler and improves the estimates slightly. The proof will use an even and diﬀerentiable tapering function Gρ : R [0, 1] with three properties: → (i) G (x)=1 for x ρ ; ρ | | ≤ 2 (ii) G (x)=0 for x ρ; ρ | | ≥ ′ ′ 1 −1 (iii) The Fourier transform Gρ of the derivative Gρ has an L -norm bounded by 8ρ .

Such a function is explicitly constructedc in [9] where it is also shown (using [6, Lemma 10.15]) that [G (D), A] 8 ρ−1 D, A] . k ρ k ≤ k k ′ D 0 0 A ′ ′ 0 [D, A] Setting D = and H = ∗ , one has D H + HD = ∗ and, 0 D A 0 [D, A] 0 due to G ( D)= G−(D), also ρ − ρ [G (D′),H] 8 ρ−1 [D, A] . (8) k ρ k ≤ k k In order to connect radii ρ and ρ′ ρ, let us introduce ≥ ′ ∗ ∗ ′ ′ ′ Lκ,ρ,ρ (λ) = κ πρ D πρ′ + πρ Gλ,ρ HGλ,ρ πρ′ , acting on ′ ′ where 0 λ 1 and Hρ ⊕ Hρ ≤ ≤ G = (1 λ)1 + λG (D′) . λ,ρ − ρ

Also (5) is supposed to hold. Notice that Lκ,ρ,ρ′ (0) = Lκ,ρ′ . The ﬁrst goal is to show that g2 Lκ,ρ,ρ′ (λ) is always invertible and that its square is bounded below by 4 1ρ′ when λ = 0. The square of Lκ,ρ,ρ′ (λ) becomes

2 2 ′ 2 ∗ ∗ 2 ′ ′ ∗ ′ ′ ′ ′ Lκ,ρ,ρ (λ) = κ πρ (D ) πρ′ + πρ Gλ,ρ HGλ,ρπρ′ + κ πρ Gλ,ρ(D H + HD )Gλ,ρπρ′ . Let us begin with the second summand:

∗ 2 2 ∗ ′ ′ πρ Gλ,ρ HGλ,ρπρ′ = πρ Gλ,ρ HGλ,ρ HGλ,ρπρ′ ′ 2 ∗ ′ πρ Gλ,ρ HGρ(D ) HGλ,ρπρ′ ≥ ′ 2 ′ ∗ ′ ′ ∗ ′ ′ = πρ Gλ,ρGρ(D ) H Gρ(D )Gλ,ρπρ′ + πρ Gλ,ρ [Gρ(D ) H, [Gρ(D ),H]] Gλ,ρπρ′ 2 2 ′ 2 ∗ ′ ′ ∗ ′ ′ > g πρ Gλ,ρGρ(D ) πρ′ + πρ Gλ,ρ [Gρ(D ) H, [Gρ(D ),H]] Gλ,ρπρ′ 2 ′ 4 ∗ ′ ′ ∗ g π ′ G (D ) π ′ + π ′ G [G (D )H, [G (D ),H]] G π ′ . ≥ ρ ρ ρ ρ λ,ρ ρ ρ λ,ρ ρ 4 For the special case of λ = 0 one has the better estimate

∗ 2 2 ′ 2 ∗ ′ ′ ∗ π ′ G0 HG0 π ′ g π ′ G (D ) π ′ + π ′ G [G (D ) H, [G (D ),H]] G π ′ . ρ ,ρ ,ρ ρ ≥ ρ ρ ρ ρ λ,ρ ρ ρ λ,ρ ρ Furthermore, by spectral calculus of D′ one has the bound

2 ′ 2 ∗ 2 ′ 2 ∗ κ π ′ (D ) π ′ g π ′ (1 G (D ) )π ′ , ρ ρ ≥ ρ − ρ ρ 1 ′ ′ 2 because the bound holds for spectral parameters in [ 2 ρ, ρ ] due to (5) and 1 Gρ(D ) 1, 1 − ≤ while it holds trivially on [0, 2 ρ]. Since

1 G (D′)2 + G (D′)4 3 1 , − ρ ρ ≥ 4 it thus follows

2 3 2 ′ ′ ′ ′ ∗ L ′ (λ) g 1 ′ + π ′ G [G (D ) H, [G (D ),H]] + κ(D H + HD ) G π ′ , κ,ρ,ρ ≥ 4 ρ ρ λ,ρ ρ ρ λ,ρ ρ and in the special case λ = 0,

2 2 ′ ′ ′ ′ ∗ L ′ (0) g 1 ′ + π ′ G [G (D ) H, [G (D ),H]] + κ(D H + HD ) G π ′ . κ,ρ,ρ ≥ ρ ρ λ,ρ ρ ρ λ,ρ ρ Finally the error term is bounded using the tapering estimate (8):

[G (D′) H, [G (D′),H]] + κ(HD′ + D′H) 2 G (D′) H 8(ρ)−1 + κ [A, D] ρ ρ ≤ k ρ k k k −1 < A 8(g) + 1 κ [A, D] k k k k A 9 g−1 κ [A, D] ≤ k k k k 3 g2 , ≤ 4 ′ where the second inequality used the second inequality in (5) as well as Gρ(D ) 1, the third one A 1 and g 1, and ﬁnally the last inequality follows from thek ﬁrst inequalityk ≤ in k k ≥ ≤ 2 2 1 2 (5). Together one infers L ′ (λ) > 0 and L ′ (0) g 1 ′ . κ,ρ,ρ κ,ρ,ρ ≥ 4 ρ Finally, let us show that Sig (Lκ,ρ) = Sig (Lκ′,ρ′ ) , for pairs κ, ρ and κ′, ρ′ in the permitted range of parameters. Without loss of generality let ′ ρ ρ . Clearly Lκ,ρ is continuous in κ, a homotopy argument allows to reduce to the case κ =≤ κ′. Thus one needs to show

Sig (Lκ,ρ,ρ(0)) = Sig (Lκ,ρ,ρ′ (0)) ,

′ when ρ ρ and (5) is true for κ and ρ. Clearly L ′ (λ) is continuous in λ, so it suﬃces to ≤ κ,ρ,ρ prove Sig (Lκ,ρ,ρ(1)) = Sig (Lκ,ρ,ρ′ (1)) .

5 Consider ′ ∗ ′ ′ ∗ ′ ′ ′ Lκ,ρ,ρ (1) = κπρ D πρ′ + πρ Gρ(D ) HGρ(D )πρ′ . ′ ∗ Now D commutes with π ′ π ′ so that L ′ (1) decomposes into a direct sum. Let π ′ = π ′ π ρ ρ κ,ρ,ρ ρ ,ρ ρ ⊖ ρ be the surjective partial isometry onto ( ) ′ ( ) . Then H ⊕ H ρ ⊖ H ⊕ H ρ ′ ∗ L ′ (1) = L (1) π ′ κ D π ′ . κ,ρ,ρ κ,ρ,ρ ⊕ ρ ,ρ ρ ,ρ ′ ∗ ′ The signature of πρ ,ρ D πρ′,ρ vanishes so that

Sig(Lκ,ρ,ρ′ (1)) = Sig(Lκ,ρ,ρ(1)) .

3 The proof of index = half-signature via spectral ﬂow

Now let us turn to the main objective of this note, namely the proof of (7) using only the well-known properties of the spectral flow listed in the appendix. Due to the stability result of the signature proved in the last section, it is therefore sufficient to show that for some κ and ρ satisfying (5) the equality (7) holds. In the argument below, we will first choose κ sufficiently small, and then ρ sufficiently large. The starting point is the fundamental connection, described in the Appendix, between an index pairing and the spectral flow:

Ind(T ) = Ind(ΠUΠ+ 1 Π) = Sf(U ∗(2Π 1)U, 2Π 1) . (9) − − − Here the unitary U = A A −1 is the polar of A. Deforming 2Π 1 into κ D does not lead to | | − spectral flow and therefore Ind(T ) = Sf(κ U ∗DU, κ D) . (10) On the r.h.s. appears the spectral flow along the straight-line path t [0, 1] (1 t)κ U ∗DU + t κ D between two unbounded operators. As U ∗DU D = U ∗[D, U]∈ is bounded,7→ it− is intuitively clear that this only invokes low lying spectrum of D−. Technically, the spectral flow between unbounded operators can be defined (see the Appendix) by replacing D by an increasing bounded function of D such as tanh(D). To be precise, let Fρ : R R be an increasing smooth function with F (x)= x for x ρ and F (x)=2ρ = F ( x) for→ x 2ρ. Then ρ | | ≤ ρ − ρ − ≥ ∗ ∗ Sf(κ U DU, κ D) = Sf(κ U Fρ(D)U, κ Fρ(D)) , where on the r.h.s. appears the spectral flow for the straight line between two bounded operators. ∗ Note that indeed Fρ(D) 2ρ(2Π 1) is compact so that also the difference U Fρ(D)U Fρ(D) − − ∗ − is compact, and hence the straight line from κ U Fρ(D)U to κ Fρ(D) is indeed inside the selfadjoint Fredholm operators. Furthermore, this straight-line path can be deformed into the one on the r.h.s. of (9) (still within the selfadjoint Fredholm operators) which shows (10). In the following, one should strictly speaking always replace D by Fρ(D). As only the equality Fρ(D) = D on ρ is relevant in the following, we decided to stick with the formulation with unbounded operatorsH for sake of clarity of the argument.

6 Hence let us continue from (10). Using the additivity of the spectral flow leads to ∗ U 0 κ D 0 U 0 κ D 0 Ind(T ) = Sf , 0 1 0 κ D 0 1 0 κ D − − ∗ κ D 1 U 0 κ D 0 U 0 = Sf , 1 κ D 0 1 0 κ D 0 1 − − ∗ κ D 0 U 0 κ D 1 U 0 = Sf , 0 κ D 0 1 1 κ D 0 1 − − κ D 0 κUDU ∗ U = Sf , ∗ , (11) 0 κ D U κ D − − where in the second to last step a mass term was added. This only further opens the spectral gap of the end points of the path and hence does not modify the spectral flow. Now the right entry in (11) of the spectral flow in the last equation is essentially the spectral localizer of U w.r.t. D, provided that one can replace the upper left entry by κD. The spectral localizer of U has a gap by the argument above (which also covers the case ρ = ). As UDU ∗ = D [D, U]U ∗ and [D, U] is bounded, this gap does not close along the linear∞ path connecting −UDU ∗ to D for κ sufficiently small, so that also the above left entry remains invertible, namely κUDU ∗ U κ D U Sf ∗ , ∗ = 0 , U κ D U κ D − − where again the straight line path between the arguments is taken. Due to the concatenation property of the spectral flow one concludes that, for κ sufficiently small κ D 0 κ D U Ind(T ) = Sf , ∗ . 0 κ D U κ D − − Deforming U to A via polar decomposition shows κ D 0 κ D A Ind(T ) = Sf , ∗ . 0 κ D A κ D − − The right entry is the spectral localizer. The final step is to localize this formula on in Hρ the decomposition = ρ ρc . Let πρc be the surjective partial isometry onto ρc . For ∗ ∗ H H ⊕ H c c c H any operator A, let Aρ = πρAπρ and Aρ = πρ A(πρ ) be the two restrictions with Dirichlet boundary conditions. The Dirac operator D is diagonal w.r.t. this decomposition, that is D = D D c . The spectral localizer is not diagonal in this grading, but we will homotopically ρ ⊕ ρ deform it into something diagonal by ∗ 0 πρH(πρc ) Lκ(t) = Lκ,ρ Lκ,ρc + t ∗ . (12) ⊕ πρc Hπ 0 ρ

Note that the perturbation on the r.h.s. of (12) is compact. For t = 1 one has Lκ(1) = Lκ. We will now show that the path t [0, 1] L (t) is within the invertibles, provided ρ is suﬃciently ∈ 7→ κ 7 large. Indeed, Lκ,ρc is invertible and has a central gap around 0 growing linearly in ρ because 2 2 ∗ c c c c c c 2 κ Dρc + Aρ (Aρ ) κ(Dρ Aρ Aρ Dρ ) (Lκ,ρc ) = ∗ 2 2 ∗ c c c c − c c κ(Dρ Aρ Aρ Dρ ) κ Dρc + (Aρ ) Aρ 2 2 − (κ ρ 12 κ [D, A] ) 1 c . (13) ≥ − k k ρ 1 2 2 This remains strictly positive and actually larger than 2 κ ρ if (4) holds. Moreover, the other g diagonal entry Lκ,ρ has a gap bounded below by 2 . Hence Lκ(t) is given by

− 1 ∗ − 1 1 2 c c 2 1 2 0 Lκ,ρ πρHπρ Lκ,ρ 2 Lκ,ρ Lκ,ρc S + t − 1 ∗ − 1 | | | | Lκ,ρ Lκ,ρc , | ⊕ | L c 2 π c Hπ L 2 0 | ⊕ | | κ,ρ | ρ ρ| κ,ρ| !! c where S is a diagonal selfadjoint unitary (diagonal in the direct sum = ρ ρ). As the oﬀ-diagonal entries satisfy H H ⊕ H

− 1 ∗ − 1 C 2 2 Lκ,ρc πρc Hπ Lκ,ρ | | ρ| | ≤ √κρg for some constant C, they are smaller than 1 in norm for ρ suﬃciently large. Thus one concludes that that L (t) is invertible for all t [0, 1], provided ρ is suﬃciently large. Consequently, as κ ∈ always for the straight line path,

Sf(L , L L c ) =0 . κ κ,ρ ⊕ κ,ρ Therefore by concatenation and the additivity of the spectral ﬂow

κ D 0 κ Dρ 0 Ind(T ) = Sf , Lκ,ρ Lκ,ρc = Sf , Lκ,ρ , 0 κ D ⊕ 0 κ Dρ − − because the above inequality (13) shows that there is no spectral flow on ρc . Now for finite dimensional selfadjoint matrices, the spectral flow is the difference of signaturesH of the matrices divided by 2. As the signature of diag(D , D ) vanishes, the equality (7) follows. ρ − ρ A Review of spectral flow

For the convenience of the reader, we briefly collect the main relevant information from [10, 11, 5] on the spectral flow. Let t [0, 1] F be a continuous path of bounded selfadjoint Fredholm ∈ 7→ t operators. Then there is an ǫ > 0 such that Ft has no essential spectrum in ( ǫ, ǫ) for all t [0, 1]. There may, however, be discrete spectrum (isolated eigenvalues of finite− multiplicity) of∈T in the interval ( ǫ, ǫ). Intuitively, the spectral flow is then the number of eigenvalues t − moving past 0 in the positive direction minus the number of those eigenvalues moving past 0 in the negative direction. For an analytic path t [0, 1] F , a rigorous definition of the spectral ∈ 7→ t flow Sf(t [0, 1] Ft) can immediately spelled out due to analytic perturbation theory [1]. For merely∈ continuous7→ paths, Phillips give a careful definition of the spectral flow by using spectral projections at discrete times [10]. While the reader is referred to [10] for the definition, let us recall the main basic properties of the spectral flow that are used in the proof above:

8 (i) (Homotopy invariance) Let s [0, 1] F (s) be a homotopy of paths with ﬁxed end ∈ 7→ t points F0(s) and F1(s). Then

Sf(t [0, 1] F (0)) = Sf(t [0, 1] F (1)) . ∈ 7→ t ∈ 7→ t (ii) (Concatenation) For paths t [0, 1] F and t [1, 2] F , ∈ 7→ t ∈ 7→ t Sf(t [0, 1] F ) + Sf(t [1, 2] F ) = Sf(t [0, 2] F ) . ∈ 7→ t ∈ 7→ t ∈ 7→ t (iii) (Unitary invariance) For any unitary U,

Sf(t [0, 1] U ∗F U) = Sf(t [0, 1] F ) . ∈ 7→ t ∈ 7→ t (iv) (Additivity) For paths t [0, 1] F and t [0, 1] F ′, ∈ 7→ t ∈ 7→ t ′ ′ Sf(t [0, 1] F F ) = Sf(t [0, 1] F ) + Sf(t [0, 1] F ) . ∈ 7→ t ⊕ t ∈ 7→ t ∈ 7→ t (v) For a path t [0, 1] F with 0 not in the spectrum σ(F ) of F for all t [0, 1], ∈ 7→ t t t ∈ Sf(t [0, 1] F )= 0 . ∈ 7→ t The starting point for the proof in Section 3 is the connection between spectral ﬂow and index pairings. Let P be a projection and U a unitary such that [U, P ] is a compact operator, then PUP + 1 P is a Fredholm operator (with pseudo-inverse P U ∗P + 1 P ). Associated is − − ∗ a selfadjoint Fredholm operator F = 2P 1 as well as a path t [0, 1] t F + (1 t)U F U of selfadjoint Fredholm operators, with end− points U ∗F U and F .∈ Then 7→ −

Ind(PUP + 1 P ) = Sf(t [0, 1] t F + (1 t)U ∗F U) . (14) − ∈ 7→ − The first proof of the equality (14) seems to be in [11], an alternative homotopy argument can be found in [5]. For sake of brevity, the spectral flow on the r.h.s. (14) is also denoted by ∗ Sf(U F U, F ). More generally, given two selfadjoint Fredholm operator F0 to F1 with compact difference F1 F0, we will always write Sf(F0, F1) for the spectral flow of the straight-line path, − that is, Sf(F0, F1) = Sf(t [0, 1] (1 t) F0 + t F1) . ∈ 7→ −

Finally let us add a comment on the spectral flow of paths t [0, 1] Dt of unbounded selfadjoint Fredholm operators (continuity w.r.t. the graph topolo∈gy [2]).7→ One then obtains a path t [0, 1] F = tanh(D ) of bounded selfadjoint Fredholm operators and can use its ∈ 7→ t t spectral flow to define the spectral flow of t [0, 1] Dt. Instead of tanh any increasing smooth function F with F (0) = 0 and F ′(0)∈> 0 can7→ be used. All of the above properties naturally transpose to the unbounded case.

9 References

[1] M. F. Atiyah, V. K. Patodi, I. M. Singer, Spectral asymmetry and Riemannian geometry. III, Mathematical Proceedings of the Cambridge Philosophical Society, vol. 79, pp. 71-99. Cambridge University Press, 1976.

[2] M. T. Benameur, A. L. Carey, J. Phillips, A. Rennie, F. A. Sukochev, K. P. Wojciechowski, An analytic approach to spectral ﬂow in von Neumann algebras, pp. 297-352, in: Analysis, Geometry and Topology of Elliptic Operators (World Scientiﬁc, Singapure, 2006).

[3] A. L. Carey, J. Phillips, Spectral ﬂow in Fredholm modules, eta invariants and the JLO cocycle, K-Theory 31, 135-194 (2004).

[4] A. Connes, Noncommutative Geometry, (Academic Press, San Diego, 1994).

[5] G. De Nittis, H. Schulz-Baldes Spectral ﬂows of dilations of Fredholm operators, Canad. Math. Bulletin 58, 51-68 (2015).

[6] J. M. Gracia-Bond´ıa, J. C. V´arilly, H. Figueroa, Elements of noncommutative geometry, (Springer Science & Business Media, 2013).

[7] J. Grossmann, H. Schulz-Baldes, Index pairings in presence of symmetries with applications to topological insulators, Commun. Math. Phys. 343, 477-513 (2016).

[8] T. A. Loring, K-theory and pseudospectra for topological insulators, Annals of Physics 356, 383-416 (2015).

[9] T. A. Loring, H. Schulz-Baldes, Finite volume calculations of K-theory invariants, New York J. Math. 22, 1111-1140 (2017).

[10] J. Phillips, Self-adjoint Fredholm operators and spectral ﬂow, Canad. Math. Bull. 39, 460-467 (1996).

[11] J. Phillips, Spectral Flow in Type I and Type II factors-a New Approach, Fields Institute Communications vol. 17, 137-153 (1997).

[12] E. Prodan, H. Schulz-Baldes, Bulk and boundary invariants for complex topological insulators: From K-theory to physics, (Springer Int. Pub., Szwitzerland, 2016).