Multiple operator integrals, pseudodifferential calculus, and asymptotic expansions
Eva-Maria Hekkelman Edward McDonald Teun D. H. van Nuland


Learning Syntax Without Planting Trees: Understanding When and Why Transformers Generalize Hierarchically
    Kabir Ahuja1  Vidhisha Balachandran2   Madhur Panwar3    Tianxing He1   Noah A. Smith1   Navin Goyal3   Yulia Tsvetkov1 



1 University of Washington 
2 Carnegie Mellon University 
3 Microsoft Research





    
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    We push the definition of multiple operator integrals (MOIs) into the realm of unbounded operators, using the pseudodifferential calculus from the works of Connes and Moscovici, Higson, and Guillemin. This in particular provides a natural language for operator integrals in noncommutative geometry. For this purpose, we develop a functional calculus for these pseudodifferential operators. 
    To illustrate the power of this framework, we provide a pertubative expansion of the spectral action for regular s-summable spectral triples (𝒜, ℋ, D), and an asymptotic expansion of Tr(P e^-t(D+V)^2) as t ↓ 0, where P and V belong to the algebra generated by 𝒜 and D, and V is bounded and self-adjoint.







§ INTRODUCTION



 §.§ Operator Integrals

[2010 Mathematics Subject Classification:47A55; 47A60, 47G30, 58B34, 47F10.]
[Key words and phrases: Multiple Operator Integrals, Noncommutative Geometry]
Operator integrals appear in various areas of noncommutative geometry (NCG) <cit.>, one of the most prominent examples of which is the JLO cocycle <cit.>

    ∫_Σ^n(η a^0 e^-t_0 D^2 [D,a_1] e^-t_1D^2⋯  [D,a_n]e^-t_nD^2)dt .

The expansions and other identities derived for these operator integrals are often very similar in nature, and it would therefore be useful to have a framework to systematize these techniques. The language tailored to provide such a framework is that of multiple operator integrals (MOIs) <cit.>. However, the existing literature on MOIs is not equipped to make sense of multiple operator integrals as pseudodifferential operators in the style of Connes–Moscovici, Higson and Guillemin <cit.>, which would be necessary for advanced applications in NCG.

We therefore first generalise the MOI construction of <cit.>, realising the operator integral as an abstract pseudodifferential operator on the condition that the operator integrand is itself an abstract pseudodifferential operator. Then we develop a functional calculus for elliptic pseudodifferential operators, which completes the technical framework. Finally, we derive various identities for these MOIs that are often used in the NCG literature, and conclude by showing that these identities can be used to provide a perturbative expansion of the spectral action and prove a new result on the existence of asymptotic expansions in regular s-summable spectral triples (, ,̋ D) of

    (P e^-t(D+V)^2)

as t↓ 0, where V is some self-adjoint bounded element in the algebra generated by D and , and P is an arbitrary element in this algebra. The assumptions needed to deduce the existence of these asymptotic expansions was an open problem, communicated to the authors by Bruno Iochum.

Multiple operator integrals find their origin in the work of Daletskiĭ–Kreĭn <cit.>, Birman–Solomyak <cit.>, Pavlov <cit.> and Sten'kin <cit.>. In the theory that we adapt from <cit.>, classical MOIs have a symbol ϕ: ^n+1→ which can be written as 

    ϕ(λ_0, …, λ_n) = ∫_Ω a_0(λ_0, ω) ⋯ a_n(λ_n, ω) dν(ω),

for some finite measure space Ω and bounded measurable functions a_i: ×Ω→. A MOI with such a symbol is the well-defined operator 

    T_ϕ^H_0, …, H_n(X_1, …, X_n) := ∫_Ω a_0(H_0, ω) X_1 a_1(H_1, ω) ⋯ X_n a_n(H_n ,ω) dν(ω),

where H_i are closed densely defined self-adjoint operators on a separable Hilbert space, and X_i are bounded operators. For n=0 this is simply functional calculus, but for n > 0 MOIs are multilinear maps

    T_ϕ^H_0, …, H_n   : B()̋×⋯× B()̋_n→ B()̋.

Here, and throughout the paper, B()̋ denotes the algebra of all bounded linear operators on a separable Hilbert space $̋. The notationT_ϕ^H_0, …, H_nis justified since this multilinear map does not depend on the chosen representation (<ref>) (see <cit.>). To ease notation, ifH_0 = ⋯= H_n =: Hwe simply writeT_ϕ^H := T_ϕ^H_0, …, H_n.

The quintessential symbol for MOIs is a divided difference, namely, forf:→regular enough we can recursively define

    f^[0](λ_0):=f(λ_0),    f^[n](λ_0, …, λ_n): = f^[n-1](λ_1, …, λ_n) - f^[n-1](λ_0, …, λ_n-1)/λ_0 - λ_n,   λ_0 ≠λ_n,

replacing the above by a limit in caseλ_0 = λ_n, so that

    f^[n](λ, …, λ) = 1/n!f^(n)(λ).

For a nice enough functionfthe divided differencef^[n]:^n+1→admits a decomposition of the required form, and it then holds that

    1/n!d^n/dt^n|_t=0 f(H+tV) = T_f^[n]^H(V, …, V) ∈ B()̋.

For commuting operators, this formula reduces to the chain rule.

MOIs with divided differences as their symbols appear in NCG in the context of spectral flow <cit.>, spectral shift <cit.>, the spectral action <cit.>, the heat trace expansion <cit.>, and cyclic cocycles <cit.>. The JLO cocycle mentioned earlier can explicitly be written as 

    ∫_Σ^n(η a^0 e^-t_0 D^2 [D,a_1] e^-t_1D^2⋯  [D,a_n]e^-t_nD^2)dt = (η a_0 T_f^[n]^D^2([D,a_1], …, [D,a_n])),

wheref(x) = exp(-x). The properties and expansions of these cocycles are derived from a small list of MOI-identities, which have also proved instrumental in <cit.>. Under reasonable assumptions, these identities are
    
    f(H+V)-f(H)   =T_f^[1]^H+V,H(V),
    
            [f(H),a]   =T_f^[1]^H,H([H,a]),

    and their higher-order analogues, cf. Proposition <ref>.

    Applying an identity like (<ref>) to the JLO cocycle whereH= D^2gives an unbounded argument[D^2, a]in the multiple operator integral, for which the current theory of multiple operator integrals <cit.> is not applicable. In literature on noncommutative geometry, ad-hoc arguments have been made to handle this <cit.>, but never by generalising multiple operator integrals techniques to deal with unbounded arguments. 



 §.§ Summary of Main Results

    
    The pseudodifferential calculus we use in our approach was pioneered by Connes–Moscovici, Higson and Guillemin <cit.>. A succinct overview of this calculus is given in <cit.>.



    Let Θ be a possibly unbounded invertible positive self-adjoint operator on a separable Hilbert space $̋. Define the Hilbert spaces^̋s := Θ^s^·_sfors ∈ℝwhereϕ_s : = Θ^s ϕ– though taking this closure is not necessary fors≥ 0. We write^̋∞ := ⋂_s ≥ 0^̋s, which is dense in$̋. We say that a linear operator A: ^̋∞→^̋∞ is in the class op^r(Θ) (it has analytic order ≤ r) if A extends to a continuous operator
    
    A^s+r,r: ^̋s+r→^̋s

    for all s∈ℝ. If no confusion can arise, we often write
    
    A: ^̋s+r→^̋s,

    and we write ^r instead of ^r(Θ) for brevity, := ⋃_r∈ℝ^r, and ^-∞:= ∩_r∈ℝ^r. Finally, we define ^r(Θ)⊆^r(Θ) as those A ∈^r for which δ^n_Θ(A) ∈^r for each n ≥ 0, where δ_Θ(A) := [Θ, A]. 


Both(Θ)and(Θ):= ⋃_r ∈ ^r(Θ)form a filtered algebra, as^r(Θ) ·^t(Θ) ⊆^r+t(Θ)and^r(Θ)·^t(Θ) ⊆^r+t(Θ). 

Commonly, this paradigm is also used when dealing with unbounded operators

    T: (T) →.̋

In this case, one writesT ∈^rif^̋∞⊆(T),T(^̋∞) ⊆^̋∞, and

    T|_^̋∞∈^r.

Conversely, forT ∈^rwithr>0,

    T^r,0: ^̋r ⊆→̋
    can be interpreted as an unbounded operator.
    Furthermore, note that ^-r⊆()̋ for r ≥ 0 in the sense that for A ∈^-r we have A^-r,0|_∈ B()̋. Similarly if Θ^-1∈ℒ_s()̋, a Schatten–von Neumann class, then ^-s⊆ℒ_1()̋.
    
    
    Let us provide a few examples of this abstract pseudodifferential calculus.
    
      *  Taking Θ = (1-Δ)^1/2 on L_2(ℝ^d), where Δ=∑_j=1^d ∂_j^2 is the Laplace operator, gives the classical (Bessel potential) Sobolev spaces ^̋s = W_2^s(ℝ^d). (Pseudo)differential operators of order k, and (unbounded) Fourier multipliers T_ϕ with symbols |ϕ(ξ)|≲ (1+|ξ|)^k, ξ∈ℝ^d are contained in ^k((1-Δ)^1/2). Note though that ((1-Δ)^1/2) is a larger class than this, which for example also contains translation operators.
        
      *   For spectral triples (, ,̋ D) taking Θ= (1+D^2)^1/2 recovers the calculus of Connes and Moscovici as used in noncommutative geometry.
        
      *  Taking Θ = (1-Δ)^1/2, where Δ is the sub-Laplacian on a stratified Lie group, gives the Sobolev spaces defined by Folland and Stein <cit.>. 
        
      *  Related to the previous example is the anharmonic oscillator Θ^2 = 1- Δ^2l + |x|^2k on ^d for integers l,k≥ 1 and generalisations thereof, which define Sobolev spaces and a pseudodifferential calculus that appear in the study of sub-Laplacian operators on stratified Lie groups too <cit.>. The special case where Θ^2 is the harmonic oscillator gives Shubin's Sobolev spaces Q^s(^d) <cit.>, see also <cit.>.
        
      *  In recent work by Androulidakis, Mohsen and Yuncken, Sobolev spaces are constructed with a similar procedure in order to prove the Helffer–Nourrigat conjecture <cit.>. 
        
      *  A similar calculus has been constructed for quantum Euclidean spaces <cit.>.
        
      *  Finally, we note the case where Θ = 1_, which gives that ^̋∞ = ^̋s = $̋ for alls ∈, and^r = B()̋,r ∈.
    
    
    In this formalism, we prove a generalisation of the MOI framework by Peller and Azamov–Carey–Dodds–Sukochev <cit.>. Recall the following definition, cf. <cit.>, <cit.>. Throughout this paper, we use the notation⟨ x ⟩ := (1+|x|^2)^1/2. 
    
    
    Let E be a spectral measure on  with the Borel sigma algebra. For a Borel measurable function f:→ we define the essential supremum seminorm
    f_L_∞(E):=sup{y∈ : E(|f|^-1((y,∞)))=0},
    which defines L_∞(E) in the usual way, namely as the quotient of the set of measurable functions with finite seminorm, by the set of those of zero seminorm. In the same way we define L_∞^β(E) by the seminorm
    f_L_∞^β(E):=x↦ f(x)⟨ x⟩^-β_L_∞(E),
    for any β∈, where ⟨ x ⟩ = (1+x^2)^1/2.
    
    Let H_0,… H_n be self-adjoint operators on $̋ with spectral measuresE_j, and letϕ:^n+1→be of the form
    
        ϕ(λ_0,…,λ_n)=∫_Ω a_0(λ_0,ω)⋯ a_n(λ_n,ω) dν(ω),
    
    for a finite measure space(Ω,ν)and measurable functionsa_j:×Ω→such that(x,ω)↦ a_j(x,ω)⟨ x ⟩^-β_jisE_j ×ν-a.e. bounded forβ_j ∈. Suppose that we havea_j(H_j,ω)∈^k_j,k_j ∈, and 
    
        a_j(H_j,ω)_^s+k_j→^s≤ C_s,H_ja_j(·,ω)_L^β_j_∞(E_j)
    
    for everyj ∈{0, …, n},s∈, andω∈Ω, and certain constantsC_s,H_j∈. Then the integral
    
        T_ϕ^H_0,…,H_n(X_1,…,X_n)ψ:=∫_Ω a_0(H_0,ω)X_1 a_1(H_1,ω)⋯ X_n a_n(H_n,ω)ψ dν(ω),   ψ∈^̋∞,
    
    forX_1,…,X_n∈, converges as a Bochner integral in^̋sfor everys ∈, and defines ann-multilinear mapT_ϕ^H_0,…,H_n:^r_1×⋯×^r_n→^∑_j r_j + ∑_j k_jdepending onΩanda_0,…,a_nonly through the symbolϕ. Specifically, fors∈we have the estimate
       
        T_ϕ^H_0,…, H_n(X_1, …,X_n)_^̋s+∑_jr_j + ∑_j k_j→^̋s≲∏_j=1^n X_j_^̋s_j+r_j→^̋s_j∫_Ω∏_j=0^n  a_j(·, ω)_L^β_j_∞(E_j) d|ν|(ω),
    
        for somes_1, …, s_n ∈.
    
    
    ForΘ = 1_andβ_0 = ⋯ = β_n = 0, it is immediate thata_j(H_j, ω) ∈(1_)= B()̋and
    
        a_j(H_j, ω)_^̋s→^̋s = a_j(·, ω)_L^0_∞(E_j),
    
    and the above theorem reduces to Peller's and Azamov–Carey–Dodds–Sukochev's construction of MOIs <cit.>. For generalΘ, the question is for which self-adjoint operatorsHand functionsfwe havef(H) ∈(Θ)with the required norm estimate (<ref>). In particular, forH ∈^h,h>0, the extensionH^r,0can be considered an unbounded operator on$̋ and hence (if it is self-adjoint) we might ask whether f(H^r,0) ∈.
    We draw inspiration from the traditional theory of pseudodifferential operators and consider elliptic symmetric operators in .
    
    
    We say that an operator A ∈^r, r≥0, is symmetric or self-adjoint with domain ^̋r respectively, if the extension
    
        A^r,0: ^̋r ⊆^̋0 →^̋0,
    
    regarded as an unbounded operator on ^̋0, is a symmetric or self-adjoint operator. See Section <ref> and Appendix <ref> for an investigation of these properties.
    
    We say that A∈^r(Θ) is elliptic if there exists a parametrix P ∈^-r(Θ) such that
    
        AP    = 1_^̋∞ + R_1;
        
            PA    = 1_^̋∞ + R_2,
    
    where R_1, R_2 ∈^-∞(Θ). This is similar to <cit.>. The notion of ellipticity depends on Θ and on the order r ∈, however, we simply abbreviate this to `A ∈^r is elliptic'.
    
    Elliptic operators are discussed in more depth in Section <ref>, where it is shown that symmetric elliptic operators in ^r, r≥ 0 are self-adjoint with domain ^̋r. Therefore, when we mention the spectrum or spectral measure of such operators, we are referring to the spectrum or spectral measure of the extension A^r,0.
    
    
    
    
       Let A∈^r, r >0, be elliptic and symmetric, and let E denote its spectral measure. If f ∈ L^β_∞(E), β∈, then 
            
        f(A^r,0) ∈^β r,
    
            and we simply write f(A) := f(A^r,0).
            More precisely,
            
        f(A)_^̋s+β r→^̋s≤ C_s,Af _L^β_∞(E).
    
    
    This theorem is proved in Section <ref>. The result might be surprising: for traditional elliptic pseudodifferential operators, a similar functional calculus is only constructed with smooth functions f in S^β() <cit.><cit.><cit.>. A functional calculus for ^0(Θ) is developed in Appendix <ref>.
    
     For I⊆ an interval and β∈, we define S^β(I) as the class of smooth functions f: I → such that
    
        f_S^β(I), k := sup_x∈ I |f^(k)(x)| ⟨ x ⟩^k-β < ∞,    k ∈_≥ 0.
    
    The quantities above are seminorms. Similarly, we define T^β(I) as the space of smooth functions f: I → such that 
        
        f_T^β(I), k:= ∫_I |f^(k)(x)|⟨ x⟩^k-β-1dx < ∞,    k ∈_≥ 0.
    
        In case I =, we note the inclusions
        
        ⋃_α<β S^α() ⊊ T^β() ⊊ S^β() ⊊ L^β_∞(E)
    
        for any spectral measure E.
    
    
    
        Let H_0, …, H_n be such that each H_i ∈^h_i, h_i > 0, is symmetric and elliptic with spectral measure E_i.
        Let ϕ: ℝ^n+1→ℂ be such that
        
        ϕ(λ_0, …, λ_n) = ∫_Ω a_0(λ_0, ω) ⋯ a_n(λ_n, ω) dν(ω),
    
        for a finite measure space Ω and measurable functions a_i: ×Ω→ such that (x,ω)↦ a_i(x,ω)⟨ x ⟩^-β_i is E_i ×ν-a.e. bounded for β_i ∈.
        Then Theorem <ref> applies, and for operators X_i ∈^r_i, r:= ∑_i=1^n r_i, we have that 
        
        T_ϕ^H_0, …, H_n(X_1, …, X_n) ∈^r+∑_i=0^n β_i h_i
    
        independent of the chosen representation of ϕ.
         
         In particular, if h_0 = ⋯ = h_n=:h and f ∈ C^n+2() such that f_T^β(), k <∞, k=0, …, n+2 for some β∈, then
         
        T^H_0, …, H_n_f^[n](X_1, …, X_n) ∈^r+(β-n)h,
    
         with the estimate
         
        T^H_0, …, H_n_f^[n](X_1, …, X_n)_^̋s+r+ (β-n)h→^̋s≤ C_s,H_0, …, H_n(∑_k=0^n+2f_T^β(), k) ∏_i=1^n X_i_^̋s_i+r_i→^̋s_i
    
         for some s_1, …, s_n ∈.    
    
        The first part of the theorem is a combination of Theorem <ref> and Theorem <ref>. The second part is a consequence of the fact that under the listed assumption for f∈ C^n+2(), the divided difference f^[n] satisfies the conditions required of a symbol in Theorem <ref>, which is proved through Lemma <ref> and Remark <ref>.
    
        The functional calculus for elliptic symmetric operators in (Θ) in Theorem <ref> can be extended to self-adjoint operators H (not necessarily in (Θ)) that strongly commute with some elliptic operator in (Θ) of positive order. See Proposition <ref> for details. The multiple operator integral theory presented in Theorem <ref> therefore also extends to the case where the operators H_i are of this type.
    
    
    
    Theorem <ref> can of course be applied when all operators H_i and X_i are of the class (Θ) ⊆(Θ), and it may be of interest to know when the resulting operator is again of the class (Θ). In case the symbol is a divided difference we have the following result, proved in Section <ref>.
    
    
    
        If H_0, …, H_n ∈^h(Θ) are symmetric and elliptic of the same order h>0 and X_j ∈^r_j(Θ), r:=∑_j=1^n r_j, and if f ∈ T^β(), then the multiple operator integral T^H_0, …, H_n_f^[n](X_1, …, X_n) ∈^(β - n)h + r(Θ) whose existence follows from Theorem <ref>, is again an element of (Θ), i.e.
        
        T^H_0, …, H_n_f^[n](X_1, …, X_n) ∈^(β - n)h + r (Θ).
    
        In particular, the n=0 case gives that
        
        f(H_0) ∈^β h(Θ).
    
        The operators T_ϕ^H_0,…, H_n(X_1, …, X_n) constructed in Theorem <ref>, Theorem <ref> and Theorem <ref> depend on the symbol ϕ only through the function value of ϕ on the spectra σ(H_0) ×⋯×σ(H_n) ⊆^n. Hence all statements regarding the operators of the form T^H_f^[n](X_1, …, X_n) where f ∈ C^n+2() or f ∈ T^β() remain valid if f is in C^n+2(I) or T^β(I) respectively, where I⊆ is an open neighbourhood of the spectrum of H.
    
    
    
    For a regular spectral triple, which means that , [D,] ⊆^0((1+D^2)^1/2), the applicability of the above theorems is obvious.
    
    Given these constructions for multiple operator integrals, various identities follow quite immediately, as will be shown in Section <ref>. Of particular interest are the following expansions. We say that <cit.>
        A ∼∑_k=0^∞ A_k
    
    for A, A_k ∈, if we have
    
        A - ∑_k=0^N A_k ∈^m_N
    
    with m_N ↓ -∞.
    
    
      *  Let f ∈ T^β(), take H ∈^h, h >0 elliptic and symmetric, let V ∈^r be symmetric. If the order of the perturbation V is strictly smaller than that of H, i.e. r < h, we have
        
        f(H + V) ∼   ∑_n=0^∞ T^H_f^[n](V, …, V).
    
      *  Let f ∈ T^β(), take H ∈^h, h >0 elliptic and symmetric, and take X_i ∈^r_i. If there exists some ε > 0 such that δ^n_H(X_i)∈^r_i + n (h-ε) for all i and all n≥ 0, then we have an asymptotic expansion
        
        T^H_f^[n](X_1, …, X_n) ∼∑_m=0^∞∑_m_1 + … + m_n = mC_m_1, …, m_n/(n+m)!δ_H^m_1(X_1) ⋯δ_H^m_n(X_n) f^(n+m)(H) ,
    
        where 
        
        C_m_1, …, m_n :=  ∏_j=1^nj+m_1 + … + m_j-1m_j.
    
      *  Under the assumptions of the first two parts combined, we obtain
        
        f(H+V) ∼∑_n,m=0^∞∑_m_1 + … + m_n = mC_m_1, …, m_n/(n+m)!δ_H^m_1(V) ⋯δ_H^m_n(V) f^(n+m)(H).
    
    
    This theorem is proved through Theorem <ref>, Proposition <ref> and Corollary <ref>. The first part of the theorem above should be interpreted as a Taylor expansion. Special cases of the second part of the theorem have proved essential in proofs of the local index formula <cit.>, where they have been derived in a case by case basis. A version for C^*-algebras has also appeared in <cit.>, and a version for classical pseudodifferential operators in <cit.>. The condition that δ^n_H(X_i) ∈^r_i + n(h-ε) is for example satisfied for H = Θ and X_i ∈(Θ), with ε = 1.
    
    Apart from its use in finding asymptotic expansions of trace formulas, a seemingly disparate application of the above theorem is given by the following corollary which is also used throughout the literature on noncommutative geometry. In particular, a version appears as Theorem B.1 in Connes–Moscovici <cit.>. It has been used repeatedly in the noncommutative geometry literature, see for example <cit.>, and for a more recent example <cit.>.
    
    
        For X ∈^r(Θ) and f ∈ T^β(_>ε) we have
        
        [f(Θ), X] ∼∑_k=1^∞1/k!δ_Θ^k(X) f^(k)(Θ).
    
        In particular,
        
        [Θ^α, X] ∼∑_k=1^∞αkδ_Θ^k(X) Θ^α - k,   α∈,
    
        and
        
        [log(Θ), X] ∼∑_k=1^∞ (-1)^k-1/kδ_Θ^k(X) Θ^- k,
    
        and we have that [Θ^α, X] ∈^r+(α)-1(Θ) and [log(Θ),X] ∈^r-1(Θ).
    
        This is a simple combination of Theorem <ref> with the identity
    
        [f(Θ), X] = T^Θ_f^[1]([Θ, X])
    
    mentioned earlier, which will be proved in Proposition <ref>.
    
    
    Finally, the expansions in Theorem <ref> can be refined to provide trace expansions. We apply this to prove a pertubative expansion of the spectral action in noncommutative geometry <cit.><cit.>, and a result on the existence of asymptotic expansions of the heat trace of the perturbed Dirac operator.
    For the purpose of understanding the following theorem, one only needs to know that a regular s-summable spectral triple (, ,̋ D) consists of a Hilbert space $̋, an algebra of bounded operatorsand an unbounded self-adjoint operatorD. The triple being regular (sometimes called smooth orQC^∞) means that if we writeΘ = (1+D^2)^1/2, we have
    
        , [D,] ⊆^0(Θ),
    
    and thes-summability means thatΘ^-1∈ℒ_s. For a more indepth overview, see <cit.>.
    
    
    Let (,,̋D) be a regular s-summable spectral triple, s> 0. Let V, P ∈, V self-adjoint and bounded, where  is the algebra generated by  and D. 
       Then for all f ∈ T^β() with β < -s, the expressions
    
        (f(tD + tV)),     (Pe^-t(D+V)^2)   and  (Pe^-t|D+V|)
    
    admit an asymptotic expansion as t ↓ 0 given respectively by
    
        (f(tD+tV)) = ∑_n=0^N∑_m=0^N∑_m_1 + … + m_n = m t^n+mC_m_1, …, m_n/(n+m)!( δ_H^m_1(V) ⋯δ_H^m_n(V) f^(n+m)(tD)) + O(t^N+1-s),
    
    where C_m_1, …, m_n is the same as in Theorem <ref>, 
    
        (Pe^-t(D+V)^2) = ∑_n = 0^N∑_m=0^N∑_m_1 + … + m_n = m  (-t)^n+mC_m_1, …, m_n/(n+m)!(PA^(m_1)⋯ A^(m_n)exp(-tD^2)) + O(t^N+1-s/2),
    
    where A := DV + VD + V^2, and A^(m) := δ^m_D^2(A), 
    and
    
        (Pe^-t|D+V|) 
           = ∑_n = 0^N∑_m=0^N∑_m_1 + … + m_n = m  (-t)^n+mC_m_1, …, m_n/(n+m)!(Pδ_|D|^m_1(B)⋯δ_|D|^m_n(B)exp(-t|D|)) + O(t^(N+1)(1-ε) - s),
    
    where B := |D+V| - |D| and ε > 0 can be picked arbitrarily small.
    
    
    This theorem is proved in Section <ref>. In the context the spectral action, this expansion is closely related to <cit.>. Both these cited papers and Theorem <ref> are based on a Taylor expansion of the spectral action, but the expansion in Theorem <ref> goes one step further by applying the second part of Theorem <ref>. Observe that we do not assume thatfis the Laplace transform of a measure on_+, which is a common condition in the literature.
    
    
    In Section <ref> we cover some preliminary material for this paper and prove Theorem <ref>. Section <ref> proves the existence of a functional calculus for elliptic symmetric pseudodifferential operators, and Section <ref> shows that divided differences of functions inT^β()satisfy the conditions required for symbols by Theorem <ref>, thus completing the proof of Theorem <ref>. Finally, in Section <ref> we prove some identities for our construction of a MOI and we finish with a number of results on asymptotic expansions. Appendix <ref> contains a brief study on adjoints of abstract pseudodifferential operators, Appendix <ref> gives a functional calculus for^0(Θ)and Appendix <ref> contains the (combinatorial) proof of one of the expansions in Section <ref>.
    
    Acknowledgements The authors wish to thank Fedor Sukochev and Dmitriy Zanin for helpful discussions and comments. We thank Bruno Iochum for communicating the question which inspired part of this paper, and for correspondence which improved the manuscript. Dmitriy Zanin provided help with the proof of Proposition <ref>.
    
    
    thmsection
    
    § MULTIPLE OPERATOR INTEGRALS AS PSEUDODIFFERENTIAL OPERATORS
    
    In this section we prove Theorem <ref>. The proof is a subtle modification of the proof presented in <cit.>, so the material presented here is mostly an adaptation of known results.
    
    
    
     §.§ Operator Integrals
    
    We start with standard definitions and results on measurability and integrability of operator valued functions. Throughout the paper we fix a separable Hilbert space$̋ and a positive invertible operator Θ on $̋, which yields separable Hilbert spaces^̋sby Definition <ref>.
    
        Let _̋0, _̋1 be separable Hilbert spaces and let (Ω, Σ, ν) be a measure space with complex measure. A function f: Ω→ B(_̋1, _̋0) is called weak operator topology measurable (weakly measurable for short) if for all η∈_̋0, ξ∈_̋1 the scalar-valued function
        
        ω↦⟨η, f(ω) ξ⟩__̋0,   ω∈Ω,
    
        is measurable. Similarly, f is said to be weak operator topology integrable if for all ξ and η the above map is integrable.
    <cit.>
    Let _̋0, _̋1, (Ω, Σ, ν) be as above, and let f: Ω→ B(_̋1, _̋0) be weakly measurable. Then the norm function
        
        ω↦f(ω)__̋1 →_̋0,   ω∈Ω,
    
        is measurable. If moreover
        
        ∫_Ω f(ω)__̋1 →_̋0 d|ν|(ω) < ∞,
    
        then there exists a unique I_f ∈  B(_̋1, _̋0) such that
        
        ⟨η, I_f ξ⟩__̋0 = ∫_Ω⟨η, f(ω)ξ⟩__̋0 dν(ω),   η∈_̋0, ξ∈_̋1,
    
        and
        
        I_f __̋1 →_̋0≤∫_Ω f(ω)__̋1 →_̋0 d|ν|(ω).
    
        We then write I_f = ∫_Ω f(ω) dν(ω).
    <cit.>
        Let (Ω, Σ, ν) be a σ-finite measure space, let a:×Ω→ be measurable and bounded, and let H be an (unbounded) self-adjoint operator on $̋. Then
        
        ω↦ a(H,ω)
    
        is weakly measurable.
    
    Though <cit.> is only formulated for bounded H, the unbounded case follows with the same proof. 
    
        Let _̋0, …, _̋2n+1 be separable Hilbert spaces, X_i ∈ B(_̋2i, _̋2i-1), and let f_i:Ω→ B(_̋2i+1, _̋2i ) be weakly measurable functions. Then
        
        Ω   → B(_̋2n+1, _̋0)
        ω   ↦ f_0(ω) X_1 f_1(ω) ⋯ X_n f_n(ω)
    
        is weakly measurable. Furthermore, if 
        
        ∫_Ω f_0(ω)__̋1 →_̋0⋯f_n(ω)__̋2n+1→_̋2n d|ν|(ω) < ∞,
    
        the map
        
        B(_̋1, _̋0) ×⋯× B(_̋2n-1, _̋2n-2)    → B(_̋2n, _̋0)
        
            (X_1, …, X_n)      ↦∫_Ω f_0(ω) X_1f_1(ω) ⋯ X_n f_n(ω) dν(ω),
    
    whose existence follows from Lemma <ref>, is so-continuous when restricted to the unit ball in each argument B(_̋2i, _̋2i-1).
    
        The first part of the lemma is a consequence of the fact that the pointwise product of weakly measurable functions is weakly measurable, see <cit.>.
        
        The so-continuity follows from the joint continuity of the multiplication
        
        (X_1, …, X_n) ↦  a_0(H_0, ω) X_1 a_1(H_1, ω) ⋯ X_n a_n(H_n, ω)
    
        in the strong operator topology when restricting to the unit balls <cit.>, in combination with the Dominated Convergence Theorem for the Bochner integral of Hilbert space-valued functions <cit.>.
    
    
    We use the notation·_∞for the essential supremum.
    
    
        Let (Ω, Σ, ν) be a σ-finite measure space, let a:×Ω→ be measurable and bounded, and let E be a spectral measure on $̋. Then the functions
        
        ω   ↦a(·, ω)_∞,     ω↦a(·, ω)_L^0_∞(E)
    
        are measurable.
    
        Both claims can be proved with the Fubini–Tonelli Theorem or by combining Lemma <ref> and Proposition <ref>. It is vital that ·_∞ is the essential supremum, and that E is a spectral measure on a separable Hilbert space as pointed out in <cit.>.
    
    
     §.§ Abstract Pseudodifferential Calculus
    
    In this subsection we cover some basics on the pseudodifferential calculus given in Definition <ref>. See also <cit.>.
    
    A quick first observation is that the embedding^̋s↪^̋tis continuous for alls ≥ t. It follows that^r ⊆^tforr ≤ t.
    
    Fors > 0, there is a pairing between^̋sand^̋-sgiven by
    
        ⟨ u, v⟩_(^̋s, ^̋-s) :=⟨Θ^s u, Θ^-s v ⟩_,̋   u ∈^̋s, v∈^̋-s.
    
    This pairing identifies^̋-swith the (continuous) anti-linear dual space of^̋sand vice-versa.
    
    The space^̋∞ = ⋂_s∈^̋sis a Fréchet space equipped with the norms·_s, s∈ℝ. By construction,^̋∞⊆^̋sfor anys ∈, and in fact^̋∞is dense in^̋s. Since a subspace of a separable metric space is itself separable it follows that every^̋sadmits an orthonormal basis consisting of vectors in^̋∞.
    
    
    We define^̋-∞as the continuous anti-linear dual space of^̋∞, which can be identified with
    
        ^̋-∞ = ⋃_s∈ℝ^̋s.
    
    This is anLF-space, in the sense of <cit.>. From this perspective^̋∞can be interpreted as a Schwartz space and^̋-∞as a space of distributions. 
    Givenu ∈^̋∞andv ∈^̋-∞it follows from (<ref>) thatv ∈^̋-sfor some particulars ∈. It is immediate thatu ∈^̋s, and we have
    
        ⟨ u, v ⟩_(^̋∞, ^̋-∞) = ⟨ u, v ⟩_(^̋s, ^̋-s).
    
        The Sobolev spaces ^̋s in Definition <ref> form an interpolation scale. That is, let s_0 ≤ s_1, r_0,r_1∈, and let 0<θ<1. Set
            
        s_θ := (1-θ)s_0+θ s_1,   r_θ = (1-θ)r_0+θ r_1.
    
            If T is a bounded linear map 
            
        T:^̋s_0→^̋r_0,   T|_^̋s_1:^̋s_1→^̋r_1,
    
            then T|_^̋s_θ is bounded from ^̋s_θ to ^̋r_θ for every θ. Moreover we have
            
        T_^̋s_θ→^̋r_θ≤T_^̋s_0→^̋r_0^1-θT_^̋s_1→^̋r_1^θ.
    
        After identifying ^̋s with a weighted L_2-space through the spectral theorem, this follows from the Stein–Weiss interpolation theorem for L_p-spaces <cit.>.
    
    
     §.§ Proof of Theorem <ref>
    
    
    Theorem <ref> and its proof are heavily inspired by <cit.>, see also <cit.>.
    
    
    We have self-adjoint operators H_0,…, H_n on $̋ with spectral measuresE_j, and a functionϕ:^n+1→of the form
    
        ϕ(λ_0,…,λ_n)=∫_Ω a_0(λ_0,ω)⋯ a_n(λ_n,ω) dν(ω),
    
    where(Ω,ν)is a finite measure space and the functions(x, ω) ↦ a_j(x,ω)⟨ x ⟩^-β_jare measurable andE_j ×ν-a.e. bounded. Furthermore,a_j(H_j,ω)∈^k_j,k_j ∈. 
    
    Fixω∈Ωand takeη, ξ∈^̋∞⊆ a_j(H_j, ω). Then <cit.> gives that
    
        a_j(H_j, ω) ξ = lim_n→∞ a_j(H_j, ω) χ_[-n, n](H_j) ξ,
    
    whereχ_[-n,n]is the indicator function of the interval[-n,n], becauseess sup_|λ|≤ n |a_j(λ, ω)| < ∞. Now Proposition <ref> gives that
    
        ω↦⟨η, a_j(H_j,ω) ξ⟩_^̋s =  lim_n→∞⟨Θ^2sη, a_j(H_j, ω) χ_[-n,n](H_j) ξ⟩_
    
    is measurable for alls∈.
    
    
    Let nowX_i ∈^r_i,i ∈{1, …, n}. 
     Fixs ∈and defines_0, …, s_2n+1∈with
     
        s_0:=s,    s_2n+1 := s + ∑_i=0^n k_i + ∑_i=1^n r_i,
    
     so that
        the operatorsa_j(H_j, ω)andX_jextend to bounded operators
        
        a_j(H_j, ω)    ∈ B(^̋s_2j+1, ^̋s_2j),
        
                X_j    ∈ B(^̋s_2j, ^̋s_2j-1).
    
        By the previous argument, 
        
        ω↦ a_j(H_j, ω) ∈ B(^̋s_2j+1, ^̋s_2j)
    
        is weakly measurable since^̋∞is dense in both^̋s_2jand^̋s_2j+1.
        
        Using assumption (<ref>), i.e.
        
        a_j(H_j,ω)_^s_2j+1→^s_2j≤ C_s,H_ja_j(·,ω)_L^β_j_∞(E_j),
    
        we have that
        
        ∫_Ωa_0(H_0, ω )X_1 a_1(H_1,ω)    ⋯ X_n a_n(H_n, ω) _^̋s_2n+1→^̋s_0dν(ω)
           ≲∏_j=1^n X_j_^̋s_2j→^̋s_2j-1∫_Ω∏_j=0^n  a_j(·, ω)_L^β_j_∞(E_j) d|ν|(ω) < ∞,
    
        where Lemma <ref> ensures the right-hand side is defined. This is a finite quantity sincea_j(x, ω) ⟨ x ⟩^-β_jisE_j ×ν-a.e.bounded andνis a finite measure space. Therefore, Lemma <ref>
        provides that
        
        ∫_Ω a_0(H_0, ω )X_1 a_1(H_1,ω) ⋯ X_n a_n(H_n, ω) dν(ω)
    
        defines an operator in the weak sense inB(^̋s_2n+1, ^̋s_0)with
        
        ∫_Ω a_0(H_0, ω )X_1 a_1(H_1,ω)    ⋯ X_n a_n(H_n, ω) dν(ω)_^̋s_2n+1→^̋s_0
           ≲∏_j=1^n X_j_^̋s_2j→^̋s_2j-1∫_Ω∏_j=0^n  a_j(·, ω)_L^β_j_∞(E_j) d|ν|(ω).
    
        With Pettis' theorem <cit.>, it now follows that forψ∈^̋s_2n+1, 
        
        ω↦ a_0(H_0, ω )X_1 a_1(H_1,ω) ⋯ X_n a_n(H_n, ω) ψ∈^̋s
    
        is Bochner integrable in^̋s. This holds in particular forψ∈^̋∞, and ass ∈was taken arbitrarily it follows that forψ∈^̋∞,
        
        ∫_Ω  a_0(H_0, ω )X_1 a_1(H_1,ω) ⋯ X_n a_n(H_n, ω) ψ dν(ω) ∈^̋∞.
    
        It is therefore clear that
        
        ∫_Ω  a_0(H_0, ω )X_1 a_1(H_1,ω) ⋯ X_n a_n(H_n, ω)  dν(ω) ∈^∑_j r_j + ∑_j k_j.
    
        
    That this operator is independent of the chosen representation ofϕϕ(λ_0,…,λ_n)=∫_Ω a_0(λ_0,ω)⋯ a_n(λ_n,ω) dν(ω)
    
    follows from the proof of <cit.>. Namely, givenη, ξ∈^̋∞, it is easy to check thatθ_η, ξ:^̋∞→^̋∞defined by
    
        θ_η, ξ (ψ) := ⟨η, ψ⟩_ξ,   ψ∈^̋∞,
    
    is an element of^-∞.
    The computations in <cit.> give that, forη_k, ξ_k ∈^̋∞, the integral
    
        ∫_Ω a_0(H_0, ω) θ_η_1, ξ_1 a_1(H_1, ω) ⋯θ_η_n, ξ_n a_n(H_n, ω) dν(ω) ∈ B()̋
    
    does not depend on the chosen representation ofϕ, and so neither does
    
        ∫_Ω a_0(H_0, ω) θ_η_1, ξ_1 a_1(H_1, ω) ⋯θ_η_n, ξ_n a_n(H_n, ω) dν(ω) |_^̋∞∈^-∞.
    
    Theso-density of the span of{θ_η, ξ : η, ξ∈^̋∞}inB(^̋s_2i, ^̋s_2i-1)combined with Lemma <ref> concludes the proof.
    
        The MOI constructed in Theorem <ref> is linear in its symbol:
        
        T_αϕ + βψ^H_0,…, H_n(X_1, …, X_n) = α T_ϕ^H_0,…, H_n(X_1, …, X_n) + β T_ψ^H_0,…, H_n(X_1, …, X_n),   α, β∈.
    
        If both ϕ, ψ: ^n+1→ have an integral representation of the required form over measure spaces Ω and Σ respectively, then αϕ + βψ can be decomposed appropriately as an integral over the disjoint union Ω⊔Σ. The assertion then follows by elementary arguments.
    
        The MOI constructed in Theorem <ref> is independent of the operator Θ defining the abstract pseudodifferential calculus in the following sense. If H_i and X_i are operators on $̋ such thatX_i |_^̋∞∈^r_i(Θ)anda_i(H_i, ω)|_^̋∞(Θ)∈^k_i(Θ)satisfying the conditions of Theorem <ref>, then the proof of Theorem <ref> shows that
        we can defineT_ϕ^H_0,…, H_n(X_1,…, X_n): ^̋∑_i r_i + ∑_i k_i→$̋ by
        
        T_ϕ^H_0, …, H_n(X_1,…, X_n) ψ = ∫_Ω a_0(H_0, ω) 
         V_1 a_1(H_1, ω) ⋯ V_n a_n(H_n, ω) ψ dν(ω) ∈,̋  ψ∈^̋∑_i r_i + ∑_i k_i,
    
        which is a map that, apart from the definition of its domain, does not depend on Θ.
    
    
    § FUNCTIONAL CALCULUS FOR ABSTRACT PSEUDODIFFERENTIAL OPERATORS
    
    The construction of multiple operator integrals as abstract pseudodifferential operators now hinges on finding a class of self-adjoint operatorsAon$̋ and a class of functions f such that f(A) ∈(Θ) with the appropriate estimate of norms (<ref>). To accomplish this, and in particular prove Theorem <ref>, we will take A ∈^r, r ≥ 0, such that A^r,0: ^̋r ⊆→̋$̋ is self-adjoint and study whenf(A^r,0)defined via the Borel functional calculus is an operator in(Θ). An obstacle for a naive approach is thatf(A^s+r,s)is not easily defined fors ≠0, asA^s+r,sis generally not normal or symmetric. 
    As for classical pseudodifferential operators, ellipticity provides the right notion for developing a functional calculus.
    
    
    
     §.§ Elliptic Operators
    
    To prepare the way for a functional calculus on the Sobolev scale, we will show in this subsection that forA ∈^r,r ≥ 0, elliptic and symmetric we have thatAis self-adjoint with domain^̋r. Furthermore, ifAis invertible in an appropriate sense, thenA^-1∈^-r. 
    
    First recall the definition of an elliptic operator in(Θ).
    
        We say that an operator A ∈^r is elliptic if there exists a parametrix for A of order -r, that is, there exist operators P_1, P_2 ∈^-r such that
    
        AP    = 1_^̋∞ + R_1;
        
            PA    = 1_^̋∞ + R_2,
    
    where R_1, R_2 ∈^-∞ = ⋂_s ∈^s.
    
    As emphasised in the introduction, the notion of ellipticity depends onΘand on the orderr ∈. 
    
    We quickly note that the definition of ellipticity above is equivalent to the formally weaker condition ofAhaving a right-parametrixP_1and a left-parametrixP_2, as it would then follow thatP_1 - P_2 ∈^-∞and henceP_1andP_2are both left- and right-parametrices.
    
    
        Let A ∈^r be elliptic. If the bounded extension
        
        A: ^̋s_0+r→^̋s_0
    
        admits a bounded inverse
        
        A^-1: ^̋s_0→^̋s_0+r,
    
        for a specific s_0 ∈, then A^-1|_^̋∞∈^-r. Simply writing A^-1 = A^-1|_^̋∞, we have that
        
        A^-1 A = A A^-1 = 1_^̋∞.
    
        Let P be a parametrix of A and take x ∈^̋∞, so that
        
        A^-1x    = (PA - R_2) A^-1(AP - R_1) x
           = PAPx - R_2 P x- PR_1 x + R_2 A^-1 R_1 x.
    
        Observe that for y ∈^̋t, t∈, we have A^-1R_1 y ∈^̋s_0+r, so that R_2 A^-1R_1 y  ∈^̋∞.
        Hence,
        
        R_2 A^-1 R_1 ∈^-∞,
    
        and therefore
        
        A^-1   = PAP - R_2 P - PR_1 + R_2 A^-1 R_1 
           ∈^-r + ^-∞ = ^-r.
    
    
    
    Recall the notion of an asymptotic expansion of pseudodifferential operators (<ref>).
    
        
            Let {A_k}_k=0^∞ be a sequence of linear operators from ^̋∞ to ^̋∞ for which A_k ∈^m_k(Θ) such that m_k↓ -∞
            as k→∞. There exists a linear operator A∈^m_0(Θ) such that
            
        A ∼∑_k=0^∞ A_k.
    
            Let η∈ C^∞_c() be equal to 1 in a neighbourhood of zero, and let {ε_k}_k=0^∞ be a sequence of positive numbers tending
            to zero in a manner to be determined shortly. Formally we define
            
        A := ∑_k=0^∞ A_k(1-η(ε_k Θ)).
    
            We will prove that {ε_k}_k=0^∞ can be chosen such that this series makes sense and A∈^m_0(Θ) with the desired asymptotic expansion. 
            
            Let ξ∈^̋∞. Then for every k ≥ 0 and n∈, we have
            
        A_k(1-η(ε_k Θ))ξ_^̋n   ≤A_k_^̋n+m_k→^̋n(1-η(ε_kΘ))ξ_^̋n+m_k
           ≤A_k_^̋n+m_k→^̋n1-η(ε_kΘ)_^̋n+m_0→^̋n+m_kξ_^̋n+m_0.
    
            Let a>0 be a number such that a < Θ. The norm of 1-η(ε_k Θ) from ^̋n+m_0 to ^̋n+m_k is determined by functional calculus as
            
        sup_t > a t^m_k-m_0 (1-η(ε_k t)) ≤ε_k^m_0 - m_ksup_s > 0 s^m_k - m_0  (1-η(s)) ≤ C_ηε_k.
    
            for some constant C_η, and for k sufficiently large so that m_0 - m_k ≥ 1. Now we choose ε_k sufficiently small such that
            
        0 ≤ε_kC_ηmax_|n|≤ k{A_k_^̋n+m_k→ H^n} < 2^-k.
    
            With this choice of sequence {ε_k}_k=0^∞, we have just proved that the series
            
        Aξ = ∑_k=0^∞ A_k(1-η(ε_k Θ))ξ
    
            converges in every ^̋n, and defines a bounded linear operator
            
        A:^̋n+m_0→^̋n,   n∈.
    
            Since this holds for every n∈, it follows that A:^̋∞→^̋∞ and by interpolation (Proposition <ref>) A ∈^m_0(Θ).
            Note that with this fixed choice of {ε_k}_k=0^∞, we have proved the stronger result that the “tail" of A
        ∑_k=N+1^∞ A_k(1-η(ε_k Θ))
    
            converges in every ^̋s and defines a linear operator in ^m_N+1.
            
            Now we prove that A has the desired asymptotic expansion.
            For every N>0 we have
            
        A-∑_k=0^N A_k = -∑_k=0^N A_kη(ε_k Θ) + ∑_k=N+1^∞ A_k(1-η(ε_k Θ))
     
            Since η is compactly supported, it is easy to see that the first summand has order -∞ for every N≥ 0, and the second summand
            has order at most m_N+1 due to the result just proved. 
        
        Suppose that A ∈^r has an inverse B∈^-r up to order -1. That is, 
            
        AB = 1_^̋∞+R_1;
        
                    BA = 1_^̋∞+R_2
     
            where R_1, R_2 have order -1. Then A is elliptic.
    
        Since R_1^j has order at most -j, we can use the Borel lemma to construct an operator B' such that
        
        B' ∼∑_k=0^∞ BR_1^j.
    
            Then AB' -1 has order -∞. Similarly we can construct a left inverse.
    
        Let A ∈^r be elliptic of order r ≥ 0. Then the unbounded operator
        
        A: ^̋s+r⊆^̋s →^̋s
    
        is closed for each s ∈.
    
        Define the graph norm of A on (A) = ^̋s+r as 
        
        x _G(A) := Ax _s +  x_s,    x ∈^̋s+r.
    
        By definition, A is a closed operator if and only if (A) is complete with respect to this graph norm. We will show that for elliptic operators, the graph norm is equivalent to ·_s+r, which immediately implies that ^̋s+r is complete with respect to the graph norm. First, we have that
        
        Ax _s + x_s    ≤A_^̋s+r→^̋sx_s+r + Θ^-r_^̋s →^̋sΘ^r x _s
           ≲ x_s+r.
    
        Next, let P be a parametrix for A, and take x ∈^̋∞ so that
        
        x _s+r   ≤ PA x _s+r +  R_2 x_s+r
           ≤ P _^̋s →^̋s+r Ax _s +  R_2 _^̋s →^̋s+r x _s
           ≲Ax _s + x_s.
    
        The assertion of the proposition is now immediate.
    
    
    
    Elliptic operators have a property which is often called elliptic regularity, or maximal subellipticity.
    
        Let A ∈^r be an elliptic operator. If x ∈^̋-∞ is such that Ax ∈^̋s for an s ∈, then x ∈^̋s+r.
    
        Take x ∈^̋-∞, and suppose that Ax ∈^̋s. Then PAx∈^̋s+r,
        which implies that
        
        x = PAx - R_2x ∈^̋s+r.
    
    
    Finally we will now show that ifA ∈^r,r ≥ 0is elliptic and symmetric, thenAis self-adjoint with domain^̋r. 
    
        For A ∈^r, we define the adjoint A^†:^̋-∞→^̋-∞ by
        
        ⟨ Au, v ⟩_(^̋∞, ^̋-∞) = ⟨ u, A^† v ⟩_(^̋∞, ^̋-∞)   u ∈^̋∞, v∈^̋-∞.
    
    It is shown in Appendix <ref> thatA^†|_^̋∞∈^r, and for anys∈, we have that
    
        ⟨ A u, v ⟩_(^̋s, ^̋-s) = ⟨ u, A^† v ⟩_(^̋s+r, ^̋-s-r),    u ∈^̋s+r, v∈^̋-s.
    
    Furthermore, forr ≥ 0Proposition <ref> gives thatA = A^†if and only ifA^r,0: ^̋r ⊆→̋$̋ is symmetric.
    
    
        Let A ∈^r, r ≥ 0, be an elliptic and symmetric operator. Then A
        is self-adjoint with domain ^̋r.
    
        To prove that A is self-adjoint, we need to show that the Hermitian adjoint of the closed operator A: ^̋r⊆^̋0 →^̋0, writing A^*_0:= (A^r,0)^*,
        
        A^*_0: (A^*_0) ⊆^̋0 →^̋0,
    
        has domain (A^*_0) = ^̋r. 
        Recall that, by definition,
        
        (A^*_0) := {u ∈^̋0 : ∃ v ∈^̋0  such that ∀ϕ∈^̋r  ⟨ u, Aϕ⟩_^̋0=⟨ v, ϕ⟩_^̋0}.
    
        If u,v ∈^̋0 and ϕ∈^̋r, then by (<ref>) and Proposition <ref>,
        
        ⟨ u, Aϕ⟩_^̋0   =⟨ A^† u, ϕ⟩_(^̋-r, ^̋r)
           = ⟨ A u, ϕ⟩_(^̋-r, ^̋r);
        ⟨ v, ϕ⟩_^̋0   = ⟨ v, ϕ⟩_(^̋-r, ^̋r).
    
        Since ^̋r separates the points of ^̋-r, we have that for u, v ∈^̋0,
        
        ⟨ A u, ϕ⟩_(^̋-r, ^̋r) = ⟨ v, ϕ⟩_(^̋-r, ^̋r),   ∀ϕ∈^̋r,
    
        if and only if
        
        Au = v ∈^̋-r.
    
        Hence, 
        
        (A^*_0)    = {u ∈^̋0 : ∃ v ∈^̋0  such that   A u = v }
           = {u ∈^̋0 :  A u ∈^̋0}.
    
        By elliptic regularity (Proposition <ref>) it follows that (A^*_0) = ^̋r,
        completing the proof.
    
    
     §.§ Functional Calculus for Elliptic Operators
    
        Let A∈^r(Θ), r>0 be elliptic and symmetric. Then ⟨ A ⟩^1/r extends to an invertible positive self-adjoint operator on $̋, and 
        
        Θ^s = (⟨ A ⟩^1/r)^s.
    
        The normsΘ^s ξ_and⟨ A ⟩^s/rξ_are equivalent on this subspace of$̋. Therefore, Θ and ⟨ A ⟩^1/r define the same Sobolev scale
        
        ^̋s(Θ) = ^̋s(⟨ A ⟩^1/r),
    
        and we have
        
        ^t(Θ)  = ^t (⟨ A ⟩^1/r).
    
    The first statement follows from Proposition <ref>. For the remaining statements, it suffices to prove that 
        
        (1+A^2)^αΘ^-2α r
    
        extends to a bounded operator on $̋ for allα∈, as this would imply forξ∈^̋∞,
        
        ⟨ A ⟩^s/rξ_≤ (1+A^2)^s/2rΘ^-s_∞Θ^s ξ_≲Θ^s ξ_,
    
        and an analogous estimate in the other direction.
    
    
    LetPbe a parametrix forA, so thatAP = 1 + RwithR ∈^-∞. Since
        
        (1+A^2)P^2 = P^2 + A(1+R)P = 1 + P^2 + R + ARP
    
        andP^2 + R + ARP ∈^-2rand similarly for(1+A^2)P^2, it follows that the operator1+A^2is also elliptic due to Corollary <ref>. SinceAis self-adjoint with domain^̋r, applying Proposition <ref> gives that(1+A^2)^-1∈^-2r. We therefore have(1+A^2)^k ∈^2kr,k ∈. This in turn gives that^̋∞⊆ (1+A^2)^zfor anyz∈.
        
        We use the Hadamard three-line theorem, so define the function
        
        F(z) := ⟨ x, (1+A^2)^mzΘ^-2 mz r y ⟩_,    z ∈,
    
        wherem ∈andx, y ∈^̋∞are fixed.     
        Let{e_n}_n∈⊆^̋∞be an orthonormal basis of$̋, then
        
        F(z) = ∑_n=1^∞⟨ x, (1+A^2)^mz e_n ⟩_⟨̋e_n , Θ^-2mzr y⟩_.
    
        Using the dominated convergence theorem, it can be seen that z ↦⟨ x, (1+A^2)^mz e_n ⟩_$̋ andz ↦⟨ e_n , Θ^-2mzr y⟩_are continuous maps. Applying the Cauchy–Schwarz inequality to the series (<ref>) yields
        
        ∑_n=1^∞ |⟨ x,(1+A^2)^mze_n⟩_||⟨ e_n,Θ^-2mzry⟩_| ≤(1+A^2)^mzx_Θ^-2mzry_,
    
        which is uniformly bounded on compact subsets ofdue to the continuity of the right-hand side. 
        We can therefore apply the dominated convergence theorem again to deduce thatF(z)is a continuous function itself. 
        Furthermore, this uniform boundedness yields through Fubini's theorem that ifγis a closed loop
        inℂthen
        
        ∫_γ F(z) dz = ∑_n=1^∞∫_γ⟨ x,(1+A^2)^mze_n⟩_⟨ e_n,Θ^-2mzry⟩_ dz.
    
        Using Fubini's theorem once more,
        we have that 
        ∫_γ F(z) dz = ∑_n=1^∞∫_σ(1+A^2)∫_σ(Θ^-2r)∫_γ (λμ)^mz dz ⟨ x,dE^1+A^2e_n⟩_⟨ e_n,dE^Θ^-2ry⟩_ = 0,
     so that we can conclude by Morera's theorem thatF(z)is holomorphic.    
        
        Since1+A^2andΘare positive operators and
        
        sup_x > 0 |x^it| = 1,
    
        it follows from the Borel functional calculus that fors ∈,|F(is)|    = |⟨  (1+A^2)^-ims x, Θ^-2 imsr y ⟩_|
           ≤ x _y_.
     
        Likewise,
        
        |F(1+is)|    = |⟨  (1+A^2)^-ims x, (1+A^2)^m Θ^-2mrΘ^-2imsr y ⟩_|
           ≤ (1+A^2)^m Θ^-2mr_→̋x _y_,
    
        which we know to be finite sincemis an integer.
    
        The Hadamard three-line theorem (see e.g. <cit.>) now gives that forα∈ (0,1),
        
        |F(α)|    ≤max_s∈ |F(α + is)|
           ≤( max_s∈ |F(is)|)^(1-α)( max_s∈ |F(1 + is)|)^α
           ≤ (1+A^2)^m Θ^-2mr_→̋^αx_y_.
    
        Hence, withα∈ (0,1)andm ∈,
        
        (1+A^2)^mαΘ^-2m α r_→̋≤ (1+A^2)^m Θ^-2mr_→̋^α,
    
        which proves that(1+A^2)^mαΘ^-2mα rextends to a bounded operator on$̋ for all m∈, α∈ [0,1].
    
        Let A ∈^r(Θ), r>0 be elliptic and symmetric. Using Proposition <ref>, we replace Θ by ⟨ A ⟩^1/r so that A ∈^r(⟨ A ⟩^1/r) is elliptic and symmetric. By Proposition <ref>, the operator
        
        A^r,0: ^̋r(⟨ A ⟩^1/r) ⊆→̋
            is self-adjoint; we denote its spectral measure by E. Then for f ∈ L^β_∞(E), using Borel functional calculus to define f(A^r,0), we have
            
            ⟨ A ⟩^s/r f(A^r,0)⟨ A ⟩^-s/r - β_→̋ =  f(A^r,0)⟨ A ⟩^- β_→̋ = f_L^β_∞(E)<∞,
        
            which shows that f(A) := f(A^r,0)|_^̋∞∈^rβ(⟨ A ⟩^1/r). Converting this estimate back into an estimate on the spaces ^̋s(Θ) introduces the constant C_s,A.
        
        
        
        Theorem <ref> has a converse in the following sense. If A ∈^r, r>0 is an arbitrary elliptic symmetric operator and if f: → is such that f(A) ∈^β r(Θ), then the proof of Proposition <ref> gives that f(A) (1+A^2)^-β/2 is a bounded operator on $̋. This happens if and only iff ⟨ x ⟩^-β∈ L^0_∞(E)<cit.>, i.e.f∈ L^β_∞(E).
        
        
        
            If A∈^r, r>0 is symmetric and elliptic and if f :→ is a bounded Borel measurable function, then for any t∈ we have
            
            f(tA)_^̋s →^̋s≤ C_s,Af_∞,
        
            independent of t∈.
        
            Like in the proof Theorem <ref>, we have
            
            f(tA)_^̋s(⟨ A ⟩^1/r) →^̋s(⟨ A ⟩^1/r)    = f_t_L^0_∞(E)≤f_∞,
        
            where we wrote f_t(x) := f(tx).
        
        
        The functional calculus constructed in Theorem <ref> can easily be extended to a larger class of operators. For example, on^dwithΘ = (1-Δ)^1/2we have thatid/dxis not elliptic, but it does commute strongly with a symmetric elliptic operator. The following proposition shows that a functional calculus forid/dxdoes exist in(Θ)for this reason.
        
            Let A be a self-adjoint operator on $̋ with spectral measureE. If there exists an elliptic symmetric operatorH ∈^h(Θ),h>0such thatAstrongly commutes withH^h,0: ^̋h ⊆→̋$̋, then for f ∈ L^0_∞(E) we have that f(A) ∈^0(Θ). If A ∈^r itself for some r ∈, we have that f(A) ∈^β r(Θ) for f ∈ L^β_∞(E), β≥ 0.
        
            In light of Proposition <ref>, we can assume without loss of generality that H = Θ. If f ∈ L^0_∞(E), then f(A):→̋$̋ is a bounded operator, and forξ∈^̋∞we have
            
            f(A) ξ_^̋k = Θ^k f(A) ξ_≤f(A)_∞ξ_^̋k,    k ∈,
        
            which shows thatf(A) ∈^0(Θ)through interpolation (Proposition <ref>). The second part of the proposition is proved similarly, after the observation that the Hadamard three-line argument in the proof of Proposition <ref> goes through forAifm ∈_≥ 0, i.e.(1+A^2)^αΘ^-2α ris bounded forα≥ 0. 
        
        
        § DIVIDED DIFFERENCES
        
        The condition that appears in Theorem <ref> on the symbolϕ: ^n+1→needs to be analysed more closely in order to prepare this multiple operator integral construction for practical applications. The main result of this section is Lemma <ref>, which gives that forf ∈ T^β()the divided differencef^[n]has an integral representation satisfying the conditions of Theorem <ref>.
        
        First of all, for functionsϕ:^n+1→it is an equivalent condition to admit a representation
        
            ϕ(λ_0,…,λ_n)=∫_Ω a_0(λ_0,ω)⋯ a_n(λ_n,ω) dν(ω)
        
        for a finite complex measure space (technically: of finite variation)(Ω,ν)and measurable functionsa_j:×Ω→such(x,ω)↦ a_j(x,ω)⟨ x ⟩^-β_jisE_j ×ν-a.e. bounded forβ_j ∈,
        or a representation
        
            ϕ(λ_0,…,λ_n)=∫_Σ b_0(λ_0,σ)⋯ b_n(λ_n,σ) dμ(σ),
        
        whereΣis aσ-finite measure space,b_j: ×Ω→measurable, and
        
            ∫_Σb_0(·, σ)_L^β_0_∞(E_0)⋯b_n(·, σ)_L^β_n_∞(E_n)d|μ|(σ)<∞.
        
        Namely, given the second representation, a representation of the first type can be obtained by putting <cit.>
            a_j(λ_j, σ) := b_j(λ_j, σ)/b_j(·, σ)_L_∞^β_j(E_j),   ν(σ):= b_0(·, σ)_L^β_0_∞(E_0)⋯b_n(·, σ)_L^β_n_∞(E_n)μ(σ).
        
        
        Mainly for notational purposes, we introduce the following definition. It is inspired by the integral projective tensor product appearing in <cit.>, a precise study can be found in <cit.>.
        
        
        
            Let Γ_0, …, Γ_n be function spaces of bounded measurable functions → equipped with (semi)norms ·_Γ_i, k.
            We define Γ_0 ⊠_i ⋯⊠_i Γ_n as the set of functions ϕ: ℝ^n+1→ℂ for which there exists a decomposition
            
            ϕ(λ_0, …, λ_n) = ∫_Ω a_0(λ_0, ω) ⋯ a_n(λ_n, ω) dν(ω)
        
            where (Ω, ν) is a σ-finite measure space, a_i : ℝ×Ω→ℂ is measurable, a_i(·, ω) ∈Γ_i, the functions ω↦a_i(·, ω)_Γ_i, k are measurable for each i and k, and
            
            ∫_Ωa_0(·, ω)_Γ_0, k_0⋯a_n(·, ω)_Γ_n, k_n d|ν|(ω) < ∞.
        
            We define the seminorm
            
            ϕ_Γ_0 ⊠_i ⋯⊠_i Γ_n, k_0, …, k_n
        
            to be the infimum of the quantity (<ref>) over all representations (<ref>).
        
        We have that 
        
            Γ_0 ⊗⋯⊗Γ_n ⊆Γ_0 ⊠_i ⋯⊠_i Γ_n,
        
        where ⊗ denotes the algebraic tensor product for topological vector spaces.
        
        
        Note that due to Lemma <ref>, ifa: ×Ω→is measurable anda(·, ω) ∈ S^β(), we have thatω↦a(·, ω)_S^β(), kis measurable, and the same claim holds forT^β()andL_∞^β(E). Hence the construction in Definition <ref> can be applied without this extra assumption.
        
        ForL_∞^β(E)withEa spectral measure, this gives the integral projective tensor product
        
            L_∞^β_0(E_0) ⊠_i ⋯⊠_i L_∞^β_n(E_n) = L_∞^β_0(E_0) ⊗̂_i ⋯⊗̂_i L_∞^β_n(E_n),
        
        which appears in particular forβ_0 = ⋯ = β_n = 0in the works by Peller <cit.>.
        
        We refrain from answering the question whether the space in Definition <ref> is in general the completion of the algebraic tensor product under the given seminorms. 
        
            Observe that
            
            S^β_0() ⊠_i ⋯⊠_i S^β_n() ⊆  L_∞^β_0(E_0) ⊗̂_i ⋯⊗̂_i L_∞^β_n(E_n)
        
            no matter what spectral measures E_i are taken. 
            By Theorem <ref> we have a well-defined multiple operator integral precisely for symbols in the latter space.
        
        If ϕ∈ S^α_0() ⊠_i ⋯⊠_i S^α_n() and ψ∈ S^β_0() ⊠_i ⋯⊠_i S^β_n(), then
        
            Φ(λ_0, …, λ_n):= ϕ(λ_0, …, λ_n) ψ(λ_0, …, λ_n)∈ S^α_0 + β_0() ⊠_i ⋯⊠_i S^α_n + β_n().
        
        An analogous statement holds for the spaces L^β_∞(E).
        
            According to Definition <ref>, we can find σ-finite measure spaces (Ω, ν), (Σ, μ) and measurable functions a_i: ×Ω→, b_i: ×Σ→ such that
            
            ϕ(λ_0, …, λ_n)    = ∫_Ω a_0(λ_0, ω) ⋯ a_n(λ_n, ω) dν(ω);
            ψ(λ_0, …, λ_n)    = ∫_Σ b_0(λ_0 ,σ) ⋯ b_n(λ_n, σ) dμ(σ).
        
        As observed above, the maps
            
            ω↦ a_i(·, ω)_S^α_i(), k
        
            are measurable, similarly for the functions b_i.
            
            Using Tonelli's theorem,
            
            ∫_Ω×Σ|a_0(λ_0, ω)b_0(λ_0, σ)   ⋯ a_n(λ_n, ω)b_n(λ_n, σ)| d(ν×μ)(ω, σ)
               ≤⟨λ_0 ⟩^α_0+β_0⋯⟨λ_n ⟩^α_n + β_n∫_Ω a_0(·, ω)_S^α_0,0⋯a_n(·, ω)_S^α_n, 0 dν(ω) 
                 ×∫_Σb_0(·, σ)_S^β_0, 0⋯b_n(·, σ)_S^β_n,0 dμ( σ)<∞.
        
            Hence, by Fubini's theorem
            
            Φ(λ_0, …, λ_n) = ∫_Ω×Σ a_0(λ_0, ω)b_0(λ_0, σ) ⋯ a_n(λ_n, ω) b_n(λ_n, σ) d(ν×μ)(ω, σ).
        
            The fact that Φ∈ S^α_0+β_0() ⊠_i ⋯⊠_i S^α_n + β_n() now follows from the computation
            
            a_k(·, ω) b_k(·, σ)_S^α_k + β_k, m≤∑_j=0^m mja_k(·, ω)_S^α_k, j b_k(·, σ)_S^β_k, m-j.
        
        
        We now use the construction of an almost analytic extension to provide an explicit integral representation forf∈ T^β(). The technique was introduced by Hörmander <cit.>, and subsequently used in various contexts by many authors. For detailed notes on the historical origins, see <cit.>. In Appendix <ref> we use it to develop a functional calculus for^0(Θ), based on the results of <cit.>. 
        
        For the details of this construction we follow Davies <cit.><cit.>.
        
            Let f ∈ C^N(ℝ), N ∈. We define an extension f̃: ℂ→ℂ by
            
            f̃(x+iy) := τ(y/⟨ x ⟩ ) ∑_r=0^N f^(r)(x) (iy)^r/r!,
        
            where τ:ℝ→ℝ is a smooth bump function with τ(s) = 0 for |s|>2, τ(s) =1 for |s|<1. If f ∈ C_c^∞(), we have
            
            f(x) = -1/π∫_ℂ( ∂f̃/∂z(z)) (z-x)^-1 dz,    x ∈ℝ,
        
            independent of the choice of τ and N. We refer to f̃ as an almost analytic extension of f.
        
          *  Take n ∈_≥ 0, let α be some real number with -1 ≤α≤ n, and consider any collection of real numbers -1 ≤β_0, …, β_n ≤ 0 such that ∑β_j = α - n. Then
            
            f ∈ T^α() ⇒ f^[n]∈ S^β_0() ⊠_i ⋯⊠_i S^β_n(),
        
        where for each k_0, …, k_n ≥ 0 we have
        
            f^[n](λ_0, …, λ_n)_S^β_0() ⊠_i ⋯⊠_i S^β_n(), k_0, …, k_n≲∑_r=0^n+∑_j=0^n k_j+2 f _T^α(), r.
        
          *  Let α≤ n. Then
        
            f ∈ T^α() ⇒ f^[n]∈∑_β_0, …, β_n ≤ 0 
            ∑β_j= α -nS^β_0() ⊠_i ⋯⊠_i S^β_n().
        
        For each component ϕ∈ S^β_0() ⊠_i ⋯⊠_i S^β_n() in the (finite) decomposition, we have
        
            ϕ_S^β_0() ⊠_i ⋯⊠_i S^β_n(), k_0, …, k_n≲∑_r=0^n+∑_j=0^n k_j+2 f _T^α(), r.
        
          *  Let α≥ n. Then
        
            f ∈ T^α() ⇒ f^[n]∈∑_∑β_j= α -nS^β_0() ⊠_i ⋯⊠_i S^β_n().
        
        For each component ϕ∈ S^β_0() ⊠_i ⋯⊠_i S^β_n() in the (finite) decomposition, we have
        
            ϕ_S^β_0() ⊠_i ⋯⊠_i S^β_n(), k_0, …, k_n≲∑_r=0^n+∑_j=0^n k_j+2 f _T^α(), r.
        
          *  For g ∈ C^∞_c(ℝ) with almost analytic extension g̃, we have
        
            g(x) = -1/π∫_ℂ∂g̃/∂z (z-x)^-1dz,
        
        and hence
        
            g^[n](λ_0, …, λ_n) = (-1)^n/π∫_ℂ∂g̃/∂z (z-λ_0)^-1⋯ (z-λ_n)^-1dz.
        
        
        Now take f ∈ T^α() with α≤ n and with almost analytic extension f̃. Directly from Definition <ref>, writing σ(z) := τ((z)/⟨(z)⟩), it follows that (cf. <cit.>)
        
            ∂f̃/∂z = 1/2(∑_r=0^N f^(r)((z)) (i(z))^r/r!)(σ_x(z) + iσ_y(z) ) + 1/2 f^(N+1)((z)) (i(z))^N/n!σ(z).
        
        We define
        
            U:={z ∈ : ⟨(z) ⟩ < |(z)| < 2 ⟨(z) ⟩},      V:={z ∈ : 0 ≤ |(z)| < 2 ⟨(z) ⟩},
        
        and note that the support of σ is contained in V, while the support of σ_x and σ_y are contained in U. More precisely,
        
            |σ_x(z) + i σ_y(z)| ≲1/⟨(z) ⟩χ_U(z).
        
        Therefore we have the estimate <cit.>
            ∫_ℂ|    ∂f̃/∂z| |z-λ_0|^-1⋯ |z-λ_n|^-1dz
               ≲∑_r=0^N ∫_U |f^(r)((z))| |(z)|^r-n-1⟨(z) ⟩^-1 dz + ∫_V |f^(N+1)((z))| |(z)|^N-n-1 dz
               ≲∑_r=0^N+1∫_ |f^(r)(x)|⟨ x ⟩^r-n-1dx = ∑_r=0^N+1f_T^n(), r,
        
        where the last estimate (integration over the imaginary direction) is justified when N ≥ n+1.
        
        Hence the integral
        
            (-1)^n/π∫_ℂ∂f̃/∂z (z-λ_0)^-1⋯ (z-λ_n)^-1dz
        
        converges.
        
        Since <cit.> gives that C^∞_c() is dense in T^α(), the Lebesgue dominated convergence theorem then gives that the identity (<ref>) extends to all f ∈ T^α(), α≤ n, i.e.
        
            f^[n](λ_0, …, λ_n) = (-1)^n/π∫_ℂ∂f̃/∂z (z-λ_0)^-1⋯ (z-λ_n)^-1dz.
        
        In order to show that this is a decomposition as described in Definition <ref>, with a_j(λ_j, z) = (z-λ_j)^-1, we will now estimate the expressions
        
            (z-·)^-1_S^β(), k   =sup_λ∈ℝ⟨λ⟩^k-β|∂^k/∂λ^k (z-λ)^-1| 
               ≲(sup_λ∈ℝ⟨λ⟩^-β |z-λ|^-1) ·(sup_λ∈ℝ⟨λ⟩^k |z-λ|^-k).
        
        
        
        Note that, for λ∈ and z ∈∖,
        
            ⟨λ⟩/|z-λ| = |λ± i|/|z-λ|   ≤ 1 + |z± i|/|z-λ|
               ≤ 1+ ⟨ z ⟩/|(z)|,
        
        as  min(|z+i|,|z-i|)≤⟨ z⟩. Next, we estimate
        
            sup_λ∈⟨λ⟩^-β |z-λ|^-1
        
        for -1 ≤β≤ 0. We estimate the supremum over λ > 1, |λ| ≤ 1 and λ < -1 separately. First, for |λ| ≤ 1 we have 1 ≤⟨λ⟩≤ 2^1/2, and so
        
            sup_|λ|≤ 1⟨λ⟩^-β |z-λ|^-1≲1/|(z)|.
        
        For λ > 1, we have ⟨λ⟩≤ 2^1/2λ, so that
        
            sup_λ > 1⟨λ⟩^-β |z-λ|^-1   ≲sup_λ > 1-(z)(λ+(z))^-β/(λ^2+(z)^2)^1/2.
        
        Writing v = (λ, (z)) ∈^2, then by using Cauchy–Schwarz we have
        
            sup_λ > 1-(z)(λ+(z))^-β/(λ^2+(z)^2)^1/2    = sup_λ > 1-(x)(v · (1, (z)/(z)))^-β/v
               ≤sup_λ > 1-(z) (1, (z)/(z))^-β/v^1+β
               ≤|z|^-β/ |(z)|.
        
        
        For λ < -1 we have a similar estimate, and hence combined we have
        
            sup_λ∈⟨λ⟩^-β |z-λ|^-1≲1/|(z)|max(1, |z|^-β) ≤⟨ z ⟩^-β/|(z)|.
        
        
        Combining (<ref>), (<ref>) and (<ref>) we get an estimate 
        
            sup_λ∈ℝ⟨λ⟩^k-β|∂^k/∂λ^k (z-λ)^-1| ≲⟨ z ⟩^-β/|(z)|(1+⟨ z ⟩/|(z)|)^k.
        
        
        Let -1 ≤β_0, …, β_n ≤ 0. Taking the inequality above and proceeding as before with N ≥ n+1+∑_j=0^n k_j, we have
        
            ∫_ℂ| ∂f̃/∂z|   (sup_λ_0 ∈ℝ⟨λ_0 ⟩^k_0-β_0|∂^k_0/∂λ_0^k_0 (z-λ_0)^-1| ) ⋯(sup_λ_n ∈ℝ⟨λ_n ⟩^k_n-β_n|∂^k_n/∂λ_n^k_n (z-λ_0)^-1| )dz 
               ≲∑_r=0^N+1∫_ℝ |f^(r)(x)|  ⟨ x ⟩^r-n-1-∑_j=0^n β_j dx < ∞.
        
        This converges in particular for ∑_j=0^n β_j = α - n. Since -n-1 ≤∑_j=0^n β_j ≤ 0, this choice is possible if -1 ≤α≤ n. We have therefore proved for -1 ≤α≤ n, and -1 ≤β_0, …, β_n ≤ 0 such that ∑_j=0^n β_j = α - n, that 
        
            f^[n](λ_0, …, λ_n)   _S^β_0() ⊠_i ⋯⊠_i S^β_n(), k_0, …, k_n
               = ∫_ℂ  (z-·)^-1_S^β_0(), k_0⋯(z-·)^-1_S^β_n(), k_n| ∂f̃/∂z|dz 
               ≲∑_r=0^n+∑_j=0^n k_j+2 f _T^n(), r≤∑_r=0^n+∑_j=0^n k_j+2f_T^α(),r.
        
          *  For f ∈ T^α(), -1 ≤α≤ n, we have by the first part of the lemma that for each n ∈,
        
            f^[n]∈ S^β_0() ⊠_i ⋯⊠_i S^β_n()
        
        where each β_j can be chosen to lie in the interval [-1, 0], and ∑β_j = α - n. 
        
        For f ∈ T^α() with α≤ -1, we can write f  = g · (x+i)^-k where g ∈ T^β(), -1 ≤β≤ 0 and k ∈.
        The Leibniz rule for divided differences dictates
        
            f^[n](λ_0, …, λ_n) = ∑_l=0^n g^[l](λ_0, …, λ_l) ((x+i)^-k)^[n-l](λ_l, …, λ_n).
        
        From part 1 and the explicit form of the divided differences of ((x+i)^-k)^[n] we therefore conclude that each term is an element of
        
            ∑_β_0, …, β_n ≤ 0 
            ∑β_j= α -k-nS^β_0() ⊠_i ⋯⊠_i S^β_n(),
        
        with the required estimate of norms.
        
        
          *  This follows analogously to assertion 2, by analysing (g(x+i)^k)^[n] for g ∈ T^α() with 
        -1 ≤α≤ 0.
        
            The proof of Lemma <ref> in fact shows that if f ∈ C^n+2() such that f_T^β(), k < ∞ for k=0, …, n+2, then given any spectral measures E_0, …, E_n, we have
            
            f^[n]_L_∞^β_0(E_0) ⊗̂_i ⋯⊗̂_i L_∞^β_n(E_n)≤f^[n]_S^β_0() ⊠_i ⋯⊠_i S^β_n(), 0, …, 0≤∑_k=0^n+2 f_T^β(), k < ∞.
        
            For n = 0, the space of functions that satisfy this condition closely resembles the space 𝔉_m() used in <cit.> in the context of double operator integrals.
        
        Lemma <ref> and Remark <ref>, combined with Theorem <ref>, finish the proof of Theorem <ref>.
        
        
        
        
        
        § MOI IDENTITIES AND ASYMPTOTIC EXPANSIONS
        
        
         §.§ MOI Identities and Applications
        
        The most important identities for our applications of our multiple operator integrals are the following. These are higher-order analogues of (<ref>) and (<ref>), seeing thatf(H) = T_f^[0]^H().
        
        
        Let a, X_1,…,X_n ∈, let H_i∈^h_i, h_i > 0 be symmetric and elliptic and let f ∈ T^β(), β∈. Then
        
            T^H_0, …, H_n_f^[n](X_1,…,X_j,aX_j+1,…,X_n)-T_f^[n]^H_0, …, H_n   (X_1,…,X_ja,X_j+1,…,X_n)
               =T_f^[n+1]^H_0, …, H_j, H_j, …, H_n(X_1,…,X_j,[H_j,a],X_j+1,…,X_n);
            
                T^H_0, …, H_n_f^[n](aX_1,…,X_n) - a T^H_0, …, H_n_f^[n](X_1,…,X_n)    = T^H_0, H_0, H_1, …, H_n_f^[n+1]([H_0,a],X_1,…,X_n) ;
            
                T^H_0, …, H_n_f^[n](X_1,…,X_n)a -  T^H_0, …, H_n_f^[n](X_1,…,X_na)    = T^H_0, …, H_n, H_n_f^[n+1](X_1,…,X_n,[H_n,a]).
        
        Moreover, for A ∈^a, a>0, B ∈^b, b>0 symmetric and elliptic,
        
            T^H_0, …, H_j-1, A, H_j+1, …, H_n_f^[n](X_1, …, X_n) - T^H_0, …, H_j-1, B, H_j+1, …, H_n_f^[n](X_1, …, X_n)
               =T^H_0, …, H_j-1, A, B, H_j+1, …, H_n_f^[n+1](X_1, …, X_j, A-B, X_j+1, …, X_n).
        
        Note that for f∈ T^β(), the multiple operator integrals appearing above are well-defined through Theorem <ref>.
        
        We prove equation <ref>, the others follow analogously. Write
        
            F_j(λ_0, …, λ_n+1)    := f^[n](λ_0, …, λ_j-1, λ_j+1, …λ_n+1);
            
                F_j+1(λ_0, …, λ_n+1)    := f^[n](λ_0, …, λ_j, λ_j+2, …λ_n+1).
        
        Observe that
        
            F_j+1(λ_0, …, λ_n+1) - F_j(λ_0, …, λ_n+1) = (λ_j - λ_j+1) f^[n+1](λ_0, …, λ_n+1).
        
        Hence, 
        
            T_f^[n+1]^H_0, …, H_j, H_j, …, H_n   (X_1,…,X_j,[H_j,a],X_j+1,…,X_n)
               = T_(λ_j - λ_j+1) f^[n+1]^H_0, …, H_j, H_j, …, H_n(X_1,…,X_j,a,X_j+1,…,X_n)
               = T_F_j+1^H_0, …, H_j, H_j, …, H_n(X_1,…,X_j,a,X_j+1,…,X_n) 
                  - T_F_j^H_0, …, H_j, H_j, …, H_n(X_1,…,X_j,a,X_j+1,…,X_n) 
               = T^H_0, …, H_n_f^[n](X_1,…,X_j,aX_j+1,…,X_n)-T_f^[n]^H_0, …, H_n(X_1,…,X_ja,X_j+1,…,X_n).
        
        
        
        With these identities in hand, we can show that the MOI constructed in the previous section is an element of(Θ)if all its components are and the symbol is a divided difference.
        
             Let H_i ∈^h, h>0, be symmetric and elliptic and take f ∈ T^β(), β∈. For operators X_i ∈^r_i, r:= ∑_i=1^n r_i, we have that
             
            T_f^[n]^H_0, …, H_n(X_1, …, X_n) ∈^h(β-n)+ r
        
             due to Lemma <ref>.
             Now, taking n=1 to ease notation, using Proposition <ref> gives that
             
            [Θ, T^H_0, H_1_f^[1](X_1)] = T^H_0, H_0, H_1_f^[2]([Θ, H_0],X_1) + T^H_0, H_1_f^[1]([Θ,X_1]) + T^H_0, H_1, H_1_f^[2](X_1, [Θ, H_1]).
        
             As [Θ, H_i]∈^h and [Θ, X_1] ∈^r, Lemma <ref> combined with Theorem <ref> gives that
             
            T^H_0, H_0, H_1_f^[2]([Θ, H_0],X_1), T^H_0,H_1_f^[1]([Θ,X_1]), T^H_0, H_1, H_1_f^[2](X_1, [Θ, H_1]) ∈^h(β-n)+r.
        
             Higher commutators and n > 1 follow analogously. Hence
             
            T_f^[n]^H_0,…, H_n(X_1, …, X_n) ∈^h(β-n)+ r.
        
        
        In the setting thatΘ^-1∈ℒ_s,s>0, it is immediate from Theorem <ref> and Theorem <ref> that forH_0, …, H_n ∈^h, h >0symmetric and elliptic, andf∈ T^β(), the multiple operator integral
        
            T_f^[n]^H_0, …, H_n(X_1, …, X_n) ∈^(β-n)h + r
        
        can be considered to be a trace-class operator on$̋ if β is small enough. Namely, we have
        
            A_1≤Θ^-s_1 A_^̋0 →^̋s.
         
        For asymptotic expansions of trace formulas, it will be useful to make a more detailed analysis.
        
        
        
        Let H_i ∈^h for a fixed h > 0 be symmetric and elliptic operators. If X_i ∈^r_i, r:= ∑_i=1^n r_i, f ∈ T^α(ℝ) with α≤ n, t≤ 1, then
            
            T^tH_0, …, tH_n_f^[n](X_1, …, X_n) _^̋q+r+(α-n)h→^̋q≲ t^α-n,    q∈.
        
        Rephrased, if f ∈ T^u-r/h+n, u≤ r, then
        
            T^tH_0, …, tH_n_f^[n](X_1, …, X_n) _^̋q+u→^̋q≲ t^u-r/h,    q∈.
        
        Lemma <ref> gives that
        
            f^[n]∈∑_β_0 + ⋯ + β_n = α-n S^β_0() ⊠_i ⋯⊠_i S^β_n(),
        
        where each β_i ≤ 0 since α≤ n.
           Consider one of the summands, ϕ∈ S^β_0() ⊠_i ⋯⊠_i S^β_n(). Then 
           
            ϕ(λ_0, …, λ_n) = (λ_0 + i)^β_0⋯ (λ_n + i)^β_n·ψ(λ_0, …, λ_n),
        
           for a function ψ∈ S^0() ⊠_i ⋯⊠_i S^0(), and thus, by Remark <ref> and Theorem <ref>,
        
            T^tH_0, …, tH_n_ϕ   (X_1, …, X_n)
               =  T^tH_0, …, tH_n_ψ((tH_0+i)^β_0 X_1 (tH_1+i)^β_1, X_2(tH_2+i)^β_2,…, X_n (tH_n+i)^β_n).
        
        Corollary <ref> and Theorem <ref> give that
        
            T^tH_0, …, tH_n_ψ((tH_0+i)^β_0  X_1 (tH_1+i)^β_1, X_2(tH_2+i)^β_2,…, X_n (tH_n+i)^β_n) _^̋q+r+(α-n)h→^̋q
               ≲ (tH_0+i)^β_0_^̋q_0 + β_0h→^̋q_0⋯ (tH_n+i)^β_n_^̋q_n+β_nh→^̋q_n,
        
        for q_i some real numbers. Theorem <ref> gives that 
        
            (tH_j+i)^β_ij_^̋q_j+β_jh→^̋q_j≲sup_x∈ |(tx+i)^β_j | ⟨ x⟩^-β_j≲ t^β_j.
        
        Therefore, 
        
            T^tH_0, …, tH_n_ϕ(X_1, X_2,…, X_n )   _^̋q+r+(α-n)h→^̋q≲ t^β_0 + ⋯ + β_k= t^α-n.
        
        
         §.§ Asymptotic Expansions
        
        Through the identities proved in the previous section, the theory of multiple operator integrals lends itself well for establishing asymptotic expansions of operators. As an immediate example, we first prove a Taylor expansion for pseudodifferential operators, which is the first part of Theorem <ref>.
        
             Let f ∈ T^β(), H ∈^h, h>0 elliptic and symmetric, and let V ∈^r be symmetric. If the order of the perturbation V is strictly smaller than that of H, i.e. r < h, we have
            
            f(H + V) ∼   ∑_n=0^∞ T^H_f^[n](V, …, V),
        
            in the sense that
            
            f(H+V) - ∑_n=0^N T^H_f^[n](V, …, V)∈^m_N
        
            with m_N ↓ -∞.
        
        Using the last part of Proposition <ref> with A = H+V, B = H, we have
        
            f(H+V) - f(H)    = T^H+V_f^[0]() - T^H_f^[0]()
               = T_f^[1]^H+V, H(V).
        
        Repeating the argument, we get
            
            f(H+V) -    ∑_n=0^N  T^H_f^[n](V, …, V) = T_f^[N+1]^H+V, H, …, H(V, …, V).
        
        Now, if h > r, Theorem <ref> gives that
        
            T_f^[N+1]^H+V, H, …, H(V, …, V) ∈^(β-N-1)h+Nr,
        
        with 
        
            (β-N-1)h+Nr = N(r-h) + (β-1)h ↓ -∞.
        
        
        
        Note that, if H and V are commuting numbers, (<ref>) recovers the classic Taylor expansion formula f(H+V)∼∑_n=0^∞f^(n)(H)/n!V^n. The noncommutative Taylor expansion (<ref>) features prominently in <cit.> for bounded operators V.
        
        
        Each term in the Taylor expansion can itself be expanded as follows. Recall that we write δ_H(X) := [H,X], δ_H^n(X) := δ_H(⋯δ_H(δ_H(X)) ⋯). The following proposition finishes the proof of Theorem <ref>.
        
        
         Let X_i ∈^r_i(Θ), H∈^h(Θ), h >0 symmetric and elliptic, and f∈ T^β(). Then
            
            T^H_f^[n](X_1, …, X_n)    = ∑_m=0^N∑_m_1 + … + m_n = mC_m_1, …, m_n/(n+m)!δ_H^m_1(X_1) ⋯δ_H^m_n(X_n) f^(n+m)(H) 
                  +   S^n_N(X_1, …, X_n),
        
            where 
            
            C_m_1, …, m_n :=  ∏_j=1^nj+m_1 + … + m_j-1m_j
        
            and the remainder S^n_N(X_1, …, X_n) is a sum of terms of the form
            
            δ_H^m_1(X_1)⋯δ_H^m_k(X_k)   T^H_f^[n+N+1](1, …, 1, δ_H^N+1-m_1 - ⋯ - m_k(X_k+1), 1, …, 1, X_k+2, …, X_n).
        
        If the commutators δ_H^k(X_j) have a lower order than the expected r_j+kh, explicitly if
            
            δ_H^k(X_j) ∈^r_j+k(h-ε)
        
            for some ε > 0, then the above gives an asymptotic expansion
            
            T^H_f^[n](X_1, …, X_n) ∼∑_m=0^∞∑_m_1 + … + m_n = mC_m_1, …, m_n/(n+m)!δ_H^m_1(X_1) ⋯δ_H^m_n(X_n) f^(n+m)(H),
        
            in the sense that the remainder term 
            
            S^n_N(X_1, …, X_n) ∈^m_N
        
            with m_N = ∑_j r_j + (β-n)h-ε (N+1) ↓ -∞.
        
        As the proof is a lengthy combinatorial exercise, which consists of repeatedly applying Proposition <ref>, we have elected to move the proof of this proposition to Appendix <ref>.
        
            Under the combined assumptions of Theorem <ref> and Proposition <ref>, we get
            
            f(H+V) ∼∑_n,m=0^∞∑_m_1 + … + m_n = mC_m_1, …, m_n/(n+m)!δ_H^m_1(V) ⋯δ_H^m_n(V) f^(n+m)(H).
        
            For N, M > 0, Theorem <ref> and Proposition <ref> give
            
            f(H+V)    = ∑_n=0^N∑_m=0^M∑_m_1 + … + m_n = mC_m_1, …, m_n/(n+m)!δ_H^m_1(V) ⋯δ_H^m_n(V) f^(n+m)(H)
                  + ^N(r-h)+(β-1) h + ∑_n=0^N ^nr + (β-n) h-ε(M+1).
        
            As (N,M) → (∞, ∞), we see that the order of the remainder decreases to -∞.
        
        
        A version of Proposition <ref> for classical pseudodifferential operators (i.e., on a manifold) appears as <cit.>. It should also be noted that the combinatorial or algebraic manipulations needed to prove Proposition <ref> form a part of every proof of the local index formula, see for example <cit.><cit.><cit.>. Namely, the cocycle that lies at the heart of the local index formula is the residue of the trace of an expansion of the above kind. 
        
        This observation leads us naturally to the topic of asymptotic expansions of trace formulas. In various contexts of noncommutative geometry and beyond, expansions are studied of the kind
        
            (f(tH+tV)) t↓ 0∼∑_k=0^∞ c_k t^r_k,
        
        for an increasing sequence r_k ↑∞ and constants c_k ∈ℂ, which means that as t ↓ 0
            (f(tH+tV)) = ∑_k=0^N c_k t^r_k + O(t^r_N+1)
        
        for every N ∈. Or, more generally (c.f. <cit.>),
        
            (f(tH+tV)) t↓ 0∼∑_k=0^∞ρ_k(t),
        
        where ρ_k(t) = O(t^r_k) and 
            (f(tH+tV)) = ∑_k=0^N ρ_k(t) + O(t^r_N+1).
        
        
        We will target a specific open problem posed in some form by Eckstein and Iochum in <cit.>. 
        Given a spectral triple (, ,̋ D) it is a common assumption to require the existence of an asymptotic expansion as t ↓ 0 of
        
            (ae^-tD^2),
        
        where a ∈. It is currently not clear 
        whether the existence of asymptotic expansions of
        
            (ae^-t(D+V)^2)
        
        can be deduced for suitable perturbations V from this, and whether it could be enough for (e^-tD^2) to admit an asymptotic expansion.
        
        <cit.>
            For a bounded operator a and invertible positive operator D such that D^-1∈ℒ_s, s >0, the existence of an asymptotic expansion 
            
            (ae^-tD^2) t ↓ 0∼∑_k=0^∞ρ_k(t),
        
            where 
            
            ρ_k(t) := ∑_z ∈ X_k( ∑_n=0^d c_n,klog^n t ) t^-z,
        
            with c_n,k∈ and for suitable sets X_k ⊂ (for details, see <cit.>),
            implies the existence of a meromorphic continuation of
            
            ζ_D^2,a(s) := (a |D|^-2s)
        
            to the complex plane, with poles of order at most d+1 located at points in ⋃_k=0^∞ X_k ⊂.
        
        
        We first show that in general the asymptotic expansion of (e^-tD^2) provides no control over the expansions of (a e^-tD^2).
        
            Let = ℓ_∞(_≥ 1),=̋ℓ_2(_≥ 1) where  is represented
            on $̋ by pointwise multiplication, and letDbe the diagonal operator on$̋ given by
            
            De_n = ne_n,   n≥ 1.
        
        This is a spectral triple for trivial reasons:  acts on $̋ by bounded operators,
        and[D,a]=0for alla∈.Despite being atypical,(,,̋D)satisfies most of the assumptions commonly made in the literature in terms of smoothness or summability.
        The algebrais not separable, but all of the following arguments can be performed in a separable (even finite dimensional) subalgebra of.It is a classical result that we have the asymptotic expansion
        
            (e^-tD^2)= ∑_n=1^∞ e^-tn^2t ↓ 0∼√(π)/2 t^-1/2 - 1/2,
        
        see for example <cit.>. 
        Nonetheless, the functions(a|D|^-2s)fora∈are very badly behaved. For example, let
        
            a:= ∑_n=2^∞1/log n e_n ∈,
        
        so that
        
            ζ_a,D^2(s)= (aD^-2s) = ∑_n=2^∞1/log nn^-2s,  (s)>1/2,
        
        which is holomorphic on(s) > 1/2. Now,
        
            d/dsζ_a,D^2(s)    = -2∑_n=2^∞ n^-2s
               =2-2ζ(2s),  (s)>1/2,
        
        whereζis the Riemann zeta function which has a simple pole at1. Therefore,ζ_a,D^2(s) + log(2s-1)is the antiderivative of an entire function, which implies thatζ_a,D^2(s) = -log(2s-1) + f(s)wheref(s)is entire <cit.>.
        We conclude thatζ_a,D^2does not admit a meromorphic extension to the complex plane, and thus(ae^-tD^2)does not have an asymptotic expansion ast↓ 0of the type in Theorem <ref>.
        
        An even more pathological example is 
        
            b := ∑_n=2^∞Λ(n)/log(n)e_n
        
        whereΛis the von Mangoldt function which satisfies
        
            Λ(n) :=
            log(p)     if  n=p^k  for  p  prime;
            
                0     otherwise.
         
        A classical formula asserts that <cit.>
            ζ_b,D(s)=(b|D|^-s) = ∑_n=2^∞Λ(n)/log(n)n^-s = logζ(s),   (s) > 1
        
        which is badly behaved at every zero ofζand ats= 1.
        
        
        To study the asymptotic expansion of expressions like
        
            (a e^-t(D+V)^2),
        
        we will use a modified version of Theorem <ref> and Proposition <ref>. For this purpose, we first analyse the remainder in the Taylor expansion in Theorem <ref> more precisely.
        
        Let Θ^-1∈ℒ_s, s > 0, f ∈ T^β(), H ∈^h, h> 0 elliptic and symmetric and V ∈^r symmetric. Let h > r ≥ 0 and β < -s/h.
            For every N ∈, we have as t ↓ 0,
            
            (f(tH+tV)) = ∑_n=0^N t^n( T^tH_f^[n](V, …, V) ) + O(t^(N+1)(1-r/h) - s/h).
        
            The proof of Theorem <ref> gives that
            
            f(tH + tV) = ∑_n=0^N T_f^[n]^tH(tV, …, tV) + T_f^[N+1]^tH + tV, tH, …, tH(tV, …, tV).
        
            The condition β < -s/h assures that all terms on the left and right-hand side are trace-class (cf. (<ref>)). Furthermore, we have f ∈ T^β()⊆ T^(N+1)(1-r/h)-s/h() so that Proposition <ref> provides that
            
            T_f^[N+1]^tH + tV, tH, …, tH   (tV, …, tV)_1 
               ≲ t^N+1T_f^[N+1]^tH + tV, tH, …, tH(V, …, V)_^̋-s→^̋0
               ≲ t^(N+1)(1-r/h) - s/h.
        
        
        This proposition makes it clear that to determine the coefficients of asymptotic expansions of the type (<ref>) or (<ref>), it suffices to study the asymptotic expansions of the multiple operator integral
        
            ( T^tH_f^[n](V, …, V) ),
        
        which we do with Proposition <ref>.
        
        
        
            Let Θ^-1∈ℒ_s, s > 0, f ∈ T^β(), H∈^h, h>0 symmetric and elliptic, V ∈^r symmetric. If h > r ≥0, β≤ -s/h, and δ_H^n(V) ∈^r+n(h-ε), then (f(tH+tV)) admits an asymptotic expansion as t↓ 0 of type (<ref>) given by
            
            (f(tH+tV)) = ∑_n=0^N∑_m=0^N∑_m_1 + … + m_n = m t^n+mC_m_1, …, m_n/(n+m)!( δ_H^m_1(V) ⋯δ_H^m_n(V) f^(n+m)(tH)) + O(t^m_N),
        
            where m_N := (N+1)min(ε/h, (1-r/h) ) - s/h, so that m_N ↑∞ as N →∞.
        
            Combining Propositions <ref> and <ref>, we have that
            
            (f(tH+tV))    = ∑_n=0^N∑_m=0^N  t^n+m∑_m_1 + … + m_n = mC_m_1, …, m_n/(n+m)!( δ_H^m_1(V) ⋯δ_H^m_n(V) f^(n+m)(tH))
                  + ∑_n=0^N t^n (S^n_N,t(V, …, V))+ O(t^(N+1)(1-r/h) - s/h),
        
            where S^n_N,t(V, …, V) is a sum of terms of the form
            
            t^N+1δ_H^m_1(V)⋯δ_H^m_k(V)   T^tH_f^[n+N+1](1, …, 1, δ_H^N+1-m_1 - ⋯ - m_k(V), 1, …, 1, V, …, V).
        
            We then estimate
            
            t^N+1δ_H^m_1(V)⋯δ_H^m_k(V)   T^tH_f^[n+N+1](1, …, 1, δ_H^N+1-m_1 - ⋯ - m_k(V), 1, …, 1, V, …, V)_1
               ≲ t^N+1δ_H^m_1(V)⋯δ_H^m_k(V)   T^tH_f^[n+N+1](1, …, 1, δ_H^N+1-m_1 - ⋯ - m_k(V), 1, …, 1, V, …, V)_^̋-s→^̋0
               ≤ t^N+1δ_H^m_1(V)⋯δ_H^m_k(V) _^̋kr+(m_1 + ⋯ + m_k)(h-ε)→^̋0
                 × T^tH_f^[n+N+1](1, …, 1, δ_H^N+1-m_1 - ⋯ - m_k(V), 1, …, 1, V, …, V)_^̋-s→^̋kr+(m_1 + ⋯ + m_k)(h-ε).
        
            Applying Proposition <ref> then provides that
            
            S^n_N, t(V, …, V)_1    ≲ t^N+1-s+kr+(m_1 + ⋯ + m_k)(h-ε)/h-(n-k)r+(N+1-m_1 - ⋯ - m_k)(h-ε)/h
                  = t^-s+nr/h+(N+1)ε/h,
        
            and hence
            
            ∑_n=0^N t^n (S^n_N,t(V, …, V)) ≲∑_n=0^N t^n(1-r/h)+(N+1)ε/h-s/h.
        
            Defining
            
            m_N := (N+1)min(ε/h, (1-r/h) ) - s/h
        
            concludes the proof.
        
        Given a regular s-summable spectral triple  (,,̋D), write Θ := (1+D^2)^1/2. Let V, P ∈, V self-adjoint and bounded, where  is the algebra generated by  and D. If f ∈ T^β() with β < -s, Proposition <ref> immediately gives that
          
            (f(tD+tV)) = ∑_n=0^N∑_m=0^N∑_m_1 + … + m_n = m t^n+mC_m_1, …, m_n/(n+m)!( δ_H^m_1(V) ⋯δ_H^m_n(V) f^(n+m)(tD)) + O(t^N+1-s).
        
        
        Regarding the expansion of (Pe^-t(D+V)^2) we have that D^2 ∈^2(Θ), A ∈^1(Θ) since (, ,̋ D) is regular. As [D^2, A] = [Θ^2, A] = Θ[Θ, A] + [Θ,A] Θ, we have that A^(m)∈^1+m(Θ). Furthermore, ⊆(Θ). The proof of this corollary is then the same as the proof of Proposition <ref>. Filling in
        
            m_N = (N+1)min(ε/h, (1-r/h) ) - s/h = N+1-s/2
        
        gives the order of the error term.
        
        For the expansion of (Pe^-t|D+V|), while |D| ∈^1(Θ) due to Theorem <ref>, to conclude something similar for |D+V| we have to do more work. Note that D has discrete spectrum since (1+D^2)^-1/2∈ℒ_s. If V ∈^0(Θ) is self-adjoint, then D+V has real discrete spectrum too. Hence we can modify the function x ↦ |x| slightly on a small neighbourhood around x=0 to get a smooth function f which has the property that f(x) = |x| on σ(D+V) ∪σ(D), and f ∈ S^1() since the second and higher derivatives of f are all compactly supported. Using Theorem <ref> and the observation that S^1() ⊆ T^1+ε() for all ε > 0, we have
        
            |D+V| ∈^1+ε(Θ),   |D+V|-|D| = T^D+V, D_f^[1](V) ∈^ε(Θ).
        
        Therefore, we get
        
            (Pe^-t|D+V|) 
               = ∑_n = 0^N∑_m=0^N∑_m_1 + … + m_n = m  (-t)^n+mC_m_1, …, m_n/(n+m)!(Pδ_|D|^m_1(B)⋯δ_|D|^m_n(B)exp(-t|D|)) + O(t^(N+1)(1-ε) - s),
        
        where B := |D+V|-|D| and ε > 0 can be chosen arbitrarily small.
        
        
        Apart from providing a pertubative expansion of the spectral action, Theorem <ref> shows that if for allP ∈we have an expansion
        
            (P e^-tD^2) t↓ 0∼∑_k=0^∞ c_k(P) t^r_k
        
        for constantsc_k(P) ∈, then there exist constantsc_k(P,V) ∈such that
        
            (P e^-t(D+V)^2) t↓ 0∼∑_k=0^∞ c_k(P, V) t^r_k.
        
        Similarly, if each (P e^-tD^2)admits an asymptotic expansions of the type in Theorem <ref>, then (P e^-t(D+V)^2)does too.
        
        
        
            Theorem <ref> can be modified to work for non-unital spectral triples. Given a spectral triple (, ,̋ D) with non-unital algebra  writing Θ = (1+D^2)^1/2, if one assumes instead of Θ^-1∈ℒ_s that there exists p ≥ 1 such that a Θ^-s∈ℒ_1 for all a ∈∪ [D,] and s > p as is proposed in <cit.>, then we also have a ·^-s∈ℒ_1 for s > p. It follows that as t↓ 0
            (af(tD+tV)) = ∑_n=0^N∑_m=0^N  t^n+m∑_m_1 + … + m_n = mC_m_1, …, m_n/(n+m)!(a δ_H^m_1(V) ⋯δ_H^m_n(V) f^(n+m)(tD)) + O(t^N+1-p)
        
            for a ∈∪ [D,], V ∈ bounded and self-adjoint, and f ∈ T^β(), β< -p.
        
        
        In <cit.> the question is asked when, for a spectral triple(, ,̋ D), the existence of an asymptotic expansion of(e^-t|D|)implies the existence of an expansion of(e^-t|D+V|)for a suitable perturbationV. Theorem <ref> compared with Example <ref> suggests that this is not generally possible. We illustrate this with the following example.
        
         
            Let us revisit Example <ref> where (, ,̋ D) = (ℓ_∞(_≥ 1), ℓ_2(_≥ 1), D), and D is defined by
            
            De_n = ne_n,   n≥ 1.
        
            We take as before
            
            a:= ∑_n=2^∞1/log n e_n.
        
            Noting that |D+a| - |D| = a, we can apply Theorem <ref> to get (in this situation we can choose ε = 0)
            
            (e^-t|D+a|) = (e^-tD) -t (ae^-tD) + O(t).
        
            Taking the Mellin transform we get for (s)>1 (see <cit.>)
            
            (|D+a|^-s)    = 1/Γ(s)∫_0^∞ t^s-1(e^-t|D+a|) dt
               = 1/Γ(s)∫_0^1 t^s-1(e^-t|D+a|) dt + 1/Γ(s)∫_1^∞ t^s-1(e^-t|D+a|) dt.
        
        Since
        
            (e^-t|D+a|)) ≤(e^-tD) = (e^t-1)^-1≤ 2 e^-t,   t≥ 1,
        
        we have that
        
            s↦1/Γ(s)∫_1^∞ t^s-1(e^-t|D+a|) ds
        
        is holomorphic.
        It follows that
        
            (|D+a|^-s)    = 1/Γ(s)∫_0^1 t^s-1(e^-t|D+a|) dt + holo_(s)
               =   1/Γ(s)∫_0^1 t^s-1(e^-tD) -t (ae^-tD) dt + holo_(s)>-1(s)
               = ζ_0,D(s) - s ζ_a,D(s+1) + holo_(s)>-1(s).
        
        Since s ↦ζ_a,D(s+1) does not extend holomorphically to any punctured neighbourhood of s=0 as the computations in Example <ref> show, we conclude that ζ_a,D(s) = (|D+a|^-s) does not admit a meromorphic extension to the entire complex plane. By Theorem <ref>, 
        
            (e^-t|D+a|)
        
        does not admit an asymptotic expansion of the type listed in the theorem.
        
        
        We conclude by remarking that the existence of an asymptotic expansion of( Q e^-t D^2)forQ ∈of the type in Theorem <ref> is guaranteed for commutative spectral triples <cit.>, and by the same theorem also for almost commutative spectral triples <cit.>.
        
        
        
        
        
        § ADJOINTS
        
        In this appendix we discuss various ways of defining the adjoint in ^r(Θ). 
        
            Let A ∈^r(Θ) so that A extends to a bounded operator
            
            A: ^̋s+r→^̋s
        
        for all s∈ℝ.
        
          *  The adjoint of A as an endomorphism of the topological vector space ^̋∞ we denote 
        
            A^† : ^̋-∞→^̋-∞
        
        defined by the identity
        
            ⟨ A u, v ⟩_(^̋∞, ^̋-∞) = ⟨ u, A^† v ⟩_(^̋∞, ^̋-∞),    u ∈^̋∞, v∈^̋-∞.
        
          *  In similar fashion we denote the adjoint
        
            A'^_s: ^̋-s→^̋-s-r
        
        defined by the relevant identity
        
            ⟨ Au, v⟩_(^̋s, ^̋-s) = ⟨ u, A'^_sv ⟩_(^̋s+r, ^̋-s-r),    u ∈^̋s+r, v∈^̋-s.
        
          *  We define the Hermitian adjoint
        
            A^♭_s: ^̋s→^̋s+r
        
        via the identity
        
            ⟨ Au, v ⟩_^̋s = ⟨ u, A^♭_s v ⟩_^̋s+r,    u ∈^̋s+r, v∈^̋s.
        
          *  In case r ≥ 0, the map
        
            A: ^̋s+r⊆^̋s →^̋s
        
          is an unbounded operator on the Hilbert space ^̋s, so we define another Hermitian adjoint
        
            A^*_s: 𝒟_s →^̋s,
        
        with domain
        
            𝒟_s := {u ∈^̋s  | ∃ v ∈^̋s ∀ϕ∈^̋s+r : ⟨ u, Tϕ⟩_^̋s = ⟨ v , ϕ⟩_^̋s},
        
        such that
        
            ⟨ Au, v ⟩_^̋s = ⟨ u, A^*_s v ⟩_^̋s,    u ∈^̋s+r, v∈𝒟_s.
        
        
        These adjoints are related in the following way.
        
            Let A ∈^r(Θ). Then, for all s ∈,
            
          * A'^_s = A^†|_^̋-s;
                
          * A^♭_s= Θ^-2s-2r A^†Θ^2s|_^̋s.
            
            If r≥ 0, 
            
          3.A^*_s = Θ^-2s A^†Θ^2s|_𝒟_s.
            
          *  Take u∈^̋∞⊆^̋s+r and v∈^̋-s⊆^̋-∞. Then
                
            ⟨ Au, v ⟩_(^̋s, ^̋-s)   = ⟨ u, A'^_s v⟩_(^̋s+r, ^̋-s-r)
               = ⟨ u, A'^_s v ⟩_(^̋∞, ^̋-∞).
        
                We also have
                
            ⟨ Au, v ⟩_(^̋s, ^̋-s)     = ⟨ Au, v ⟩_(^̋∞, ^̋-∞)
               = ⟨ u, A^† v ⟩_(^̋∞, ^̋-∞).
        
                Hence it follows that
                
            A'^_s v = A^† v ∈^̋-∞,    v ∈^̋-s.
        
          *  Take u ∈^̋∞, v∈^̋s. Then on the one hand,
                
            ⟨ Au, v ⟩_^̋s   = ⟨ Au, Θ^2s v ⟩_(^̋∞, ^̋-∞)
               = ⟨ u, A^†Θ^2s v⟩_(^̋∞, ^̋-∞),
        
                and on the other hand
                
            ⟨ Au, v ⟩_^̋s   = ⟨ u, A^♭_s v ⟩_^̋s+r
               = ⟨ u, Θ^2s+2r A^♭_s v ⟩_(^̋∞, ^̋-∞).
        
                We therefore find
                
            A^♭_s = Θ^-2s-2r A^†Θ^2s|_^̋s.
        
          *  Take u ∈^̋∞, v ∈𝒟_s ⊆^̋s ⊆^̋-∞. Then
                
            ⟨ Au, v⟩_^̋s   = ⟨ Au, Θ^2s v ⟩_(^̋∞, ^̋-∞)
               = ⟨ u, A^†Θ^2s v ⟩_(^̋∞, ^̋-∞),
        
                and
                
            ⟨ Au, v⟩_^̋s   = ⟨ u, A^*_s v⟩_^̋s
                =⟨ u, Θ^2s A^*_s v ⟩_(^̋∞, ^̋-∞).
        
                Hence
                
            A^*_s = Θ^-2s A^†Θ^2s|_𝒟_s.
        
        
        An important takeaway from this proposition is that if A : ^̋∞→^̋∞, we have a priori that
        
            A^†: ^̋-∞→^̋-∞,
        
        but if A ∈^r we have in fact that A^†∈^r (or, more precisely, A^†|_^̋∞∈^r). 
        
        It is now also clear that the Hermitian adjoints A^♭_s and A^*_s in general cannot be regarded as operators in (Θ) as the operators A^♭_s and A^♭_t do not agree on the intersection ^̋s ∩^̋t for s≠ t, and the same holds for A^*_s.
        
        
            If A ∈^r, r ≥ 0, then 
            
            A: ^̋r⊆^̋0 →^̋0
         is symmetric if and only if A = A^†. 
        
        Suppose that A = A^†. Let u∈^̋∞, v ∈^̋r. Then
            
            ⟨ Au, v ⟩_^̋0   = ⟨ Au, v⟩_(^̋∞, ^̋-∞)
               = ⟨ u, A^† v ⟩_(^̋∞, ^̋-∞)
               = ⟨ u, Av ⟩_^̋0.
        
         By density of ^̋∞⊆^̋r, the above equality holds for u ∈^̋r as well, and hence A: ^̋r⊆^̋0 →^̋0 is symmetric.
        
         On the other hand, if A: ^̋r⊆^̋0 →^̋0
         is symmetric, then for u,v ∈^̋∞,
            ⟨ u, A^† v ⟩_(^̋∞, ^̋-∞)   = ⟨ Au, v⟩_^̋0
               = ⟨ u, Av ⟩_^̋0
               = ⟨ u, Av ⟩_(^̋∞, ^̋-∞),
        
         showing that A^† v = Av ∈^̋∞ which implies that A=A^†∈^r.
        
        
        § FUNCTIONAL CALCULUS FOR OP0
        
        In Section <ref> we proved that symmetric elliptic operators in ^r for r>0 admit a functional calculus. The approach of that section does not apply for the case r=0. To illustrate how different the zero-order case is, consider the situation where Θ = (1+Δ)^1/2 on L_2(M) where M is a compact subset of ^d. We have that for ϕ: M →ℝ, the multiplication operator
            
            M_ϕ: L_2(M)    → L_2(M)
            ξ   ↦ϕ·ξ,
        
            where ϕ·ξ (x) = ϕ(x) ξ(x), can only be in (Θ) if ϕ is smooth. For smooth ϕ we have that M_ϕ∈^0(Θ). If f(M_ϕ) ∈(Θ) for all M_ϕ∈^0(Θ), then the identity 
            
            f(M_ϕ) = M_f ∘ϕ
        
            shows that the function f has to be smooth itself and no functional calculus with general functions in L^β_∞(E) is possible.
        
        For ^0(Θ) we therefore use a different strategy altogether. 
        An approach by Davies <cit.> on the construction of a functional calculus using almost analytic extensions directly applies. Using almost analytic extensions to obtain a functional calculus for pseudodifferential operators has precedent in the works of, amongst others,  Hörmander <cit.>, Helffer–Sjöstrand <cit.>, Dimassi–Sjöstrand <cit.>, and Bony <cit.>.
        
        
            Let f ∈ C^∞_c(ℝ). We define an extension f̃: ℂ→ℂ by
            
            f̃(x+iy) := τ(y/⟨ x ⟩ ) ∑_k=0^N f^(k)(x) (iy)^k/k!,
        
            where N ≥ 1 and τ:ℝ→ℝ is a smooth bump function with τ(s) = 0 for |s|>2, τ(s) =1 for |s|<1. Then we have
            
            f(x) = -1/π∫_ℂ( ∂f̃/∂z(z)) (z-x)^-1 dz,    x ∈ℝ,
        
            independent of the choice of τ and N. We refer to f̃ as an almost analytic extension of f.
        
            Let f ∈ C^∞_c(ℝ) with almost analytic extension f̃ as in Definition <ref>, so that
            
            f(x) = -1/π∫_ℂ( ∂f̃/∂z(z)) (z-x)^-1 dz,    x ∈ℝ.
        
            For any closed, densely defined operator H with σ(H) ⊆ℝ, if for some α∈ℝ_≥ 0 we have the estimate
            
            (z-H)^-1≤ C 1/|(z)|(⟨ z ⟩/|(z)|)^α,    z∈ℂ⊆ℝ,
        
            then we have that
            
            f(H) := -1/π∫_ℂ∂f̃/∂z (z-H)^-1 dz
        
            defines a bounded operator on $̋ independent of the choice ofN > αandτin the construction of the extensionf̃, with
            
            f(H)_∞≤∑_k=0^N+1f_T^0(), k.
        
            The integral should be interpreted as aB()̋-valued Bochner integral. In caseHis self-adjoint, this agrees with the continuous functional calculus.
        
        We thank Dmitriy Zanin for providing a key step in the following proof, which is an adaptation of an argument by Beals <cit.>.
        
        
        
            Let X ∈^r(Θ) be such that [Θ, X] ∈^r. If the extension 
            
            X: ^̋s_0+r→^̋s_0
        
            has a bounded inverse 
            
            X^-1:^̋s_0→^̋s_0+r
        
            for one particular s_0 ∈ℝ, then X^-1|_^̋∞∈^-r. We have X X^-1|_^̋∞ = X^-1X|_^̋∞ = 1_^̋∞. In particular, if X ∈^r and [Θ, X] ∈^r with r ≥ 0, then we have as (unbounded) operators 
            
            σ(X: ^̋s_0 +r⊆^̋s_0→^̋s_0) = σ(X: ^̋s+r⊆^̋s →^̋s)
        
            for all s∈ℝ, where σ denotes the spectrum of the operator.
        
        Since ^r = ^0 ·Θ^r, it suffices to prove the proposition for r=0. 
        
        Suppose that X ∈^0 is a bijection on ^̋s_0→^̋s_0 and write X^-1:^̋s_0→^̋s_0. Then X restricts to a necessarily injective map ^̋s_0+1→^̋s_0+1. We now prove that X: ^̋s_0+1→^̋s_0+1 is also surjective.
        
        We follow <cit.>, filling in some omitted details. Take v ∈^̋s_0 + 1, then there exists u ∈^̋s_0 with Xu = v. Let ε >0, then Θ/1+εΘ∈^0, and
        
            Θ/1+ εΘ u    = Θ/1+ εΘ X^-1 v 
               = X^-1Θ/1+εΘv + X^-1[Θ/1+ εΘ, X] X^-1v.
        
        Now, the ^̋s_0 norm of the right-hand side is bounded independent of ε: 
        
            Θ/1+εΘ_^̋s_0+1→^̋s_0   ≤Θ_^̋s_0+1→^̋s_0;
            [Θ/1+ εΘ, X] _^̋s_0→^̋s_0   = (1+εΘ)^-1[Θ,X](1+εΘ)^-1_^̋s_0→ H^s_0≤[Θ, X]_^̋s_0→^̋s_0.
        
        This implies that u ∈^̋s_0+1, a fact that can be quickly verified with the spectral theorem and Fatou's lemma. Therefore, 
        
            X: ^̋s_0 + 1→^̋s_0+1
        
        is a bijection. By induction and interpolation (Proposition <ref>), the same assertion holds for each ^̋s, s ≥ s_0.
        
        Finally, it is a basic fact that the adjoint of a bijective operator is bijective, i.e.
        
            X'^_s_0 = X^†|_^̋-s_0:^̋-s_0→^̋-s_0
        
        is a bijection. Since
        
            [Θ, X^†] = - [Θ, X]^†∈^0,
        
        we can apply the same arguments as above to deduce that
        
            X^†:^̋-s→^̋-s
        
        is a bijection for all -s ≥ -s_0. This implies that 
        
            X = X^††: ^̋s→^̋s
        
        is a bijection for all s ≤ s_0.
        
        This following type of estimate on the resolvent also appears inL_p-boundedness problems, see <cit.>.
        
            Let A ∈^0 be such that [Θ, A] ∈^0 and A^0,0:→̋$̋ is self-adjoint. 
            Then for alls ∈ℝ,there is a constantC_s > 0such that
            
            (z-A)^-1_^̋s→^̋s   ≤ C_s 1/|(z)|(⟨ z ⟩/|(z)|)^2^|s|-1,    z ∈ℂ∖ℝ.
        
            The proof is by induction and interpolation (Proposition <ref>). For s=0, the estimate holds by self-adjointness of A^0,0. Note that (z-A)^-1∈^0 due to Proposition <ref>.
            
            Suppose the inequality is proved for a fixed s∈_≥ 0. Then for z ∈ℂ∖ℝ,
            
            (z-A)^-1_^̋s+1→^̋s+1
               = Θ(z-A)^-1Θ^-1_^̋s→^̋s
               ≤(z-A)^-1_^̋s→^̋s+(z-A)^-1[Θ,A](z-A)^-1Θ^-1_^̋s→^̋s
               ≤(z-A)^-1_^̋s→^̋s(1+ [Θ,A](i+A)^-1_^̋s→^̋s(i+A)(z-A)^-1_^̋s→^̋sΘ^-1_^̋s →^̋s).
        
            Note that (i+A)^-1∈^0 by Proposition <ref>, so that for some constant B_s >0,
            
            [Θ,A](i+A)^-1_^̋s→^̋sΘ^-1_^̋s→^̋s≤ B_s.
        
            Using the resolvent identity, we have
            
            (i+A)(z-A)^-1 = (i+z)(z-A)^-1-1
        
            and therefore
            
            (i+A)(z-A)^-1_^̋s→^̋s   ≤ 1+ |z+i|(z-A)^-1_^̋s→^̋s.
        
            This yields
            
            (z-A)^-1_^̋s+1→^̋s+1   ≤(z-A)^-1_^̋s→^̋s (1+B_s )
                  + |z+i| B_s  (z-A)^-1_^̋s→^̋s^2 
               ≤ (1+B_s) (z-A)^-1_^̋s→^̋s· (1+ |z+i|(z-A)^-1_^̋s→^̋s).
        
            This estimate also holds with |z-i| on the right-hand side, and min(|z+i|,|z-i|)≤⟨ z⟩, so that
            
            (z-A)^-1_^̋s+1→^̋s+1   ≤  (1+B_s)  (z-A)^-1_^̋s→^̋s· (1+ ⟨ z ⟩(z-A)^-1_^̋s→^̋s),
        
            from which the claimed estimate follows. Induction and interpolation now provide the estimate for all s ≥ 0. 
        
            The case s ≤ 0 is proved in the same manner, using induction in the negative direction. Namely, the norm 
            
            (z-A)^-1_^̋s-1→^̋s-1 = Θ^-1(z-A)^-1Θ_^̋s→^̋s
        
            can be estimated as before.
        
            Let A ∈^0 be such that [Θ, A] ∈^0 and A^0,0: →̋$̋ is self-adjoint. Iff ∈ C^∞(), thenf(A) ∈^0with
            
            f(A)_^̋s →^̋s≤∑_k=0^⌈ 2^|s|⌉+1f_T^0(), k.
        
        Without loss of generality, we assume that f∈ C_c^∞(). As A∈^0, it extends to a bounded operator
        
            A^s,s: ^̋s →^̋s,    s∈.
        
        Furthermore, by Proposition <ref>, we have
        
            (A-z)^-1∈^0,    z ∈∖.
        
        Theorem <ref> and Lemma <ref> combined give that
        
            f(A^s,s): ^̋s→^̋s,    s ∈,
        
        is a bounded operator with the norm bound as claimed. By construction, for ξ∈^̋∞ we have that
        
            f(A^s,s)ξ = ∫_∂f̃/∂z (A^s,s - z)^-1ξ dz ∈^̋s,
        
        as an ^̋s-valued Bochner integral. It is clear that
        
            (A^s,s - z)^-1|_^̋∞ = (A-z)^-1,
        
        and therefore these arguments show that for ξ∈^̋∞ the integral
        
            ∫_∂f̃/∂z (A - z)^-1ξ dz
        
        can be evaluated as a Bochner integral in each Sobolev space ^̋s. Hence
        
            ξ↦∫_∂f̃/∂z (A - z)^-1ξ dz
        
        forms a bounded linear map on ^̋∞; denote this operator by f(A): ^̋∞→^̋∞. Then since f(A) agrees with f(A^s,s) on ^̋∞, we must have
        
            f(A)^s,s = f(A^s,s),
        
        and thus we have f(A) ∈^0.
        
        
        This functional calculus can be used to construct multiple operator integrals, similarly to Theorem <ref>.
        
            Let H_0, …, H_n, [Θ, H_0], …, [Θ, H_n] ∈^0(Θ) be such that each H_i^0,0 is self-adjoint. For ϕ∈ T^0() ⊠_i ⋯⊠_i T^0() (see Definition <ref>) with corresponding representation
            
            ϕ(λ_0,…,λ_n)=∫_Ω a_0(λ_0,ω)⋯ a_n(λ_n,ω) dν(ω),
        
        the integral
        
            T_ϕ^H_0,…,H_n(X_1,…,X_n)ψ:=∫_Ω a_0(H_0,ω)X_1 a_1(H_1,ω)⋯ X_n a_n(H_n,ω)ψ dν(ω),   ψ∈^̋∞
        
        for X_1,…,X_n∈, converges as a Bochner integral in ^̋s for every s ∈, and defines an n-multilinear map T_ϕ^H_0,…,H_n:^r_1×⋯×^r_n→^∑_j r_j.
        
            Recall that by definition of T^0() ⊠_i ⋯⊠_i T^0() we have for all k_0, …, k_n ≥ 0
            ∫_Ωa_0(·, ω)_T^0(), k_0⋯a_n(·, ω)_T^0(), k_n d|ν|(ω) < ∞.
        
            The proof of the theorem is then identical to the proof of Theorem <ref>, using that
            
            a_j(H_j, ω)_^̋s →^̋s≤∑_k=0^⌈ 2^|s|⌉+1a_j(·, ω)_T^0(), k
        
            instead of
            
            a_j(H_j,ω)_^s+k_j→^s≤ C_s,H_ja_j(·,ω)_L^β_j_∞(E_j).
        
        Due to Lemma <ref>, the divided differencef^[n]forf ∈ T^α()withα < nis in particular a permitted symbol.
        
        
        One may wonder how the conditionA, [Θ,A] ∈^rcompares to the condition ofAbeing elliptic withr >0. The following consequence of Proposition <ref> shows that, under the assumption thatAis self-adjoint with domain^̋r, ellipticity is a weaker condition.
        
            Let A∈^r(Θ), r>0, be such that
            
            A:^̋r⊆^̋0 →^̋0
        
            has a non-empty resolvent set (for example if A is self-adjoint with domain ^̋r), and suppose that [Θ, A] ∈^r as well. Then A is elliptic.
        
            By assumption, there exists z ∈ such that 
            
            z-A: ^̋r ⊆^̋0 →^̋0
        
            is invertible, and because [Θ, A] ∈^r it follows that (z-A)^-1∈^-r by Proposition <ref>. Now,
            
            A (z-A)^-1 = - 1 + z (z-A)^-1.
        
            In other words, -(z-A)^-1 is an inverse of A modulo ^-r. Corollary <ref> gives that A elliptic.
        
        
        § COMBINATORIAL COMPUTATIONS
        
        This appendix is dedicated to proving Proposition <ref>. The computations are standard, cf. <cit.><cit.><cit.>– the novelty is that they can be performed in the very general context of (unbounded) MOIs.
        
        
            The multiset coefficient nk for n, k ∈ℤ_≥ 0 is defined as 
            
            nk := n+k-1k.
         For f∈ T^β(), H∈^h, h> 0 symmetric and elliptic, X_i ∈^r_i(Θ), we have
        
            T^H _f^[n+j](1, …, 1_j, X_1, …, X_n ) 
               = ∑_m=0^Nm+1jδ_H^m(X_1) T^H_f^[n+j+m](1, …, 1_j+1+m, X_2, …, X_n) + R^n_j,N(X_1, …, X_n),
        
        where 
        
            R^n_j,N(X_1, …, X_n) :=  ∑_l=0^jN+1l T^H_f^[n+j+N+1](1, …, 1_j-l, δ_H^N+1(X_1), 1, …, 1_N+1+l, X_2, …, X_n).
        
            Multiset coefficients have the property that
            
            ∑_l=0^j ml = m+1j.
        
            The assertion of the lemma follows by induction on N. For N=0,
            
            T^H_f^[n+j](1, …, 1_j, X_1, …, X_n )
               = X_1 T^H_f^[n+j](1, …, 1_j+1, X_2, …, X_n ) + ∑_l=0^j T^H_f^[n+j+1](1, …, 1_j-l, δ_H(X_1), 1, …, 1_1+l, X_2, …, X_n)
        
            by applying Proposition <ref>j+1 times on X_1.
        
            Suppose that the assertion holds for N-1. Then 
            
            T^H_f^[n+j](1, …, 1_j, X_1, …, X_n ) 
               = ∑_m=0^N-1m+1jδ_H^m(X_1) T^H_f^[n+j+m](1, …, 1_j+1+m, X_2, …, X_n) 
                  +  ∑_l=0^jNl T^H_f^[n+j+N](1, …, 1_j-l, δ_H^N(X_1), 1, …, 1_N+l, X_2, …, X_n)
               =^*  ∑_m=0^N-1m+1jδ_H^m(X_1) T^H_f^[n+j+m](1, …, 1_j+1+m, X_2, …, X_n) 
                  +  ∑_l=0^jNlδ_H^N(X_1) T^H_f^[n+j+N](1, …, 1_N+1+j, X_2, …, X_n)
                  +  ∑_l=0^jNl∑_k=0^j-l T^H_f^[n+j+N+1](1, …, 1_j-l-k, δ_H^N+1(X_1), 1, …, 1_N+1+l+k, X_2, …, X_n),
        
            where in the step marked with * we applied Proposition <ref>j-l times on δ_H^N(X_1). Continuing on, 
            
            T^H_f^[n+j](1, …, 1_j, X_1, …, X_n )
               =  ∑_m=0^N-1m+1jδ_H^m(X_1) T^H_f^[n+j+m](1, …, 1_j+1+m, X_2, …, X_n) 
                  +   N+1jδ_H^N(X_1) T^H_f^[n+j+N](1, …, 1_N+1+j, X_2, …, X_n)
                  +  ∑_l=0^j∑_k=0^j-lNl T^H_f^[n+j+N+1](1, …, 1_j-l-k, δ_H^N+1(X_1), 1, …, 1_N+1+l+k, X_2, …, X_n).
        
         In the last sum, relabel r:= k+l, so that
        
            T^H_f^[n+j](1, …, 1_j, X_1, …, X_n ) 
               = ∑_m=0^Nm+1jδ_H^m(X_1) T^H_f^[n+j+m](1, …, 1_j+1+m, X_2, …, X_n) 
                  +  ∑_r=0^j∑_l=0^rNl T^H_f^[n+j+N+1](1, …, 1_j-r, δ_H^N+1(X_1), 1, …, 1_N+1+r, X_2, …, X_n)
               = ∑_m=0^Nm+1jδ_H^m(X_1) T^H_f^[n+j+m](1, …, 1_j+1+m, X_2, …, X_n) 
                  +  ∑_r=0^jN+1r T^H_f^[n+j+N+1](1, …, 1_j-r, δ_H^N+1(X_1), 1, …, 1_N+1+r, X_2, …, X_n).
        
        This concludes the induction step.
        
        Apply Lemma <ref> first to the first entry of T^H_f^[n](X_1, …, X_n),
        
            T^H_f^[n](X_1, …, X_n )    = ∑_m_1=0^Nm_1+10δ_H^m_1(X_1) T^H_f^[n+m_1](1, …, 1_m_1+1, X_2, …, X_n) + R^n_0,N(X_1, …, X_n).
        
        Apply Lemma <ref> once more, expanding up to order N-m_1 instead of N.
        
            T^H_f^[n](X_1, …, X_n ) 
               = ∑_m_1=0^N∑_m_2=0^N-m_1m_1+10m_2+1m_1+1δ_H^m_1(X_1) δ_H^m_2(X_2) T^H_f^[n+m_1+m_2](1, …, 1_2+m_1+m_2, X_3, …, X_n)
                  + ∑_m_1=0^Nm_1+10δ_H^m_1(X_1) R^n-1_1+m_1,N-m_1(X_2, …, X_n) + R^n_0,N(X_1, …, X_n).
        
        Repeating gives the formula
        
            T^H_f^[n](X_1, …, X_n )
               = ∑_m=0^N∑_m_1 + … + m_n = m∏_j=1^n  m_j+1j-1+m_1+… + m_j-1δ_H^m_1(X_1) ⋯δ_H^m_n(X_n) T^H_f^[n+m](1, …, 1_n+m)
                  + S^n_N(X_1, … X_n),
        
        where
        
            S^n_N(X_1, … X_n)   :=   ∑_k=0^n-1∑_m_1 + … + m_k ≤ N∏_j=1^km_j+1j-1+m_2+… + m_j-1
                 ×δ_H^m_1(X_1)⋯δ_H^m_k(X_k)  R^n-k_k+m_1+…+m_k,N-m_1-…-m_k(X_k+1, …, X_n).
        
        
        The observation that 
        
            nk = k+1n-1,
        
        and the definition
        
            nk = n+k-1k,
        
        finishes the proof of the proposition.