SPECTRAL AVERAGING FOR TRACE
COMPATIBLE OPERATORS
N.A.AZAMOV AND F.A. SUKOCHEV
Abstract. In this note the notions of trace compatible operators and infin-
itesimal spectral flow are introduced. We define the spectral shift function
as the integral of infinitesimal spectral flow. It is proved that the spectral
shift function thus defined is absolutely continuous and Krein’s formula is es-
tablished. Some examples of trace compatible affine spaces of operators are
given.
1. Introduction
Let H0 be a self-adjoint operator, and let V be a trace class operator on a
Hilbert space H. Then M.G.Krĕın’s famous result [13] says that there is a unique
L1 -function ξH0+V,H0(λ), known as the Krein spectral shift function, such that for
any C∞c (R) function f
Tr(f(H0 + V )− f(H0)) =
f ′(λ)ξH0+V,H0(λ) dλ.(1)
The notion of the spectral shift function was discovered by the physicist I.M. Lifshits
[15]. An excellent survey on the spectral shift function can be found in [6].
In 1975, Birman and Solomyak [7] proved the following remarkable formula for
the spectral shift function
ξ(λ) =
Tr(V EHr
(−∞,λ]) dr,(2)
where Hr = H0 + rV, r ∈ R, and EHr(−∞,λ] is the spectral projection. Birman-
Solomyak’s proof relies on double operator integrals. An elementary derivation of
(2) was obtained in [10] (without using double operator integrals).
Actually, this spectral averaging formula was discovered for the first time by
Javrjan [12] in 1971, in case of a Sturm-Liouville operator on a half-line, perturba-
tion being a perturbation of the boundary condition, so that in this case V was
one-dimensional. An important contribution to spectral averaging was made by
A.B.Alexandrov [1]. In 1998, B. Simon [18, Theorem 1] gave a simple short proof
of the Birman-Solomyak formula. He also noticed, that this formula holds for the
wide class of Schrödinger operators on Rn [18, Theorems 3,4]. The connection of
1991 Mathematics Subject Classification. Primary 47A11; Secondary 47A55 .
Key words and phrases. Spectral shift function, spectral averaging, infinitesimal spectral flow,
trace compatible operators, semifinite von Neumann algebra .
http://arxiv.org/abs/0704.1683v2
2 N.A.AZAMOV AND F.A. SUKOCHEV
this formula with the integral formula for spectral flow from non-commutative ge-
ometry is outlined in [3]. An interesting approach to spectral averaging via Herglotz
functions can be found in [11].
In this note we present an alternative viewpoint to the spectral shift function,
and generalize the result of Simon so that it becomes applicable to a class of Dirac
operators as well.
The new point of view, which the Birman-Solomyak formula suggests, is that
there is a more fundamental notion than that of the spectral shift function. We
call this notion the speed of spectral flow or infinitesimal spectral flow of a self-
adjoint operator H under perturbation by a bounded self-adjoint operator V. It
was introduced in [3] in the case of operators with compact resolvent. It is defined
by formula
ΦH(V )(ϕ) = Tr(V ϕ(H)), ϕ ∈ C∞c (R),(3)
whenever this definition makes sense. This naturally leads to the notion of trace
compatibility of two operators. We say that operators H and H + V are trace
compatible, if for all ϕ ∈ C∞c (R) the operator V ϕ(H) is trace class. The spectral
shift function between two trace compatible operators is then considered as the
integral of infinitesimal spectral flow. It turns out that the spectral shift function
does not depend on the path connecting the initial and final operators, a fact
which follows from the aforementioned result of B. Simon in the case of trace class
perturbations.
The results of this note are summarized in Theorem 2.9. This theorem extends
formulae (1) and (2) to the setting of trace compatible pairs (H,H + V ) and also
strengthens [18, Theorems 3,4] in the sense that it does not require H to be a
positive operator and maximally weakens conditions on the path H+rV, r ∈ [0, 1].
Our results also hold for a more general setting, when H = H∗ is affiliated with a
semifinite von Neumann algebra N and V = V ∗ ∈ N .
Our investigation here also strengthens the link between the theory of the Krein
spectral shift function and that of spectral flow firstly discovered in [2]. For exposi-
tion of the latter theory we refer to [5] and a detailed discussion of the connection
between the two theories in the situation where the resolvent of H is τ -compact
(here, τ is an arbitrary faithful normal semifinite trace on N ) is contained in [3].
It should be pointed out here that the idea of viewing the spectral shift function
as the integral of infinitesimal spectral flow is akin to I.M. Singer’s ICM-1974 pro-
posal to define the η invariant (and hence spectral flow) as the integral of a one
form. Very general formulae of that type have been produced in the framework
of noncommutative geometry (see [5] and references therein). We believe that our
present approach will have applications to noncommutative geometry, in particular,
it may be useful in avoiding ”summability constraints” on H customarily used in
that theory.
In semifinite von Neumann algebras N Krein’s formula (1) was proved for the
first time in [9] in case of a bounded self-adjoint operator H ∈ N and a trace class
perturbation V = V ∗ ∈ L1(N , τ) and in [4] for self-adjoint operators H affiliated
with N .
SPECTRAL AVERAGING 3
An additional reason to call ΦH(V ) the speed of spectral flow is the follow-
ing observation. Let H be the operator of multiplication by λ on L2(R, dρ(λ))
with some measure ρ and let the perturbation V be an integral operator with a
sufficiently regular (for example C1 ) kernel k(λ′, λ). Then for any test function
ϕ ∈ C∞c (R)
ΦH(V )(ϕ) = Tr(V ϕ(H)) =
k(λ, λ)ϕ(λ) dρ(λ).
Hence, the infinitesimal spectral flow of H under perturbation by V is the measure
on the spectrum of H with density k(λ, λ) dρ(λ). We note that this agrees with
the classical formula [14, (38.6)]
n = Vnn
from formal perturbation theory. Here E
n is the n -th eigenvalue of the unper-
turbed operator H0, E
n , j = 1, 2, . . . is the j -th correction term for the n -th
eigenvalue En of the perturbed operator H = H0 + V in the formal perturbation
series En = E
n + E
n + E
n + . . . , and Vmn =
m |V |ψ(0)n
is the matrix
element of the perturbation operator V with respect to the eigenfunctions ψ
and ψ
n of the unperturbed operator H0 [14].
Acknowledgement. We thank Alan Carey for useful comments and criticism.
2. Results
Let N be a von Neumann algebra on a Hilbert space H with faithful normal
semifinite trace τ. Let A = H0 + A0 be an affine space of self-adjoint operators
affiliated with N , where H0 is a self-adjoint operator affiliated with N and A0
is a vector subspace of the real Banach space of all self-adjoint operators from N .
We say that A is trace compatible, if for all ϕ ∈ C∞c (R), V ∈ A0 and H ∈ A
V ϕ(H) ∈ L1(N , τ),(4)
where L1(N , τ) is the ideal of trace class operators from N , and if A0 is endowed
with a locally convex topology which coincides with or is stronger than the uniform
topology, such that the map (V1, V2) ∈ A20 7→ V1ϕ(H0 + V2) is L1 continuous for
all H0 ∈ A and ϕ ∈ C∞c (R). In particular, A is a locally convex affine space.
The ideal property of L1(N , τ) and [16, Theorem VIII.20(a)] imply that, in the
definition of trace compatibility, the condition ϕ ∈ C∞c (R) may be replaced by
ϕ ∈ Cc(R). It follows from the definition of the topology on A0 that H ∈ A 7→ eitH
is norm continuous.
If A = H0+A0 is a trace compatible affine space then we define a (generalized)
one-form (on A ) of infinitesimal spectral flow or speed of spectral flow by the
formula
ΦH(V ) = τ(V δ(H)), H ∈ A, V ∈ A0,(5)
where δ is Dirac’s delta function. The last formula is to be understood in a
generalized function sense, i.e. ΦH(V )(ϕ) = τ(V ϕ(H)), ϕ ∈ C∞c (R). Φ is a
generalized function, since if ϕn → 0 in C∞c (R) such that supp(ϕn) ⊆ ∆ then
|τ(V ϕn(H))| 6
∥V EH∆
‖ϕn(H)‖ → 0. Here ‖A‖1 = τ(|A|).
4 N.A.AZAMOV AND F.A. SUKOCHEV
Since ϕ can be taken from Cc(R), for each V ∈ A0 the infinitesimal spectral
flow ΦH(V ) is actually a measure on the spectrum of H.
By a smooth path {Hr}r∈R in A, we mean a differentiable path, such that its
derivative dHr
∈ A0 is continuous.
Let Π =
(s0, s1) ∈ R2 : s0s1 > 0, |s1| 6 |s0|
, and let
dνf (s0, s1) = sgn(s0)
f̂(s0) ds0 ds1.
If f ∈ C2c (R) then (Π, νf ) is a finite measure space [2]. For any H0, H1 ∈ A, any
X ∈ A0 and any non-negative f ∈ C∞c (R) set by definition
(6) T
H1,H0
f [1]
(X) =
ei(s0−s1)H1
f(H1)Xe
is1H0
+ ei(s0−s1)H1X
f(H0)e
is1H0
dν√f (s0, s1),
where the integral is taken in the so∗ -topology. For justification of this notation
and details see [3].
Lemma 2.1. If {Hr} ⊂ A is a path, continuous (smooth) in the topology of A0,
and if f ∈ C2c (R) then
r 7→ f(Hr)− f(H0)
takes values in L1(N , τ) and it is L1(N , τ) continuous (smooth).
Proof. We can assume that f is non-negative and that
f ∈ C2c (R). It is proved
in [3] that
f(Hr)− f(H0) = THr ,H0f [1] (Hr −H0).(7)
Since ei(s0−s1)x
f(x), eis1x
f(x) ∈ C2c (R), trace compatibility implies that the
integrand of the right hand side of (6) takes values in L1 and is L1 -continuous
(smooth), so the dominated convergence theorem completes the proof. �
If Γ = {Hr}r∈[0,1] is a smooth path in A, then we define the spectral shift
function ξ along this path as the integral of infinitesimal spectral flow: ξ =
ξ(ϕ) =
ϕ(Hr)
dr, ϕ ∈ C∞c (R).(8)
Now we prove that the spectral shift function is well-defined in the sense that it
does not depend on the path of integration.
A one-form αH(V ) on an affine space A is called exact if there exists a zero-
form θH on A such that dθ = α, i.e.
αH(V ) =
θH+rV
We say that the generalized one-form Φ is exact if Φ(ϕ) is an exact form for any
ϕ ∈ C∞c (R).
SPECTRAL AVERAGING 5
The proof of the following proposition follows the lines of the proof of [3, Propo-
sition 3.5].
Proposition 2.2. The infinitesimal spectral flow Φ is exact.
Proof. Let V ∈ A0, Hr = H0 + rV, r ∈ [0, 1], and let f ∈ C∞c (R). By (7)
(9) f(Hr)− f(H0) = THr ,H0f [1] (rV )
ei(s0−s1)Hr
f(Hr)rV e
is1H0 + ei(s0−s1)HrrV
f(H0)e
is1H0
dν√f (s0, s1),
ei(s0−s1)H0
f(H0)rV e
is1H0 + ei(s0−s1)H0rV
f(H0)e
is1H0
dν√f (s0, s1)
(ei(s0−s1)Hr
f(Hr)− ei(s0−s1)H0
f(H0))rV e
is1H0
+ (ei(s0−s1)Hr − ei(s0−s1)H0)rV
f(H0)e
is1H0
dν√f (s0, s1)
H0,H0
f [1]
(rV ) +R1 +R2.
All three summands here are trace class by the trace compatibility assumption. So,
for any S ∈ N
τ(S(f(Hr)− f(H0))) = rτ
H0,H0
f [1]
+ τ(SR1) + τ(SR2).
Now, Duhamel’s formula and (7) show that τ(SR1) = o(r) and τ(SR2) = o(r).
Hence,
τ(S(f(Hr)− f(H0))) = τ
Hr,Hr
f [1]
.(10)
This implies that for any S ∈ N
τ(S(f(H1)− f(H0))) = τ
Hr ,Hr
f [1]
(V ) dr
Now let H0 ∈ A be a fixed operator and for any f ∈ C∞c (R) let
τ(V f(Hr)) dr,
where Hr = H0 + rV, H = H1. We are going to show that dθ
H(X) = ΦH(X)(f)
for any X ∈ A0.
Following the proof of [3, Proposition 3.5] we have
(A) := dθ
H(X) = lim
Xf(Hr + srX)
+ lim
f(Hr + srX)− f(Hr)
By definition of A0 topology the integrand of the first summand is continuous with
respect to r and s. So, the first summand is equal to
Xf(Hr)
6 N.A.AZAMOV AND F.A. SUKOCHEV
By [2, Theorem 5.3] the second summand is equal to
Hr+srX,Hr
f [1]
(srX)
dr = lim
Hr+srX,Hr
f [1]
Hr,Hr
f [1]
Hr ,Hr
f [1]
r dr,
where the second equality follows from the definition of A0 -topology and the last
equality follows from [3, Lemma 3.2]. Now, using (10) and integrating by parts we
(11) (A) = τ (Xf(H1)−Xf(H0)) +
(τ(Xf(Hr))− τ(X [f(Hr)− f(H0)])) dr
= τ (Xf(H1)) .
The argument before [8, Proposition 1.5] now implies
Corollary 2.3. The spectral shift function given by (8) is well-defined.
Proposition 2.4. If r ∈ R 7→ Hr ∈ A is smooth then the equality
df(Hr)
ϕ(Hr)
′(Hr)ϕ(Hr)
holds for any f ∈ C2c (R) and any bounded measurable function ϕ.
Proof. Without loss of generality, we can assume that f > 0 and
f ∈ C2c (R). We
prove the above equality at r = 0. The formula (7) and the dominated convergence
theorem imply that
df(Hr)
ϕ(H0)
ϕ(H0)
ei(s0−s1)Hr
f(Hr)
Hr −H0
eis1H0
+ ei(s0−s1)Hr
Hr −H0
f(H0)e
is1H0
dν√f (s0, s1)
where the limit is taken in L1(N , τ). By the A0 -smoothness of {Hr} , we have
df(Hr)
ϕ(H0)
ϕ(H0)
ei(s0−s1)H0
f(H0)Ḣr
eis1H0
+ ei(s0−s1)H0Ḣr
f(H0)e
is1H0
dν√f (s0, s1)
so that by [2, Lemmas 3.7, 3.10] and letting A = Ḣr
df(Hr)
ϕ(H0)
ϕ(H0)e
is0H0
f(H0)A
dν√f (s0, s1)
Aϕ(H0)
f(H0)
eis0H0
f)(s0) ds0
Aϕ(H0)f
′(H0)
SPECTRAL AVERAGING 7
Proposition 2.5. The spectral shift function given by (8) satisfies Krein’s formula,
i.e. for any f ∈ C∞c , H0, H1 ∈ A
τ (f(H1)− f(H0)) = ξ(f ′).
Proof. Taking the integral of (12) with ϕ = 1 we have
d(f(Hr)− f(H0))
′(Hr)
The right hand side is ξ(f ′) by definition. It follows from Lemma 2.1 that one can
interchange the trace and the derivative in the left hand side. �
Corollary 2.6. In case of trace class perturbations, the spectral shift function ξ
defined by (8) coincides with classical definition, given by [4, Theorem 3.1].
Proof. This follows from Theorem 6.3 and Corollary 6.4 of [2]. �
For trace class perturbations the absolute continuity of the spectral shift function
is established in [13] (see also [11]). For the general semifinite case we refer to [4, 2].
Lemma 2.7. Let A be a trace compatible affine space and let f ∈ C∞c (R). Let
H0, H1 ∈ A, let ξ and ξf be the spectral shift distributions of the pairs (H0, H1)
and f(H0), f(H1) respectively. Then for any ϕ ∈ C∞c (R)
ξf (ϕ) = ξ(ϕ ◦ f · f ′).(13)
Proof. By Proposition 2.4 for any f, ϕ ∈ C∞c (R)
df(Hr)
ϕ(f(Hr))
′(Hr)ϕ(f(Hr))
Ḣr(F ◦ f)′(Hr)
where F ′ = ϕ. Hence, for any smooth path Γ = {Hr}r∈[0,1] ⊆ A
df(Hr)
ϕ(f(Hr))
Ḣr(F ◦ f)′(Hr)
dr,(14)
which is (13). �
Proposition 2.8. Let A = H0 + A0 be a trace compatible affine space and let
A0 be such that for any V ∈ A0 there exist positive V1, V2 ∈ A such that V =
V1−V2. Then the spectral shift function ξ of any pair H,H+V ∈ A is absolutely
continuous.
Proof. Since the map (V1, V2) 7→ τ (V1ϕ(H + V2)) is L1(N , τ) -continuous (by
definition), it follows that the infinitesimal spectral flow is a uniformly locally finite
measure with respect to the path parameter. Hence, the spectral shift function is
also a locally finite measure being the integral of locally finite measures, which are
uniformly bounded on every segment.
If, for H0, H1, H2 ∈ A, the spectral shift functions from H0 to H1 and from H1
to H2 are absolutely continuous, then evidently the spectral shift function from H0
to H2 is also absolutely continuous. Hence, if V = V1 − V2 with 0 6 V1, V2 ∈ A0,
then representing the spectral shift function from H to H + V as the sum of
8 N.A.AZAMOV AND F.A. SUKOCHEV
spectral shift function from H to H + V1 and from H + V1 to H + V, we see
that we can assume that the perturbation V is positive.
By Lemma 2.1 and [4, Theorem 3.1] the spectral shift function ξf of the pair
(f(H), f(H + V )) is absolutely continuous. Let us suppose that the spectral shift
function ξ of the pair (H,H + V ) has non-absolutely continuous part µ.
Without loss of generality, we can assume that there exists a set of Lebesgue
measure zero E ⊂ (ε, 1− ε) such that µ(E) > 0. For any a, b ∈ R with b− a > 2
let us consider a ”cap”-function fa,b, i.e. f is a smooth function which is zero on
(−∞, a) and (b,∞), it is 1 on (a+1, b−1) and its derivatives on (a+ε, a+1−ε)
and (b− 1 + ε, b− ε) is 1 and −1 respectively.
Let U be an open set of Lebesgue measure < δ such that E ⊂ U and let ϕ
be a smoothed indicator of U. Then (13), applied to functions ϕ and f0,b and to
functions ϕ and fa,b (with big enough b ) implies that µ(E) = µ(a+ E), i.e. µ
is translation invariant. Since it is also locally finite it is some multiple of Lebesgue
measure. This yields a contradiction. �
We summarize the results in the following theorem.
Theorem 2.9. Let A be a trace compatible affine space of operators in a semifinite
von Neumann algebra N with a normal semifinite faithful trace τ. Let H and H+
V be two operators from A. Let the spectral shift (generalized) function ξH,H+V
be defined as the integral of infinitesimal spectral flow by the formula
ξH,H+V (ϕ) =
Φ(ϕ) =
ΦHr (Ḣr)(ϕ) dr, ϕ ∈ C∞c ,
where Γ = {Hr}r∈[0,1] is any piecewise smooth path in A connecting H and
H+V. Then the spectral shift function is well-defined in the sense that the integral
does not depend on the choice of the piecewise smooth path Γ connecting H and
H + V, and it satisfies Krein’s formula
τ(f(H + V )− f(H)) = ξ(f ′), f ∈ C∞c .
Moreover, if for any V ∈ A0 there exist V1, V2 ∈ A0 such that V = V1 −V2, then
ξH,H+V is an absolutely continuous measure.
Two extreme examples of trace compatible affine spaces are H0 + L1sa(N , τ),
H0 = H
0ηN , with the topology induced by L1(N , τ) [2, 4], and D0 + Nsa,
where (D0 − i)−1 is τ -compact, with the topology induced by operator norm
[3]. In particular, the space −∆+C(M), where (M, g) is a compact Riemannian
manifold, and ∆ is the Laplacian, is trace compatible.
As an example of an intermediate trace compatible affine space one can consider
Schrödinger operators −∆+ Cc(Rn) with the inductive topology of uniform con-
vergence. It is proved in [19, Section B9] that for this example the condition (4)
holds. It also follows from [19, Section B9] that ‖gf(H)‖1 6 C ‖g‖2 , where C
depends only on f, on the support of g and on ‖V−‖∞ , where V− is the negative
part of V, H = −∆+V. So, the condition on the topology of A0 is fulfilled by (7).
SPECTRAL AVERAGING 9
Another example is given by Dirac operators of the form D +A0, where D =
, α1, . . . , αn are m×m -matrices such that αjαk + αkαj = −2δjk, and
a = a∗ ∈ Cc(Rn,Mm(R)) : ∃ϕ = ϕ∗ ∈ C1(Rn) iDϕ = a
with the inductive topology of uniform convergence. A proof that the space D+A0
is trace compatible can be reduced to [17, Theorem 4.5] via the gauge transforma-
tion ψ 7→ e−iϕ(x)ψ. We have
(D + a)(e−iϕ(x)u) =
−ie−iϕ(x) ∂
ϕ(x)αju+ e
−iϕ(x)αj
+ ae−iϕ(x)u,
where u is an m -column of C∞c -functions. So, if iDϕ = a then
eiϕ(x)(D + a)(e−iϕ(x)u) = Du.
Hence, (D+a)2 = e−iϕ(x)D2eiϕ(x). This shows that gf((D+a)2), g, a ∈ A0, f ∈
C∞c (R), is trace class iff gf(e
−iϕ(x)D2eiϕ(x)) = e−iϕ(x)geiϕ(x)f(e−iϕ(x)D2eiϕ(x)) is
trace class. But the last operator is unitarily equivalent to gf(D2), which is trace
class by [17, Theorem 4.5]. So, if f > 0 then by the same theorem
‖gf(D + a)‖1 6 C ‖g‖∞ ‖f‖∞ ,(15)
where C depends on supports of g and f. Hence, for g, g1, a, a1 ∈ A0, we have
‖gf(D + a)− g1f(D + a1)‖1 6 ‖(g − g1)f(D + a)‖1 + ‖g1(f(D + a)− f(D + a1)‖1 .
So, the condition on the topology of A0 is fulfilled by (7) and (15).
In case n = m = 1, we have D +A0 = 1i
+ Cc(R).
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[19] B. Simon, Schrödinger semigroups, Bull. Amer. Math. Soc. 7 (1982), 447–526.
School of Informatics and Engineering, Flinders University of South Australia,
Bedford Park, 5042, SA Australia.
E-mail address: azam0001@infoeng.flinders.edu.au, sukochev@infoeng.flinders.edu.au
	1. Introduction
	2. Results
	References

  In this note the notions of trace compatible operators and infinitesimal
spectral flow are introduced. We define the spectral shift function as the
integral of infinitesimal spectral flow. It is proved that the spectral shift
function thus defined is absolutely continuous and Krein's formula is
established. Some examples of trace compatible affine spaces of operators are
given.

Introduction
Let H0 be a self-adjoint operator, and let V be a trace class operator on a
Hilbert space H. Then M.G.Krĕın’s famous result [13] says that there is a unique
L1 -function ξH0+V,H0(λ), known as the Krein spectral shift function, such that for
any C∞c (R) function f
Tr(f(H0 + V )− f(H0)) =
f ′(λ)ξH0+V,H0(λ) dλ.(1)
The notion of the spectral shift function was discovered by the physicist I.M. Lifshits
[15]. An excellent survey on the spectral shift function can be found in [6].
In 1975, Birman and Solomyak [7] proved the following remarkable formula for
the spectral shift function
ξ(λ) =
Tr(V EHr
(−∞,λ]) dr,(2)
where Hr = H0 + rV, r ∈ R, and EHr(−∞,λ] is the spectral projection. Birman-
Solomyak’s proof relies on double operator integrals. An elementary derivation of
(2) was obtained in [10] (without using double operator integrals).
Actually, this spectral averaging formula was discovered for the first time by
Javrjan [12] in 1971, in case of a Sturm-Liouville operator on a half-line, perturba-
tion being a perturbation of the boundary condition, so that in this case V was
one-dimensional. An important contribution to spectral averaging was made by
A.B.Alexandrov [1]. In 1998, B. Simon [18, Theorem 1] gave a simple short proof
of the Birman-Solomyak formula. He also noticed, that this formula holds for the
wide class of Schrödinger operators on Rn [18, Theorems 3,4]. The connection of
1991 Mathematics Subject Classification. Primary 47A11; Secondary 47A55 .
Key words and phrases. Spectral shift function, spectral averaging, infinitesimal spectral flow,
trace compatible operators, semifinite von Neumann algebra .
http://arxiv.org/abs/0704.1683v2
2 N.A.AZAMOV AND F.A. SUKOCHEV
this formula with the integral formula for spectral flow from non-commutative ge-
ometry is outlined in [3]. An interesting approach to spectral averaging via Herglotz
functions can be found in [11].
In this note we present an alternative viewpoint to the spectral shift function,
and generalize the result of Simon so that it becomes applicable to a class of Dirac
operators as well.
The new point of view, which the Birman-Solomyak formula suggests, is that
there is a more fundamental notion than that of the spectral shift function. We
call this notion the speed of spectral flow or infinitesimal spectral flow of a self-
adjoint operator H under perturbation by a bounded self-adjoint operator V. It
was introduced in [3] in the case of operators with compact resolvent. It is defined
by formula
ΦH(V )(ϕ) = Tr(V ϕ(H)), ϕ ∈ C∞c (R),(3)
whenever this definition makes sense. This naturally leads to the notion of trace
compatibility of two operators. We say that operators H and H + V are trace
compatible, if for all ϕ ∈ C∞c (R) the operator V ϕ(H) is trace class. The spectral
shift function between two trace compatible operators is then considered as the
integral of infinitesimal spectral flow. It turns out that the spectral shift function
does not depend on the path connecting the initial and final operators, a fact
which follows from the aforementioned result of B. Simon in the case of trace class
perturbations.
The results of this note are summarized in Theorem 2.9. This theorem extends
formulae (1) and (2) to the setting of trace compatible pairs (H,H + V ) and also
strengthens [18, Theorems 3,4] in the sense that it does not require H to be a
positive operator and maximally weakens conditions on the path H+rV, r ∈ [0, 1].
Our results also hold for a more general setting, when H = H∗ is affiliated with a
semifinite von Neumann algebra N and V = V ∗ ∈ N .
Our investigation here also strengthens the link between the theory of the Krein
spectral shift function and that of spectral flow firstly discovered in [2]. For exposi-
tion of the latter theory we refer to [5] and a detailed discussion of the connection
between the two theories in the situation where the resolvent of H is τ -compact
(here, τ is an arbitrary faithful normal semifinite trace on N ) is contained in [3].
It should be pointed out here that the idea of viewing the spectral shift function
as the integral of infinitesimal spectral flow is akin to I.M. Singer’s ICM-1974 pro-
posal to define the η invariant (and hence spectral flow) as the integral of a one
form. Very general formulae of that type have been produced in the framework
of noncommutative geometry (see [5] and references therein). We believe that our
present approach will have applications to noncommutative geometry, in particular,
it may be useful in avoiding ”summability constraints” on H customarily used in
that theory.
In semifinite von Neumann algebras N Krein’s formula (1) was proved for the
first time in [9] in case of a bounded self-adjoint operator H ∈ N and a trace class
perturbation V = V ∗ ∈ L1(N , τ) and in [4] for self-adjoint operators H affiliated
with N .
SPECTRAL AVERAGING 3
An additional reason to call ΦH(V ) the speed of spectral flow is the follow-
ing observation. Let H be the operator of multiplication by λ on L2(R, dρ(λ))
with some measure ρ and let the perturbation V be an integral operator with a
sufficiently regular (for example C1 ) kernel k(λ′, λ). Then for any test function
ϕ ∈ C∞c (R)
ΦH(V )(ϕ) = Tr(V ϕ(H)) =
k(λ, λ)ϕ(λ) dρ(λ).
Hence, the infinitesimal spectral flow of H under perturbation by V is the measure
on the spectrum of H with density k(λ, λ) dρ(λ). We note that this agrees with
the classical formula [14, (38.6)]
n = Vnn
from formal perturbation theory. Here E
n is the n -th eigenvalue of the unper-
turbed operator H0, E
n , j = 1, 2, . . . is the j -th correction term for the n -th
eigenvalue En of the perturbed operator H = H0 + V in the formal perturbation
series En = E
n + E
n + E
n + . . . , and Vmn =
m |V |ψ(0)n
is the matrix
element of the perturbation operator V with respect to the eigenfunctions ψ
and ψ
n of the unperturbed operator H0 [14].
Acknowledgement. We thank Alan Carey for useful comments and criticism.
2. Results
Let N be a von Neumann algebra on a Hilbert space H with faithful normal
semifinite trace τ. Let A = H0 + A0 be an affine space of self-adjoint operators
affiliated with N , where H0 is a self-adjoint operator affiliated with N and A0
is a vector subspace of the real Banach space of all self-adjoint operators from N .
We say that A is trace compatible, if for all ϕ ∈ C∞c (R), V ∈ A0 and H ∈ A
V ϕ(H) ∈ L1(N , τ),(4)
where L1(N , τ) is the ideal of trace class operators from N , and if A0 is endowed
with a locally convex topology which coincides with or is stronger than the uniform
topology, such that the map (V1, V2) ∈ A20 7→ V1ϕ(H0 + V2) is L1 continuous for
all H0 ∈ A and ϕ ∈ C∞c (R). In particular, A is a locally convex affine space.
The ideal property of L1(N , τ) and [16, Theorem VIII.20(a)] imply that, in the
definition of trace compatibility, the condition ϕ ∈ C∞c (R) may be replaced by
ϕ ∈ Cc(R). It follows from the definition of the topology on A0 that H ∈ A 7→ eitH
is norm continuous.
If A = H0+A0 is a trace compatible affine space then we define a (generalized)
one-form (on A ) of infinitesimal spectral flow or speed of spectral flow by the
formula
ΦH(V ) = τ(V δ(H)), H ∈ A, V ∈ A0,(5)
where δ is Dirac’s delta function. The last formula is to be understood in a
generalized function sense, i.e. ΦH(V )(ϕ) = τ(V ϕ(H)), ϕ ∈ C∞c (R). Φ is a
generalized function, since if ϕn → 0 in C∞c (R) such that supp(ϕn) ⊆ ∆ then
|τ(V ϕn(H))| 6
∥V EH∆
‖ϕn(H)‖ → 0. Here ‖A‖1 = τ(|A|).
4 N.A.AZAMOV AND F.A. SUKOCHEV
Since ϕ can be taken from Cc(R), for each V ∈ A0 the infinitesimal spectral
flow ΦH(V ) is actually a measure on the spectrum of H.
By a smooth path {Hr}r∈R in A, we mean a differentiable path, such that its
derivative dHr
∈ A0 is continuous.
Let Π =
(s0, s1) ∈ R2 : s0s1 > 0, |s1| 6 |s0|
, and let
dνf (s0, s1) = sgn(s0)
f̂(s0) ds0 ds1.
If f ∈ C2c (R) then (Π, νf ) is a finite measure space [2]. For any H0, H1 ∈ A, any
X ∈ A0 and any non-negative f ∈ C∞c (R) set by definition
(6) T
H1,H0
f [1]
(X) =
ei(s0−s1)H1
f(H1)Xe
is1H0
+ ei(s0−s1)H1X
f(H0)e
is1H0
dν√f (s0, s1),
where the integral is taken in the so∗ -topology. For justification of this notation
and details see [3].
Lemma 2.1. If {Hr} ⊂ A is a path, continuous (smooth) in the topology of A0,
and if f ∈ C2c (R) then
r 7→ f(Hr)− f(H0)
takes values in L1(N , τ) and it is L1(N , τ) continuous (smooth).
Proof. We can assume that f is non-negative and that
f ∈ C2c (R). It is proved
in [3] that
f(Hr)− f(H0) = THr ,H0f [1] (Hr −H0).(7)
Since ei(s0−s1)x
f(x), eis1x
f(x) ∈ C2c (R), trace compatibility implies that the
integrand of the right hand side of (6) takes values in L1 and is L1 -continuous
(smooth), so the dominated convergence theorem completes the proof. �
If Γ = {Hr}r∈[0,1] is a smooth path in A, then we define the spectral shift
function ξ along this path as the integral of infinitesimal spectral flow: ξ =
ξ(ϕ) =
ϕ(Hr)
dr, ϕ ∈ C∞c (R).(8)
Now we prove that the spectral shift function is well-defined in the sense that it
does not depend on the path of integration.
A one-form αH(V ) on an affine space A is called exact if there exists a zero-
form θH on A such that dθ = α, i.e.
αH(V ) =
θH+rV
We say that the generalized one-form Φ is exact if Φ(ϕ) is an exact form for any
ϕ ∈ C∞c (R).
SPECTRAL AVERAGING 5
The proof of the following proposition follows the lines of the proof of [3, Propo-
sition 3.5].
Proposition 2.2. The infinitesimal spectral flow Φ is exact.
Proof. Let V ∈ A0, Hr = H0 + rV, r ∈ [0, 1], and let f ∈ C∞c (R). By (7)
(9) f(Hr)− f(H0) = THr ,H0f [1] (rV )
ei(s0−s1)Hr
f(Hr)rV e
is1H0 + ei(s0−s1)HrrV
f(H0)e
is1H0
dν√f (s0, s1),
ei(s0−s1)H0
f(H0)rV e
is1H0 + ei(s0−s1)H0rV
f(H0)e
is1H0
dν√f (s0, s1)
(ei(s0−s1)Hr
f(Hr)− ei(s0−s1)H0
f(H0))rV e
is1H0
+ (ei(s0−s1)Hr − ei(s0−s1)H0)rV
f(H0)e
is1H0
dν√f (s0, s1)
H0,H0
f [1]
(rV ) +R1 +R2.
All three summands here are trace class by the trace compatibility assumption. So,
for any S ∈ N
τ(S(f(Hr)− f(H0))) = rτ
H0,H0
f [1]
+ τ(SR1) + τ(SR2).
Now, Duhamel’s formula and (7) show that τ(SR1) = o(r) and τ(SR2) = o(r).
Hence,
τ(S(f(Hr)− f(H0))) = τ
Hr,Hr
f [1]
.(10)
This implies that for any S ∈ N
τ(S(f(H1)− f(H0))) = τ
Hr ,Hr
f [1]
(V ) dr
Now let H0 ∈ A be a fixed operator and for any f ∈ C∞c (R) let
τ(V f(Hr)) dr,
where Hr = H0 + rV, H = H1. We are going to show that dθ
H(X) = ΦH(X)(f)
for any X ∈ A0.
Following the proof of [3, Proposition 3.5] we have
(A) := dθ
H(X) = lim
Xf(Hr + srX)
+ lim
f(Hr + srX)− f(Hr)
By definition of A0 topology the integrand of the first summand is continuous with
respect to r and s. So, the first summand is equal to
Xf(Hr)
6 N.A.AZAMOV AND F.A. SUKOCHEV
By [2, Theorem 5.3] the second summand is equal to
Hr+srX,Hr
f [1]
(srX)
dr = lim
Hr+srX,Hr
f [1]
Hr,Hr
f [1]
Hr ,Hr
f [1]
r dr,
where the second equality follows from the definition of A0 -topology and the last
equality follows from [3, Lemma 3.2]. Now, using (10) and integrating by parts we
(11) (A) = τ (Xf(H1)−Xf(H0)) +
(τ(Xf(Hr))− τ(X [f(Hr)− f(H0)])) dr
= τ (Xf(H1)) .
The argument before [8, Proposition 1.5] now implies
Corollary 2.3. The spectral shift function given by (8) is well-defined.
Proposition 2.4. If r ∈ R 7→ Hr ∈ A is smooth then the equality
df(Hr)
ϕ(Hr)
′(Hr)ϕ(Hr)
holds for any f ∈ C2c (R) and any bounded measurable function ϕ.
Proof. Without loss of generality, we can assume that f > 0 and
f ∈ C2c (R). We
prove the above equality at r = 0. The formula (7) and the dominated convergence
theorem imply that
df(Hr)
ϕ(H0)
ϕ(H0)
ei(s0−s1)Hr
f(Hr)
Hr −H0
eis1H0
+ ei(s0−s1)Hr
Hr −H0
f(H0)e
is1H0
dν√f (s0, s1)
where the limit is taken in L1(N , τ). By the A0 -smoothness of {Hr} , we have
df(Hr)
ϕ(H0)
ϕ(H0)
ei(s0−s1)H0
f(H0)Ḣr
eis1H0
+ ei(s0−s1)H0Ḣr
f(H0)e
is1H0
dν√f (s0, s1)
so that by [2, Lemmas 3.7, 3.10] and letting A = Ḣr
df(Hr)
ϕ(H0)
ϕ(H0)e
is0H0
f(H0)A
dν√f (s0, s1)
Aϕ(H0)
f(H0)
eis0H0
f)(s0) ds0
Aϕ(H0)f
′(H0)
SPECTRAL AVERAGING 7
Proposition 2.5. The spectral shift function given by (8) satisfies Krein’s formula,
i.e. for any f ∈ C∞c , H0, H1 ∈ A
τ (f(H1)− f(H0)) = ξ(f ′).
Proof. Taking the integral of (12) with ϕ = 1 we have
d(f(Hr)− f(H0))
′(Hr)
The right hand side is ξ(f ′) by definition. It follows from Lemma 2.1 that one can
interchange the trace and the derivative in the left hand side. �
Corollary 2.6. In case of trace class perturbations, the spectral shift function ξ
defined by (8) coincides with classical definition, given by [4, Theorem 3.1].
Proof. This follows from Theorem 6.3 and Corollary 6.4 of [2]. �
For trace class perturbations the absolute continuity of the spectral shift function
is established in [13] (see also [11]). For the general semifinite case we refer to [4, 2].
Lemma 2.7. Let A be a trace compatible affine space and let f ∈ C∞c (R). Let
H0, H1 ∈ A, let ξ and ξf be the spectral shift distributions of the pairs (H0, H1)
and f(H0), f(H1) respectively. Then for any ϕ ∈ C∞c (R)
ξf (ϕ) = ξ(ϕ ◦ f · f ′).(13)
Proof. By Proposition 2.4 for any f, ϕ ∈ C∞c (R)
df(Hr)
ϕ(f(Hr))
′(Hr)ϕ(f(Hr))
Ḣr(F ◦ f)′(Hr)
where F ′ = ϕ. Hence, for any smooth path Γ = {Hr}r∈[0,1] ⊆ A
df(Hr)
ϕ(f(Hr))
Ḣr(F ◦ f)′(Hr)
dr,(14)
which is (13). �
Proposition 2.8. Let A = H0 + A0 be a trace compatible affine space and let
A0 be such that for any V ∈ A0 there exist positive V1, V2 ∈ A such that V =
V1−V2. Then the spectral shift function ξ of any pair H,H+V ∈ A is absolutely
continuous.
Proof. Since the map (V1, V2) 7→ τ (V1ϕ(H + V2)) is L1(N , τ) -continuous (by
definition), it follows that the infinitesimal spectral flow is a uniformly locally finite
measure with respect to the path parameter. Hence, the spectral shift function is
also a locally finite measure being the integral of locally finite measures, which are
uniformly bounded on every segment.
If, for H0, H1, H2 ∈ A, the spectral shift functions from H0 to H1 and from H1
to H2 are absolutely continuous, then evidently the spectral shift function from H0
to H2 is also absolutely continuous. Hence, if V = V1 − V2 with 0 6 V1, V2 ∈ A0,
then representing the spectral shift function from H to H + V as the sum of
8 N.A.AZAMOV AND F.A. SUKOCHEV
spectral shift function from H to H + V1 and from H + V1 to H + V, we see
that we can assume that the perturbation V is positive.
By Lemma 2.1 and [4, Theorem 3.1] the spectral shift function ξf of the pair
(f(H), f(H + V )) is absolutely continuous. Let us suppose that the spectral shift
function ξ of the pair (H,H + V ) has non-absolutely continuous part µ.
Without loss of generality, we can assume that there exists a set of Lebesgue
measure zero E ⊂ (ε, 1− ε) such that µ(E) > 0. For any a, b ∈ R with b− a > 2
let us consider a ”cap”-function fa,b, i.e. f is a smooth function which is zero on
(−∞, a) and (b,∞), it is 1 on (a+1, b−1) and its derivatives on (a+ε, a+1−ε)
and (b− 1 + ε, b− ε) is 1 and −1 respectively.
Let U be an open set of Lebesgue measure < δ such that E ⊂ U and let ϕ
be a smoothed indicator of U. Then (13), applied to functions ϕ and f0,b and to
functions ϕ and fa,b (with big enough b ) implies that µ(E) = µ(a+ E), i.e. µ
is translation invariant. Since it is also locally finite it is some multiple of Lebesgue
measure. This yields a contradiction. �
We summarize the results in the following theorem.
Theorem 2.9. Let A be a trace compatible affine space of operators in a semifinite
von Neumann algebra N with a normal semifinite faithful trace τ. Let H and H+
V be two operators from A. Let the spectral shift (generalized) function ξH,H+V
be defined as the integral of infinitesimal spectral flow by the formula
ξH,H+V (ϕ) =
Φ(ϕ) =
ΦHr (Ḣr)(ϕ) dr, ϕ ∈ C∞c ,
where Γ = {Hr}r∈[0,1] is any piecewise smooth path in A connecting H and
H+V. Then the spectral shift function is well-defined in the sense that the integral
does not depend on the choice of the piecewise smooth path Γ connecting H and
H + V, and it satisfies Krein’s formula
τ(f(H + V )− f(H)) = ξ(f ′), f ∈ C∞c .
Moreover, if for any V ∈ A0 there exist V1, V2 ∈ A0 such that V = V1 −V2, then
ξH,H+V is an absolutely continuous measure.
Two extreme examples of trace compatible affine spaces are H0 + L1sa(N , τ),
H0 = H
0ηN , with the topology induced by L1(N , τ) [2, 4], and D0 + Nsa,
where (D0 − i)−1 is τ -compact, with the topology induced by operator norm
[3]. In particular, the space −∆+C(M), where (M, g) is a compact Riemannian
manifold, and ∆ is the Laplacian, is trace compatible.
As an example of an intermediate trace compatible affine space one can consider
Schrödinger operators −∆+ Cc(Rn) with the inductive topology of uniform con-
vergence. It is proved in [19, Section B9] that for this example the condition (4)
holds. It also follows from [19, Section B9] that ‖gf(H)‖1 6 C ‖g‖2 , where C
depends only on f, on the support of g and on ‖V−‖∞ , where V− is the negative
part of V, H = −∆+V. So, the condition on the topology of A0 is fulfilled by (7).
SPECTRAL AVERAGING 9
Another example is given by Dirac operators of the form D +A0, where D =
, α1, . . . , αn are m×m -matrices such that αjαk + αkαj = −2δjk, and
a = a∗ ∈ Cc(Rn,Mm(R)) : ∃ϕ = ϕ∗ ∈ C1(Rn) iDϕ = a
with the inductive topology of uniform convergence. A proof that the space D+A0
is trace compatible can be reduced to [17, Theorem 4.5] via the gauge transforma-
tion ψ 7→ e−iϕ(x)ψ. We have
(D + a)(e−iϕ(x)u) =
−ie−iϕ(x) ∂
ϕ(x)αju+ e
−iϕ(x)αj
+ ae−iϕ(x)u,
where u is an m -column of C∞c -functions. So, if iDϕ = a then
eiϕ(x)(D + a)(e−iϕ(x)u) = Du.
Hence, (D+a)2 = e−iϕ(x)D2eiϕ(x). This shows that gf((D+a)2), g, a ∈ A0, f ∈
C∞c (R), is trace class iff gf(e
−iϕ(x)D2eiϕ(x)) = e−iϕ(x)geiϕ(x)f(e−iϕ(x)D2eiϕ(x)) is
trace class. But the last operator is unitarily equivalent to gf(D2), which is trace
class by [17, Theorem 4.5]. So, if f > 0 then by the same theorem
‖gf(D + a)‖1 6 C ‖g‖∞ ‖f‖∞ ,(15)
where C depends on supports of g and f. Hence, for g, g1, a, a1 ∈ A0, we have
‖gf(D + a)− g1f(D + a1)‖1 6 ‖(g − g1)f(D + a)‖1 + ‖g1(f(D + a)− f(D + a1)‖1 .
So, the condition on the topology of A0 is fulfilled by (7) and (15).
In case n = m = 1, we have D +A0 = 1i
+ Cc(R).
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School of Informatics and Engineering, Flinders University of South Australia,
Bedford Park, 5042, SA Australia.
E-mail address: azam0001@infoeng.flinders.edu.au, sukochev@infoeng.flinders.edu.au
	1. Introduction
	2. Results
	References
