On the Koplienko Spectral Shift Function. I. Basics
We study the Koplienko Spectral Shift Function (KoSSF), which is distinct from the one of Krein (KrSSF). KoSSF is defined for pairs A,B with (A - B) is in I₂, the Hilbert{Schmidt operators, while KrSSF is defined for pairs A,B with (A - B) is in I₁, the trace class operators. We review various aspec...
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irk-123456789-1064952016-09-30T03:02:45Z On the Koplienko Spectral Shift Function. I. Basics Gesztesy, F. Pushnitski, A. Simon, B. We study the Koplienko Spectral Shift Function (KoSSF), which is distinct from the one of Krein (KrSSF). KoSSF is defined for pairs A,B with (A - B) is in I₂, the Hilbert{Schmidt operators, while KrSSF is defined for pairs A,B with (A - B) is in I₁, the trace class operators. We review various aspects of the construction of both KoSSF and KrSSF. Among our new results are: (i) that any positive Riemann integrable function of compact support occurs as a KoSSF; (ii) that there exist A,B with (A - B) is in I₂ so det₂((A - z)(B - z)⁻¹) does not have nontangential boundary values; (iii) an alternative definition of KoSSF in the unitary case; and (iv) a new proof of the invariance of the a.c. spectrum under I₁-perturbations that uses the KrSSF. 2008 Article On the Koplienko Spectral Shift Function. I. Basics / F. Gesztesy, A. Pushnitski, B. Simon // Журнал математической физики, анализа, геометрии. — 2008. — Т. 4, № 1. — С. 63-107. — Бібліогр.: 71 назв. — англ. 1812-9471 http://dspace.nbuv.gov.ua/handle/123456789/106495 en Журнал математической физики, анализа, геометрии Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
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We study the Koplienko Spectral Shift Function (KoSSF), which is distinct from the one of Krein (KrSSF). KoSSF is defined for pairs A,B with (A - B) is in I₂, the Hilbert{Schmidt operators, while KrSSF is defined for pairs A,B with (A - B) is in I₁, the trace class operators. We review various aspects of the construction of both KoSSF and KrSSF. Among our new results are: (i) that any positive Riemann integrable function of compact support occurs as a KoSSF; (ii) that there exist A,B with (A - B) is in I₂ so det₂((A - z)(B - z)⁻¹) does not have nontangential boundary values; (iii) an alternative definition of KoSSF in the unitary case; and (iv) a new proof of the invariance of the a.c. spectrum under I₁-perturbations that uses the KrSSF. |
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Gesztesy, F. Pushnitski, A. Simon, B. On the Koplienko Spectral Shift Function. I. Basics Журнал математической физики, анализа, геометрии |
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Gesztesy, F. Pushnitski, A. Simon, B. |
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On the Koplienko Spectral Shift Function. I. Basics |
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On the Koplienko Spectral Shift Function. I. Basics |
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On the Koplienko Spectral Shift Function. I. Basics |
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On the Koplienko Spectral Shift Function. I. Basics |
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On the Koplienko Spectral Shift Function. I. Basics |
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on the koplienko spectral shift function. i. basics |
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Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
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On the Koplienko Spectral Shift Function. I. Basics / F. Gesztesy, A. Pushnitski, B. Simon // Журнал математической физики, анализа, геометрии. — 2008. — Т. 4, № 1. — С. 63-107. — Бібліогр.: 71 назв. — англ. |
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Журнал математической физики, анализа, геометрии |
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Journal of Mathematical Physics, Analysis, Geometry
2009, vol. 4, No. 1, pp. 63–107
On the Koplienko Spectral Shift Function. I. Basics
F. Gesztesy1
Department of Mathematics, University of Missouri
Columbia, MO 65211, USA
E-mail:fritz@math.missouri.edu
A. Pushnitski2
Department of Mathematics, King’s College London
Strand, London WC2R 2LS, England, UK
E-mail:alexander.pushnitski@kcl.ac.uk
B. Simon3
Mathematics 253-37, California Institute of Technology
Pasadena, CA 91125, USA
E-mail:bsimon@caltech.edu
Received May 16, 2007
We study the Koplienko Spectral Shift Function (KoSSF), which is dis-
tinct from the one of Krein (KrSSF). KoSSF is defined for pairs A,B with
(A − B) ∈ I2, the Hilbert–Schmidt operators, while KrSSF is defined for
pairs A,B with (A − B) ∈ I1, the trace class operators. We review vari-
ous aspects of the construction of both KoSSF and KrSSF. Among our new
results are: (i) that any positive Riemann integrable function of compact
support occurs as a KoSSF; (ii) that there exist A,B with (A−B) ∈ I2 so
det2((A − z)(B − z)−1) does not have nontangential boundary values; (iii)
an alternative definition of KoSSF in the unitary case; and (iv) a new proof
of the invariance of the a.c. spectrum under I1-perturbations that uses the
KrSSF.
Key words: Krein’s spectral shift function, Koplienko’s spectral shift
function, selfadjoint operators, trace class and Hilbert–Schmidt perturba-
tions, convexity properties, boundary values of (modified) Fredholm deter-
minants.
Mathematics Subject Classification 2000: 47A10, 81Q10 (primary); 34B27,
47A40, 81Uxx (secondary).
1 Supported in part by NSF Grant DMS-0405526.
2 Supported in part by the Leverhulme Trust.
3 Supported in part by NSF Grant DMS-0140592 and U.S. – Israel Binational Science
Foundation (BSF) Grant No. 2002068.
c© F. Gesztesy, A. Pushnitski, and B. Simon, 2009
F. Gesztesy, A. Pushnitski, and B. Simon
1. Introduction
In 1941, Titchmarsh [64] (see also [20, pp. 1564–1566] for the result) proved
that if
V ∈ L1((0,∞); dx), V real-valued,
and
Hθ = − d2
dx2
+ V, (1.1)
dom(Hθ) = {f ∈ L2((0,∞); dx) | f, f ′ ∈ AC([0, R]) for all R > 0;
sin(θ)f ′(0) + cos(θ)f(0) = 0; (−f ′′ + V f) ∈ L2((0,∞); dx)},
for some θ ∈ [0, π), then
σac(Hθ) = [0,∞).
(Actually, he explicitly computed the spectral function in terms of the inverse
square of the modulus of the Jost function for positive energies.) It was later
realized that the a.c. invariance, that is,
σac(Hθ) = σac(H0,θ) (1.2)
with
H0,θ = − d2
dx2
, (1.3)
dom(H0,θ) = {f ∈ L2((0,∞); dx) | f, f ′ ∈ AC([0, R]) for all R > 0;
sin(θ)f ′(0) + cos(θ)f(0) = 0; f ′′ ∈ L2((0,∞); dx)},
is a special case of an invariance of the absolutely continuous spectrum, σac(·) for
the passage from A to B if (A−B) ∈ I1, the trace class. In the present context of
the pair (Hθ,H0,θ) one has [(Hθ +E)−1−(H0,θ +E)−1] ∈ I1 for E > 0 sufficiently
large. The abstract trace class result is associated with Birman [10, 11], Kato
[31, 32], and Rosenblum [54].
Our original and continuing motivation is to find a suitable operator theoretic
result connected with the remarkable discovery of Deift–Killip [18] that for the
above (1.1)/(1.3) case, one has (1.2) if one only assumes V ∈ L2((0,∞); dx).
Note that V ∈ L2((0,∞); dx) implies that
[(Hθ + E)−1 − (H0,θ + E)−1] ∈ I2,
the Hilbert–Schmidt class. However, there is no totally general invariance result
for a.c. spectrum under nontrace class perturbations: It is a result of Weyl [68]
64 Journal of Mathematical Physics, Analysis, Geometry, 2009, vol. 4, No. 1
On the Koplienko Spectral Shift Function. I. Basics
and von Neumann [66] that given any selfadjoint A, there is a B with pure point
spectrum and (A−B) ∈ I2. Kuroda [43] extends this to Ip, p ∈ (1,∞), the trace
ideals. Thus, we seek general operator criteria on when (A − B) ∈ I2 but (1.2)
still holds.
We hope such a criterion will be found in the spectral shift function of
Koplienko [36] (henceforth KoSSF), an object which we believe has not received
the attention it deserves. One of our goals in the present paper is to make
propaganda for this object.
Two references for trace ideals we quote extensively are Gohberg–Krein [27]
and Simon [61]. We follow the notation of [61]. Throughout this paper all Hilbert
spaces are assumed to be complex and separable.
The KoSSF, η(λ; A,B), is defined when A and B are bounded selfadjoint
operators satisfying (A−B) ∈ I2, and is given by
∫
R
f ′′(λ)η(λ; A,B) dλ = Tr
(
f(A)− f(B)− d
dα
f(B + α(A−B))
∣∣∣∣
α=0
)
, (1.4)
where the right-hand side is sometimes (certainly if (A − B) ∈ I1) the simpler-
looking
Tr(f(A)− f(B)− (A−B)f ′(B)). (1.5)
η has two critical properties: η ∈ L1(R) and η ≥ 0. We mainly consider bounded
A,B here, but see the remarks in Section 9.
Formula (1.4) requires some assumptions on f . In Koplienko’s original paper
[36] the case f(x) = (x − z)−1 was considered and then (1.4) was extended
to the class of rational functions with poles off the real axis. Later, Peller [52]
extended the class of functions f and found sharp sufficient conditions on f which
guarantee that (1.4) holds. These conditions were stated in terms of Besov spaces.
Essentially, Peller’s construction requires that (1.4) holds for some sufficiently
wide class of functions, so that this class is dense in a certain Besov space, and
then provides an extension onto the whole of this Besov space.
We will use this aspect of Peller’s work and will not worry about the classes
of f in this paper. For the most part we will work with f ∈ C∞(R) and Peller’s
construction provides an extension to a wider function class.
The model for the KoSSF is, of course, the spectral shift function of Krein
(henceforth KrSSF), denoted by ξ(λ; A,B), and defined for A,B with (A−B) ∈ I1
by ∫
R
ξ(λ;A, B)f ′(λ) dλ = Tr(f(A)− f(B)). (1.6)
In the appendix, we recall a quick way to define ξ, its main properties and, most
importantly, present an argument that shows how it can be used to derive the
invariance of a.c. spectrum without recourse to scattering theory.
Journal of Mathematical Physics, Analysis, Geometry, 2009, vol. 4, No. 1 65
F. Gesztesy, A. Pushnitski, and B. Simon
As we will see in Section 2, it is easy to construct analogs of η for any In,
n ∈ N, but they are only tempered distributions. What makes η different is
its positivity, which also implies it lies in L1(R) (by taking f suitably). This
positivity should be thought of as a general convexity result — something hidden
in Koplienko’s paper [36].
One of our goals here is to emphasize this convexity. Another is to present
a “baby“ finite-dimensional version of the double Stieltjes operator integral of
Birman–Solomyak [12, 13, 15], essentially due to Löwner [46], whose contribution
here seems to have been overlooked.
In Section 2, we define η when (A − B) is trace class, and in Section 3, we
discuss the convexity result that is equivalent to positivity of η. In Section 4, we
prove a lovely bound of Birman–Solomyak [13, 15]:
‖f(A)− f(B)‖I2 ≤ ‖f ′‖∞‖A−B‖I2 . (1.7)
Here and in the remainder of this paper ‖ · ‖Ip denotes the norm in the trace
ideals Ip, p ∈ [1,∞). In Section 5, we use (1.7) plus positivity of η to complete
the construction of η.
We want to emphasize an important distinction between the KrSSF and the
KoSSF. The former satisfies a chain rule
ξ( · ;A, C) = ξ( · ;A, B) + ξ( · ; B,C), (1.8)
while η instead satisfies a corrected chain rule
η( · ; A,C) = η( · ;A,B) + η( · ; B,C) + δη( · ; A,B, C), (1.9)
where δη satisfies
∫
R
g′(λ)δη(λ; A,B) dλ = Tr((A−B)(g(B)− g(C))). (1.10)
(Here g corresponds to f ′ when comparing with (1.4)–(1.6).) It is in estimating
(1.10) that (1.7) will be critical.
We view Sections 2–5 as a repackaging in a prettier ribbon of Koplienko’s
construction in [36]. Section 6 explores what η’s can occur. In Sections 7 and
8, we discuss the connection to det2(·) and present a new result: an example of
(A−B) ∈ I2 where det2((A− z)(B− z)−1) does not have nontangential limits to
the real axis a.e. This is in contradistinction to the KrSSF, where (A− B) ∈ I1
implies det((A − z)(B − z)−1) has a nontangential limit z → λ for a.e. λ ∈ R.
The latter is a consequence of the formula
log(det((A− z)(B − z)−1)) =
∫
R
(λ− z)−1ξ(λ; A,B) dλ, z ∈ C\R, (1.11)
66 Journal of Mathematical Physics, Analysis, Geometry, 2009, vol. 4, No. 1
On the Koplienko Spectral Shift Function. I. Basics
since the right-hand side of (1.11) represents a difference of two Herglotz func-
tions.
Sections 9 and 10 discuss extensions of η to the case of unbounded operators
with a trace class condition on the resolvents and to unitary operators. Here a key
is that η is not determined until one makes a choice of interpolation. Section 11
discusses some conjectures.
In a future joint work, we will explore what one can learn about the KoSSF
from Szegő’s theorem [60], the work of Killip–Simon [34] and of Christ–Kiselev
[17]. This will involve the study of η for suitable Schrödinger operators and Jacobi
and CMV matrices for perturbations in Lp, respectively, `p, p ∈ [1, 2).
We are indebted to E. Lieb, K.A. Makarov, V.V. Peller, and M.B. Ruskai for
useful discussions. F. Gesztesy and A. Pushnitski wish to thank Gary Lorden
and Tom Tombrello for the hospitality of Caltech where some of this work was
done. F. Gesztesy gratefully acknowledges a research leave for the academic
year 2005/06 granted by the Research Council and the Office of Research of the
University of Missouri–Columbia. A. Pushnitski gratefully acknowledges financial
support by the Leverhulme Trust.
It is a great pleasure to dedicate this paper to the birthdays of two giants of
spectral theory: Vladimir A. Marchenko and Leonid A. Pastur.
2. The KoSSF η( · ; A,B) in the Trace Class Case
We begin with what can be said of In perturbations, n ∈ N, and then turn to
what is special for n = 1, 2. We note that our approach has common elements to
the one used by Dostanić [19].
Proposition 2.1. Let A,B be bounded selfadjoint operators with
A = B + X. (2.1)
For α, t ∈ R, define
ft(α) = eit(B+αX).
Then ft(α) is C∞ in α and
dkft
dαk
= ik
∫
0<sj<t∑k
j=1 sj<t
fs1(α)Xfs2(α) . . . fsk
(α)Xft−s1−···−sk
(α) ds1 . . . dsk. (2.2)
If X ∈ Ip for p ≥ k, k ∈ N, then dkft/dαk ∈ Ip/k and
∥∥∥∥
dkft
dαk
∥∥∥∥
Ip/k
≤ tk
k!
‖X‖k
Ip
. (2.3)
Journal of Mathematical Physics, Analysis, Geometry, 2009, vol. 4, No. 1 67
F. Gesztesy, A. Pushnitski, and B. Simon
In particular, if n ∈ N and X ∈ In, then
gt(A, B) ≡
(
eitA − eitB −
n−1∑
k=1
1
k!
(
d
dα
)k
ft(α)
∣∣∣∣
α=0
)
∈ I1, (2.4)
‖gt(A,B)‖I1 ≤
tn
n!
‖X‖n
In
. (2.5)
P r o o f. For k = 1, (2.2) comes from taking limits in DuHamel’s formula
eC − eD =
1∫
0
eβC(C −D)e(1−β)D dβ.
The general k case then follows by induction.
(2.2) implies (2.3) by Hölder’s inequality for operators (see [61, p. 21]). (2.4)
is then Taylor’s theorem with remainder and (2.5) follows from (2.3).
Theorem 2.2. Let A,B be bounded selfadjoint operators such that X = (A−
B) ∈ In for some n ∈ N. Let f be of compact support with f̂ , its Fourier
transform, satisfying ∫
R
(1 + |k|)n|f̂(k)| dk < ∞. (2.6)
Then, (
f(A)− f(B)−
n−1∑
j=1
1
k!
(
d
dα
)k
f(B + αX)
∣∣∣∣
α=0
)
∈ I1, (2.7)
and there is a distribution T with
Tr
(
f(A)− f(B)−
n−1∑
j=1
1
k!
(
d
dα
)k
f(B + αX)
∣∣∣∣
α=0
)
=
∫
R
T (λ)f (n)(λ) dλ. (2.8)
Moreover, the distribution T is such that T̂ ∈ L∞(R; dt).
P r o o f. This is immediate from the estimates in Prop. 2.1 and
f(A) = (2π)−1/2
∫
R
f̂(t) eitA dt.
For, by (2.5), we have
‖LHS of (2.7)‖I1 ≤ C
∫
R
|t|n|f̂(t)| dt = C
∫
R
|f̂ (n)(t)| dt. (2.9)
68 Journal of Mathematical Physics, Analysis, Geometry, 2009, vol. 4, No. 1
On the Koplienko Spectral Shift Function. I. Basics
Thus, (2.8) defines a distribution T with
|T (f)| ≤ C
∫
R
|f̂(t)| dt,
so T̂ is a function in L∞(R; dt).
Notice that, as we have seen,
d
dα
eit(B+αX)
∣∣∣∣
α=0
= i
t∫
0
eiβBXei(t−β)B dβ
so that formally,
Tr
(
d
dα
f(B + αX)
∣∣∣∣
α=0
−Xf ′(B)
)
= 0,
and formally, (1.5) can replace the right-hand side of (1.4). This can be proven
if the commutator [B, X] = [B, A] is trace class.
Dostanić [19, Th. 2.9] essentially proves that T is the derivative of an
L2(R; dλ)-function.
A key point for us is that in the case n = 2, the distribution T is given by
an L1(R; dλ)-function. We start this construction here by considering the trace
class case.
Lemma 2.3. Let B be a selfadjoint operator and let X = (A − B) ∈ I1.
Then there is a (complex ) measure dµB,X on R such that for any bounded Borel
function, f ,
Tr(Xf(B)) =
∫
R
f(λ) dµB,X(λ). (2.10)
Equation (2.10) defines dµB,X uniquely.
P r o o f. Equation (2.10) yields uniqueness of the measure dµB,X since it
defines the integral for all continuous functions f . Regarding existence of dµB,X ,
the spectral theorem asserts the existence of measures dµB;ϕ,ψ, such that
〈ϕ, f(B)ψ〉 =
∫
R
f(λ) dµB;ϕ,ψ(λ) (2.11)
and
‖µB;ϕ,ψ‖ ≤ ‖ϕ‖ ‖ψ‖. (2.12)
Journal of Mathematical Physics, Analysis, Geometry, 2009, vol. 4, No. 1 69
F. Gesztesy, A. Pushnitski, and B. Simon
The canonical decomposition for X (see [61, Sect. 1.2]) says (with N finite or
infinite)
X =
N∑
j=1
µj(X)〈ϕj , · 〉ψj , (2.13)
where {ψj}N
j=1 and {ϕj}N
j=1 are orthonormal sets, µj > 0, and
N∑
j=1
µj(X) = ‖X‖I1 . (2.14)
Define
dµB,X =
N∑
j=1
µj(X) dµB;ϕj ,ψj (2.15)
which converges by (2.12) and (2.14).
Theorem 2.4. Let A,B be bounded operators and X = (A − B) ∈ I1. Let
ξ(λ;A,B) be the KrSSF. Let dµB,X be given by (2.10). Define
η(λ; A,B) ≡ µB,X((−∞, λ))−
λ∫
−∞
ξ(λ′; A,B) dλ′, λ ∈ R. (2.16)
Then η( · ; A,B) has compact support and for any f ∈ C∞(R), we have
Tr
(
f(A)− f(B)− d
dα
f(B + αX)
∣∣∣∣
α=0
)
=
∫
R
f ′′(λ)η(λ; A,B) dλ. (2.17)
R e m a r k s. 1. Since
∫
R
dµB,X(λ) = Tr(X) =
∫
R
ξ(λ;A, B) dλ,
we can replace (−∞, λ) in both places in (2.16) by [λ,∞). This shows that η in
(2.16) has compact support.
2. (2.17) determines η uniquely up to an affine term. The condition that η
has compact support (as the η of (2.16) does) determines η uniquely.
P r o o f. We first claim that d
dαf(B + αX)
∣∣
α=0
is trace class and that
Tr
(
d
dα
f(B + αX)
∣∣∣∣
α=0
)
= Tr(Xf ′(B)). (2.18)
70 Journal of Mathematical Physics, Analysis, Geometry, 2009, vol. 4, No. 1
On the Koplienko Spectral Shift Function. I. Basics
This is immediate for f nice enough (e.g., f such that (2.6) holds for n = 1) since
Tr(eiαβBXeiα(1−β)B) = Tr(XeiαB). (2.19)
Thus, by (2.10),
Tr
(
d
dα
f(B + αX)
)
=
∫
R
f ′(λ) dµB,X(λ)
= −
∫
R
f ′′(λ)[µB,X((−∞, λ))].
Similarly, by (1.6),
Tr(f(A)− f(B)) =
∫
R
f ′(λ)ξ(λ;A, B) dλ
= −
∫
R
f ′′(λ)
λ∫
−∞
ξ(λ′;A, B) dλ′
dλ.
The next critical step will be to prove positivity of η.
3. Convexity of Tr(f(A))
Positivity of η is essentially equivalent to the following result:
Theorem 3.1. Let f be a convex function on R. Then the mapping
A 7→ Tr(f(A)) (3.1)
is a convex function on the m×m selfadjoint matrices for every m ∈ N.
R e m a r k s. 1. More generally, if f is convex on (a, b), (3.1) is convex
on matrices A with spectrum in (a, b). In fact, it is easy to see that any convex
function f on (a, b) is a monotone limit on (a, b) of convex functions on R. So
this more general result is a consequence of Th. 3.1.
2. We will discuss the infinite-dimensional situation below.
Journal of Mathematical Physics, Analysis, Geometry, 2009, vol. 4, No. 1 71
F. Gesztesy, A. Pushnitski, and B. Simon
Two special cases of this are widely known and used:
(a) A 7→ Tr(eA) is convex.
(b) A 7→ Tr(A log(A)) is convex on A ≥ 0.
Both of these are rather special. In the first case, one has the stronger A 7→
log(Tr(eA)) is convex and the usual proof of it is via Hölder’s inequality (cf.,
e.g., [28, pp. 19–20] or [59, p. 57]) which proves the strong convexity of the
log(·), but does not prove Th. 3.1. In the second case, by Kraus’ theorem [37, 8]
A 7→ A log(A) is operator convex. (We also note that Ar, r ∈ R, is operator
convex for A > 0 if and only if r ∈ [−1, 0] ∪ [1, 2] (cf. [9, p. 147]).) We have
found Th. 3.1 stated in Alicki–Fannes [3, Sect. 9.1] and an equivalent statement
in Ruelle [55, Sect. 2.5] (who attributes it to Klein [35] although Klein only has
the special case f(x) = x log(x) and his proof is specific to that case; Ruelle’s is
not). We have also found it in Lieb–Pedersen [45] whose proof is closer to the
one we label “Third Proof“ below. The result is also mentioned in von Neumann
[67], although the proof he gives earlier for a special f does not seem to establish
the general case.
In any event, even though this result is not hard and is known to some ex-
perts, we provide several proofs because it is not widely known and is central to
the theory of KoSSF. We provide several proofs because they illustrate different
aspects of the theorem.
F i r s t P r o o f. This uses eigenvalue perturbation theory. By a limiting
argument, it suffices to prove it for functions f ∈ C∞(R). By approximating
derivatives of f by polynomials, we see that matrix elements, and so the trace
of f(A), are C∞-functions of A. By a limiting argument, we need only to show
λ → Tr(A + λX) has a nonnegative second derivative at λ = 0 in case A has
distinct eigenvalues.
So by changing basis, we suppose A is diagonal with the eigenvalues a1 <
a2 < · · · < am. Let ej(λ) be the eigenvalue of A + λX near aj for |λ| sufficiently
small. As is well known [33, Sect. II.2], [53, Sect. XII.1],
d2ej
dλ2
∣∣∣∣
λ=0
=
m∑
k=1
k 6=j
|X2
k,j |
aj − ak
. (3.2)
Clearly,
d
dλ
[f(ej(λ))] = f ′(ej(λ))e′j(λ),
d2
dλ2
f(ej(λ)) = f ′′(ej(λ))e′j(λ)2 + f ′(ej(λ))e′′j (λ),
so
d2
dλ2
Tr(f(A(λ)))
∣∣∣∣
λ=0
= 1 + 2 , (3.3)
72 Journal of Mathematical Physics, Analysis, Geometry, 2009, vol. 4, No. 1
On the Koplienko Spectral Shift Function. I. Basics
where
1 =
m∑
j=1
f ′′(aj)e′j(0)2 ≥ 0
since f ′′ ≥ 0 and, by (3.2),
2 =
m∑
j,k=1
k 6=j
|Xk,j |2
[
f ′(aj)− f ′(ak)
aj − ak
]
≥ 0
since for x < y,
f ′(y)− f ′(x)
y − x
=
1
y − x
y∫
x
f ′′(u) du ≥ 0.
S e c o n d P r o o f. This one uses a variational principle. We consider first
the case
f+(x) = x+ ≡
{
x, x ≥ 0,
0, x < 0.
(3.4)
We claim first that
Tr(f+(A)) = max{Tr(AB) | ‖B‖ ≤ 1, B ≥ 0}, (3.5)
where ‖ · ‖ is the matrix norm on Cm with the Euclidean norm. For in an
orthonormal basis where A is a diagonal matrix,
Tr(AB) =
m∑
j=1
ajbj,j ≤
m∑
j=1
(aj)+ = Tr(f+(A))
if 0 ≤ bjj ≤ 1. On the other hand, if B is the diagonal matrix with
bjj =
{
1, aj > 0,
0, aj ≤ 0,
then B ≥ 0, ‖B‖ ≤ 1, and Tr(AB) = Tr(f+(A)). This proves (3.5).
Convexity is immediate for f+ given by (3.4) once we have (3.5), since maxima
of linear functionals are convex. Obviously, since (x − λ)+ is just a translate of
x+, we get convexity for any function of
∞∫
λ0
(x− λ)+ dµ(λ)
Journal of Mathematical Physics, Analysis, Geometry, 2009, vol. 4, No. 1 73
F. Gesztesy, A. Pushnitski, and B. Simon
for any Borel measure µ on (λ0,∞). But every convex function f with f ≡ 0 for
x ≤ λ0 has this form.
Adding ax + b to this, we get the result for any convex function f
with f ′′(x) = 0 for x ≤ λ0. Taking λ0 → −∞, we get the result for general
convex functions f .
T h i r d P r o o f (M.B. Ruskai, private communication). If f is any convex
function, C a selfadjoint m×m matrix with
Cej = λjej , (3.6)
and v ∈ Cm a unit vector, then
〈v, f(C)v〉 =
m∑
j=1
|〈v, ej〉|2f(λj)
≥ f
( m∑
j=1
λj |〈v, ej〉|2
)
(3.7)
= f(〈v, Cv〉), (3.8)
where (3.7) employs Jensen’s inequality.
Now suppose
C = θA + (1− θ)B, θ ∈ [0, 1].
Then,
Tr(f(C)) =
m∑
j=1
f(〈ej , Cej〉)
=
m∑
j=1
f(θ〈ej , Aej〉+ (1− θ)〈ej , Bej〉)
≤
m∑
j=1
[θf(〈ej , Aej〉) + (1− θ)f(〈ej , Bej〉)] (3.9)
≤ θ
m∑
j=1
〈ej , f(A)ej〉+ (1− θ)
m∑
j=1
〈ej , f(B)ej〉 (3.10)
= θTr(f(A)) + (1− θ)Tr(f(B)), (3.11)
proving convexity. In the above, (3.9) is direct convexity of f and (3.10) is (3.8)
for v = ej and C = A or B.
Corollary 2. If f ∈ C1(R) is convex and B and X are m × m selfadjoint
matrices, m ∈ N, then
Tr
(
f(B + X)− f(B)− d
dα
f(B + αX)
∣∣∣∣
α=0
)
≥ 0. (3.12)
74 Journal of Mathematical Physics, Analysis, Geometry, 2009, vol. 4, No. 1
On the Koplienko Spectral Shift Function. I. Basics
R e m a r k s. 1. It is not hard to see that (3.12) is equivalent to Th. 3.1.
2. It is in this form that the result appears in Ruelle [55, Sect. 2.5], and for
the case f(x) = x log(x), x > 0, in Klein [35].
P r o o f. If g ∈ C1(R) is a convex function,
g(x + y)− g(x)− g′(x)y ≥ 0, (3.13)
since convexity says that g lies above the tangent line at any point. (3.12) is
(3.13) for g(α) = Tr(f(B + αX)), x = 0, y = 1.
Corollary 3. For finite-dimensional matrices A and B, the KoSSF,
η( · ; A,B), satisfies η(λ; A,B) ≥ 0 for a.e. λ ∈ R.
P r o o f. Let h : R → [0,∞) be a measurable function bounded and
supported on an interval (a, b) with σ(A) ∪ σ(B) ⊂ (a, b) (so, by (2.16), η is
supported on (a, b)). Let f be the unique convex function with f = 0 near −∞
and f ′′ = h. By (3.12) and (2.17),
0 ≤
∫
R
h(λ)η(λ; A,B) dλ. (3.14)
Since h is arbitrary, η ≥ 0 a.e.
Theorem 3.4. For any finite selfadjoint matrices A,B (of the same size),
∫
R
|η(λ;A, B)| dλ = 1
2 ‖A−B‖2
I2
. (3.15)
R e m a r k s. 1. It is remarkable that we always have equality in (3.15).
The analog for the KrSSF is
∫
R
|ξ(λ; A,B)| dλ ≤ ‖A−B‖I1 , (3.16)
where equality, in general, holds if A−B is either positive or negative.
2. (3.15) emphasizes again the lack of a chain rule for η; η is nonlinear in
(A−B).
P r o o f. Take f(x) = 1
2x2 such that f ′′(x) = 1 and
f(B + X)− f(B)− d
dα
f(B + αX)
∣∣∣∣
α=0
= 1
2
[
(B + X)2 −B2 −XB −BX
]
= 1
2 X2.
Since η ≥ 0,
∫
R f ′′(λ)η(λ; A,B) dλ =
∫
R|η(λ; A,B)| dλ and (3.15) holds.
Journal of Mathematical Physics, Analysis, Geometry, 2009, vol. 4, No. 1 75
F. Gesztesy, A. Pushnitski, and B. Simon
In Section 5 we take limits from the finite-dimensional situation, but one can
easily extend Th. 3.1 in two ways and from there directly prove η ≥ 0 and (3.15)
in case (A − B) ∈ I1. Without proof, we state the extensions (the results are
simple limiting arguments from finite dimensions):
Theorem 3.5. If f is convex on R and f(0) = 0, then f(A) is trace class
for any selfadjoint trace class operator A, and for such A’s, the mapping A 7→
Tr(f(A)) is convex.
In this context we note that convex functions are Lipschitz continuous. (For
this and additional regularity results of convex functions, see, e.g., [9, pp. 145–
146].)
Theorem 3.6. For any convex function f ∈ C∞(R), any bounded selfadjoint
operator B, and any selfadjoint operator X ∈ I1,
[f(B + X)− f(B)] ∈ I1
and the mapping X 7→ Tr(f(B + X)− f(B)) is convex.
Convexity of maps of the type s 7→ Tr(f(B + X(s))− f(B)), s ∈ (s1, s2), for
convex f and certain classes of X(·) ∈ I1 was also studied in [24].
4. Löwner’s Formula and the Finite-Dimensional
Birman–Solomyak Bound
The final element needed to construct the KoSSF is the following lovely
theorem of Birman–Solomyak [13] (see also [15]):
Theorem 4.1. Let A,B be bounded selfadjoint operators with (A−B) Hilbert–
Schmidt. Let f be a function defined on an interval [a, b] ⊃ σ(A)∪σ(B). Suppose
f is uniformly Lipschitz, that is,
‖f‖L = sup
x,y∈[a,b]
x 6=y
|f(x)− f(y)|
|x− y| < ∞. (4.1)
Then [f(A)− f(B)] is also Hilbert–Schmidt and
‖f(A)− f(B)‖I2 ≤ ‖f‖L‖A−B‖I2 . (4.2)
The proof in [13] depends on the deep machinery of double Stieltjes operator
integrals. Our two points in this section are:
(1) The inequality for finite matrices is quite elementary and, by limits, ex-
tends to (4.2).
76 Journal of Mathematical Physics, Analysis, Geometry, 2009, vol. 4, No. 1
On the Koplienko Spectral Shift Function. I. Basics
(2) The key to our proof, a kind of “Double Stieltjes Operator Integral for
Dummies,” goes back to Löwner [46] in 1934 whose contributions to this
theme seem not to have been appreciated in the literature on double
Stieltjes operator integrals.
Given two finite m×m selfadjoint matrices A,B with respective eigenvectors
{ϕj}m
j=1 and {ψj}m
j=1 and eigenvalues {xj}m
j=1 and {yj}m
j=1 such that
Aϕj = xjϕj , Bψj = yjψj , (4.3)
we introduce the (modified) Löwner matrix of a function f by
Lk,` =
{
f(yk)−f(x`)
yk−x`
, yk 6= x`,
0, yk = x`,
1 ≤ k, ` ≤ m. (4.4)
(Löwner [46] originally supposed yk 6= x` for all 1 ≤ k, ` ≤ m.) Clearly, if f is
Lipschitz,
sup
1≤k,`≤m
|Lk,`| ≤ ‖f‖L. (4.5)
Löwner noted that since
f(A)ϕj = f(xj)ϕj , f(B)ψj = f(yj)ψj , (4.6)
we have Löwner’s formula:
〈ψk, [f(B)− f(A)]ϕ`〉 = Lk`〈ψk, (B −A)ϕ`〉, (4.7)
and this holds even if yk = x` (since then both matrix elements vanish). This is
the “baby” version of the double Stieltjes operator integral formula
f(B)− f(A) =
∫
σ(A)
∫
σ(B)
f(y)− f(x)
y − x
dEB(x)(B −A)dEA(y)
due to Birman and Solomyak [13, 15]. Here the integration is with respect to the
spectral measures of A and B.
Löwner’s formula immediately implies:
Proposition 4.2. (4.2) holds for finite selfadjoint matrices.
P r o o f. Hilbert–Schmidt norms can be computed in any basis, even two
different ones, that is,
‖C‖2
I2
=
m∑
`=1
‖Cϕ`‖2 =
m∑
k,`=1
|〈ψk, Cϕ`〉|2.
Journal of Mathematical Physics, Analysis, Geometry, 2009, vol. 4, No. 1 77
F. Gesztesy, A. Pushnitski, and B. Simon
Thus,
‖f(A)− f(B)‖2
I2
=
m∑
k,`=1
|〈ψk, (f(B)− f(A))ϕ`〉|2
=
m∑
k,`=1
L2
k`|〈ψk, (B −A)ϕ`〉|2 by (4.7)
≤ ‖f‖2
L
m∑
k,`=1
|〈ψk, (B −A)ϕ`〉|2 by (4.5)
= ‖f‖2
L‖A−B‖2
I2
.
P r o o f o f T h e o r e m 4.1. Let {ζj}∞j=1 be an orthonormal basis for
H and PN the orthogonal projections onto the linear span of {ζj}N
j=1. For any A
and B and Lipschitz f , by Prop. 4.2,
‖f(PNBPN )− f(PNAPN )‖I2 ≤ ‖f‖L‖PN (B −A)PN‖I2 (4.8)
≤ ‖f‖L‖B −A‖I2 (4.9)
if B −A is Hilbert–Schmidt, since
‖PN (B −A)PN‖2
I2
=
n∑
j=1
‖PN (B −A)ζj‖2 ≤
n∑
j=1
‖(B −A)ζj‖2 (4.10)
≤ ‖B −A‖2
I2
. (4.11)
Thus, for any k ∈ N,
k∑
j=1
‖[f(PNBPN )− f(PNAPN )]ζj‖2 ≤ ‖f‖2
L‖B −A‖2
I2
. (4.12)
As PNBPN −→
N→∞
B strongly, one infers by continuity of the functional cal-
culus that f(PNBPN ) −→
N→∞
f(B) strongly. Since the sum in (4.12) is finite, one
concludes that
k∑
j=1
‖(f(B)− f(A))ζj‖2 ≤ ‖f‖2
L‖B −A‖2
I2
.
Taking k →∞, we see that [f(B)− f(A)] ∈ I2 and that (4.2) holds.
78 Journal of Mathematical Physics, Analysis, Geometry, 2009, vol. 4, No. 1
On the Koplienko Spectral Shift Function. I. Basics
5. General Construction of the KoSSF η( · ; A,B)
The general construction and proof of properties of η depends first on an
approximation of trace class operators by finite rank ones and then on an appro-
ximation of Hilbert–Schmidt operators by trace class operators. In this section,
we mostly follow the approach of [36, Lem. 3.3].
Theorem 5.1. Let Bn, B, n ∈ N, be uniformly bounded selfadjoint operators
such that Bn −→
n→∞ B strongly. Let Xn, X, n ∈ N, be a sequence of selfadjoint
trace class operators such that ‖X − Xn‖I1 −→
n→∞ 0. Then for any continuous
function, g, of compact support, we conclude that
∫
R
g(λ)η(λ; Bn + Xn, Bn) dλ −→
n→∞
∫
R
g(λ)η(λ;B + X,B) dλ. (5.1)
In particular, η( · ; A,B) ≥ 0 a.e. on R if (A− B) ∈ I1 and, in that case, (3.15)
holds.
P r o o f. By Th. A.7 and
∫
R
g(λ)
( λ∫
−∞
ξ(λ′;A,B) dλ′
)
dλ =
∞∫
−∞
ξ(λ′;A,B)
( ∞∫
λ′
g(λ) dλ
)
dλ′
we get convergence of the second term in (2.16). By (2.10),
dµBn,Xn −→
n→∞ dµB,X
weakly by the strong continuity of the functional calculus since
|Tr(Xnf(Bn))− Tr(Xf(B))|
≤ |Tr(X[f(Bn)− f(B)])|+ ‖f‖∞‖X −Xn‖I1 −→n→∞ 0,
as f is continuous.
Since weak limits of positive measures are positive, the positivity follows from
positivity in the finite-dimensional case taking Bn = PnBPn and Xn = PnXPn
for finite-dimensional Pn converging strongly to I, the identity operator.
Once we have positivity, we obtain (3.15) directly by following the proof of
Th. 3.4.
Theorem 5.2. Let A, B, C be bounded selfadjoint operators such that
(A− C) ∈ I1 and (B − C) ∈ I1. Then
∫
R
|η(λ; A,C)− η(λ; B,C)| dλ ≤ ‖A−B‖I2
[
1
2 ‖A−B‖I2 + ‖B − C‖I2
]
. (5.2)
Journal of Mathematical Physics, Analysis, Geometry, 2009, vol. 4, No. 1 79
F. Gesztesy, A. Pushnitski, and B. Simon
P r o o f. We begin with (1.9) which follows from the fact that (2.17) holds
when (A−B) ∈ I1. Here (1.10) holds for nice functions g, say, g ∈ C∞(R). Thus,
LHS of (5.2) ≤
∫
R
|η(λ; A,B)| dλ +
∫
R
|δη(λ; A,B, C)| dλ. (5.3)
By (3.15),
First term on RHS of (5.3) ≤ 1
2 ‖A−B‖I2 ‖A−B‖I2 . (5.4)
As for δη, by (1.10),
∣∣∣∣
∫
R
g′(λ)δη(λ) dλ
∣∣∣∣ ≤ ‖A−B‖I2 ‖g(B)− g(C)‖I2
≤ ‖g′‖∞‖A−B‖I2 ‖B − C‖I2
by Th. 4.1. Since δη ∈ L1(R) and the bounded C∞(R)-functions are ‖ · ‖∞-dense
in the bounded continuous functions, and for h ∈ L1(R),
‖h‖1 = sup
f∈C(R)
‖f‖∞=1
∣∣∣∣
∫
R
f(x)h(x) dx
∣∣∣∣,
we conclude
‖δη‖1 ≤ ‖A−B‖I2 ‖B − C‖I2 . (5.5)
Relations (5.3)–(5.5) imply (5.2).
Here is the main theorem on the existence of the KoSSF:
Theorem 5.3. Let A, B be two bounded selfadjoint operators with (B − A)
∈ I2. Then there exists a unique L1(R; dλ)-function η( · ;A,B) supported on
(−max(‖A‖, ‖B‖), max(‖A‖, ‖B‖)) such that for any g ∈ C∞(R),
(
g(A)− g(B)− d
dα
g(B + α(A−B))
∣∣∣∣
α=0
)
∈ I1 (5.6)
and
Tr
(
g(A)− g(B)− d
dα
g(B + α(A−B))
∣∣∣∣
α=0
)
=
∫
R
g′′(λ)η(λ; A,B) dλ. (5.7)
Moreover,
η( · ; A,B) ≥ 0 a.e. on R, (5.8)
80 Journal of Mathematical Physics, Analysis, Geometry, 2009, vol. 4, No. 1
On the Koplienko Spectral Shift Function. I. Basics
∫
R
|η(λ; A,B)| dλ = 1
2 ‖A−B‖2
I2
, (5.9)
and for any bounded selfadjoint operators A, B, C with (A−C) ∈ I2 and (B−C)
∈ I2,∫
R
|η(λ;A,C)− η(λ;B, C)| dλ ≤ ‖A−B‖I2
[
1
2 ‖A−B‖I2 + ‖B − C‖I2
]
. (5.10)
R e m a r k. For a sharp condition on the class of functions η for which
Koplienko’s trace formula holds, we refer to Peller [52].
P r o o f. Let X = A − B and pick Xn ∈ I1, n ∈ N, such that
‖Xn −X‖I2 −→n→∞ 0. By (5.10), which we have proven for
η(λ;B + Xn, B)− η(λ; B + Xm, B),
we see that η( · ; B +Xn, B) is Cauchy in L1(R; dλ) and so converges a.e. to what
we will define as η( · ; A,B). (5.7) holds by taking limits; η ≥ 0 as a limit of
positive functions η. (5.9) and (5.10) hold by taking limits.
R e m a r k. Here is an alternative method of proving the estimate (5.2),
by passing Th. 5.1: In the same way as in Cor. 3.3, one can deduce positivity of
η( · ; A,B) for (A−B) ∈ I1 from Th. 3.6. The only place where Th. 5.1 is used in
the proof of Th. 5.2 is in the estimate (5.4). This estimate (see Th. 3.4) follows
directly from the positivity of η and the trace formula (2.15).
6. What Functions η Are Possible?
We introduce the classes of functions
η(I2) = {η( · ; A,B) |A,B bounded and selfadjoint, (A−B) ∈ I2},
η(I1) = {η( · ; A,B) |A,B bounded and selfadjoint, (A−B) ∈ I1}.
In this section we would like to raise the question of the description of the classes
η(I2) and η(I1).
Since, for now, we are considering A,B bounded, η has a compact support.
First we discuss the class η(I2). By Th. 5.3, all functions of this class are
nonnegative and Lebesgue integrable. It would be interesting to see if the class
η(I2) contains all nonnegative Lebesgue integrable functions.
As a step towards answering this question, we give the following elementary
result:
Theorem 6.1. The class η(I2) contains all nonnegative Riemann integrable
functions of compact support.
Journal of Mathematical Physics, Analysis, Geometry, 2009, vol. 4, No. 1 81
F. Gesztesy, A. Pushnitski, and B. Simon
R e m a r k. A contemporary account of the theory of Riemann integrable
functions can be found, for instance, in Stein–Shakarchi [63].
P r o o f. First we consider a simple example. Let a ∈ R and ε > 0; consider
the operators in C2 given by the diagonal 2 × 2 matrices B = diag(a − ε, a + ε)
and A = diag(a + ε, a− ε). Then the KoSSF for this pair is given by (cf. (2.16))
η(λ) = 2εχ(a−ε,a+ε)(λ). We note that
∫
R
η(λ) dλ = 4ε2 = 1
2Tr((A−B)2),
in agreement with (5.9).
Next, suppose that 0 ≤ η ∈ L1(R; dλ) is represented by the L1(R; dλ)-
convergent series
η(λ) =
∞∑
n=1
|In|χIn(λ), (6.1)
where In ⊂ R are (not necessarily disjoint) finite intervals and |In| is the length
of In. Denote by an the midpoint of In and let εn = 1
2 |In|. We introduce
B = ⊕∞n=1diag(an − εn, an + εn) and A = ⊕∞n=1diag(an + εn, an − εn)
in the Hilbert space ⊕∞n=1C2. Note that the L1(R; dλ)-convergence of
the series (6.1) is equivalent to the condition
∑∞
n=1 ε2
n < ∞ and so A − B =
⊕∞n=1diag(2εn,−2εn) is a Hilbert–Schmidt operator. It is clear that the KoSSF
for the pair A,B coincides with η.
Thus, it suffices to prove that any Riemann integrable function 0 ≤ η ∈
L1(R; dλ) can be represented as an L1(R; dλ)-convergent series (6.1).
Let 0 ≤ η ∈ L1(R; dλ) be Riemann integrable. According to the definition
of the Riemann integral, there exists a finite set of disjoint open squares Qn,
n ∈ {1, . . . , M}, which fit under the graph of η and
M∑
n=1
area(Qn) ≥ 1
2
∫
R
η(λ) dλ.
In other words, there exists a finite set of (not necessarily disjoint) open intervals
In ⊂ R, n ∈ {1, . . . , N}, such that
N∑
n=1
|In|χIn(λ) ≤ η(λ), λ ∈ R,
∫
R
( N∑
n=1
|In|χIn(λ)
)
dλ =
N∑
n=1
|In|2 ≥ 1
2
∫
R
η(λ) dλ.
82 Journal of Mathematical Physics, Analysis, Geometry, 2009, vol. 4, No. 1
On the Koplienko Spectral Shift Function. I. Basics
Thus, we can represent η as
η(λ) =
N1∑
n=1
|In|χIn(λ) + η1(λ),
η1(λ) ≥ 0,
∫
R
η1(λ) dλ ≤ 1
2
∫
R
η(λ) dλ,
and the sum is taken over a finite set of indices n ∈ {1, . . . , N1}. Iterating this
procedure, we see that for any m ∈ N we can represent η as
η(λ) =
Nm∑
n=1
|In|χIn(λ) + ηm(λ),
ηm(λ) ≥ 0,
∫
R
ηm(λ) dλ ≤ 2−m
∫
R
η(λ) dλ,
where εn > 0, an ∈ R, and the sum is taken over a finite set of indices n. Taking
m → ∞, it follows that η can be represented as an L1(R; dλ)-convergent series
(6.1).
Regarding the class η(I1), we note only that every function of this class is of
bounded variation. This follows from (2.16), since both terms on the right-hand
side of (2.16) are of bounded variation. We also note that it follows from the
proof of Th. 6.1 that the class η(I1) contains all functions of the type
η(λ) =
∞∑
n=1
|In|χIn(λ),
∞∑
n=1
|In| < ∞.
7. Modified Determinants and the KoSSF
In this section, as a preliminary to the next, we want to use our viewpoint to
prove a formula for modified perturbation determinants in terms of the KoSSF
originally derived by Koplienko [36]. We recall that one of Krein’s motivating
formulas for the KrSSF is (see (A.32)):
det((A− z)(B − z)−1) = exp
(∫
R
(λ− z)−1ξ(λ) dλ
)
, z ∈ C\R. (7.1)
Here det(·) is the Fredholm determinant defined on I +I1 (since A−B = X ∈ I1
implies (A− z)(B − z)−1 − I = X(B − z)−1 ∈ I1) (see [27, 61]).
Journal of Mathematical Physics, Analysis, Geometry, 2009, vol. 4, No. 1 83
F. Gesztesy, A. Pushnitski, and B. Simon
We recall that for C ∈ I1, one can define det2(·) by
det2(I + C) = det(I + C)e−Tr(C) (7.2)
and that C 7→ det2(I +C) extends uniquely and continuously to I2, the Hilbert–
Schmidt operators, although the right-hand side of (7.2) no longer makes sense
(see [27, Ch. IV], [61, Ch. 9]). Our goal in this section is to prove the following
formula first derived by Koplienko [36]:
Theorem 7.1. Let B and X be bounded selfadjoint operators and X ∈ I2.
Let A = B + X. Then for any z ∈ C\R, (A− z)(B − z)−1 ∈ I + I2 and
det2((A− z)(B − z)−1) = exp
(
−
∫
R
η(λ; A,B)
(λ− z)2
dλ
)
. (7.3)
P r o o f. It suffices to prove (7.3) for X ∈ I1 since both sides are continuous
in I2 norm and I1 is dense in I2. Continuity of the left-hand side follows from
Th. 9.2(c) of [61] and of the right-hand side by Th. 5.2 above.
When X ∈ I1, we can use (7.2). Let
g1(λ) = µB,X((−∞, λ)), g2(λ) =
λ∫
−∞
ξ(λ′;A,B) dλ′. (7.4)
By an integration by parts argument (using g′2 = ξ),
∫
R
ξ(λ)
λ− z
dλ =
∫
R
g2(λ)
(λ− z)2
dλ. (7.5)
By an integration by parts in a Stieltjes integral and by (2.10),
Tr(X(B − z)−1) =
∫
R
1
λ− z
dµB,X(λ) (7.6)
=
∫
R
g1(λ)
1
(λ− z)2
dλ. (7.7)
Thus, by (7.1) and (7.2),
det2(1 + X(B − z)−1) = exp
(
−
∫
R
(g1(λ)− g2(λ))(λ− z)−2 dλ
)
,
which, given (2.16) is (7.3).
84 Journal of Mathematical Physics, Analysis, Geometry, 2009, vol. 4, No. 1
On the Koplienko Spectral Shift Function. I. Basics
8. On Boundary Values of Modified Perturbation Determinants
det2((A− z)(B − z)−1)
By (7.1), if (A − B) ∈ I1, det((A − z)(B − z)−1) has a limit as z → λ + i0
for a.e. λ ∈ R since
∫
R(ν − z)−1ξ(ν) dν is a difference of Herglotz functions.
In this section, we will consider nontangential boundary values to the real axis
of modified perturbation determinants
det2((A− z)(B − z)−1), z ∈ C+,
where X = (A − B) ∈ I2. Unlike the trace class, we will see nontangential
boundary values may not exist a.e. on R.
For notational simplicity in the remainder of this section, we now abbreviate
KoSSF simply by η, that is, η ≡ η( · ; A,B).
In contrast to the usual (trace class) SSF theory, we have the following
nonexistence result for boundary values of modified perturbation determinants:
Theorem 8.1. There exists a pair of selfadjoint operators A, B (in a complex,
separable Hilbert space) such that X = (A−B) ∈ I2, σ(B) is an interval, and for
a.e. λ ∈ σ(B), the nontangential limit limz→λ, z∈C+ det2(I + X(B − zI)−1) does
not exist.
P r o o f. By Th. 6.1 and 7.1, the proof reduces to the following statement:
There exists a Riemann integrable 0 ≤ η ∈ L1(R; dλ) with support being an
interval such that for a.e. λ ∈ supp (η), the nontangential limit
lim
z→λ
z∈C+
∫
R
η(λ) dλ
(λ− z)2
(8.1)
does not exist.
First we note that the existence of the limit in (8.1) at the point λ depends
only on the behavior of η(t) when t varies in a small neighborhood of λ. Thus,
it suffices to construct 0 ≤ η ∈ L1(R; dλ) such that the limits (8.1) do not exist
for a.e. λ ∈ (−1, 1); by shifting and scaling such a function η, one obtains the
required statement for a.e. λ ∈ σ(B).
Let us first obtain the required example of η defined on the unit circle ∂D, and
then transplant it onto the real line. By a well-known construction employing
either lacunary series or Rademacher functions (see [21], [22, App. A], [71, I, p. 6]),
there exists a power series f(z) =
∑∞
n=1 cnzn, |z| ≤ 1, such that
∑∞
n=1|cn| < ∞
and for a.e. z ∈ ∂D, the limit limζ→z f ′(ζ) does not exist as ζ approaches z from
inside of the unit disc along any nontangential trajectory. By construction, Im(f)
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F. Gesztesy, A. Pushnitski, and B. Simon
is continuous on ∂D and
f(z) =
1
π
∫
∂D
Im(f(ζ))
(ζ − z)
dζ, f ′(z) =
1
π
∫
∂D
Im(f(ζ))
(ζ − z)2
dζ, |z| < 1.
Let a > −minζ∈∂D Im(f(ζ)) and set v(ζ) = Im(f(ζ)) + a if |arg ζ| < π/2 and
v(ζ) = 0 otherwise. Then v ≥ 0 and v is piecewise continuous (with the possible
discontinuities only for arg(ζ) = ±π/2); in particular, v is Riemann integrable.
Again by a localization argument, for a.e. θ ∈ (−π/2, π/2), the limit
lim
z→eiθ
∫
∂D
Im(f(ζ))
(ζ − z)2
dζ
does not exist as z approaches eiθ from inside of the unit disc along any nontan-
gential trajectory.
It remains to transplant v from the unit circle onto the real line. Let t = i1−ζ
1+ζ ,
w = i1−z
1+z , and η(t) = v(ζ(t)). Then 0 ≤ η ∈ L1(R; dλ), supp (η) ⊂ (−1, 1), η is
Riemann integrable, and
∫
∂D
Im(f(ζ))
(ζ − z)2
dζ = −(w + i)2
2i
1∫
−1
η(λ) dλ
(λ− w)2
.
Thus, the limit (8.1) does not exist for a.e. λ ∈ (−1, 1).
9. KoSSF for Unbounded Operators
In this section we briefly discuss the question of existence of KoSSF under
the assumption
[(A− z)−1 − (B − z)−1] ∈ I2 (9.1)
instead of (A−B) ∈ I2. This question was studied in [49] and [52] (see also [36]
for related issues).
First recall the invariance principle for the KrSSF. Assume that A,B are
bounded selfadjoint operators and (A − B) ∈ I1. Let ϕ = ϕ ∈ C∞(R), ϕ′ 6= 0
on R. Then we have
ξ(λ;A,B) = sign (ϕ′) ξ(ϕ(λ);ϕ(A), ϕ(B)) + const for a.e. λ ∈ R. (9.2)
This is a consequence of Krein’s trace formula (1.6). With an appropriate choice
of normalization of KrSSF, the constant in the right-hand side of (9.2) vanishes.
86 Journal of Mathematical Physics, Analysis, Geometry, 2009, vol. 4, No. 1
On the Koplienko Spectral Shift Function. I. Basics
When both (A − B) ∈ I1 and [ϕ(A) − ϕ(B)] ∈ I1, formula (9.2) is an easily
verifiable identity. But when [ϕ(A)− ϕ(B)] ∈ I1 yet (A−B) 6∈ I1, this formula
can be regarded as a definition of ξ( · ; A,B).
In contrast to this, no explicit formula relating η(ϕ(·);ϕ(A), ϕ(B))
to η( · ; A,B) is known. The reason is simple: The definition of η involves not
only a trace formula but a choice of interpolation A(θ) between B and A. For
bounded selfadjoint operators, the choice A(θ) = (1 − θ)B + θA, θ ∈ [0, 1], is
natural. But when one only has (9.1), what choice does one make? It is natural
to define A(θ) by
(A(θ)− z)−1 = (1− θ)(B − z)−1 + θ(A− z)−1, θ ∈ [0, 1]. (9.3)
For this to be selfadjoint, we need z ∈ R, which means we should have some real
point in the intersection of the resolvent sets for A and B. Even if there were
such a z, it is not unique and the interpolation will not be unique. Moreover,
the convexity that led to η ≥ 0 may be lost. The net result is that the situation,
both after the work of others and our work, is less than totally satisfactory.
Let us discuss a certain surrogate of (9.2) for the KoSSF. The formulas below
are a slight variation on the theme of the construction of [49].
First assume that A and B are bounded operators and X = (A − B) ∈ I2.
Let δ ⊂ R be an interval which contains the spectra of A and B and ϕ ∈ C∞(δ),
ϕ′ 6= 0. Denote a = ϕ(A), b = ϕ(B), x = a− b. By the Birman–Solomyak bound
(1.7), we have x ∈ I2 and so both η( · ; A,B) and η( · ; a, b) are well defined. Let
us display the corresponding trace formulas:
Tr
(
f(A)− f(B)− d
dα
f(B + αX)
∣∣∣∣
α=0
)
=
∫
R
η(λ;A, B)f ′′(λ) dλ, (9.4)
Tr
(
g(a)− g(b)− d
dα
g(b + αx)
∣∣∣∣
α=0
)
=
∫
R
η(µ; a, b)g′′(µ) dµ. (9.5)
Now suppose f = g ◦ ϕ. In contrast to the corresponding calculation for the
KrSSF, the left-hand sides of (9.4) and (9.5) are, in general, distinct. However, we
can make the right-hand sides look similar if we introduce the following modified
KoSSF:
η̃(λ; A,B) = η(ϕ(λ); a, b)
1
ϕ′(λ)
−
λ∫
λ0
η(ϕ(t); a, b)
(
1
ϕ′(t)
)′
dt. (9.6)
The choice of λ0 above is arbitrary; it affects only the constant term in the
definition of η̃.
Journal of Mathematical Physics, Analysis, Geometry, 2009, vol. 4, No. 1 87
F. Gesztesy, A. Pushnitski, and B. Simon
By a simple calculation involving integration by parts, we get
∫
R
η(µ; a, b)g′′(µ) dµ =
∫
R
η̃(λ; A,B)f ′′(λ) dλ, f = g ◦ ϕ ∈ C∞
0 (R). (9.7)
Combining (9.5) and (9.7), we get the modified trace formula
Tr
(
f(A)− f(B)− d
dα
f ◦ ϕ−1(b + αx)
∣∣∣∣
α=0
)
=
∫
R
η̃(λ;A,B)f ′′(λ) dλ (9.8)
for all f ∈ C∞
0 (R). Precisely as for the KrSSF, one can treat (9.6) and (9.8) as
the definition of a modified KoSSF η̃( · ;A,B).
We consider an example of this construction which might be useful in ap-
plications. Suppose that A and B are lower semibounded selfadjoint operators
such that for some (and thus for all) z ∈ C\(σ(A) ∪ σ(B)) the inclusion (9.1)
holds. Choose E ∈ R such that inf σ(A + E) > 0 and inf σ(B + E) > 0. Take
ϕ(λ) = 1
λ+E and let a = (A + E)−1, b = (B + E)−1, x = a − b. For λ > −E,
define
η̃(λ; A,B) = −η((λ + E)−1; a, b)(λ + E)2
+ 2
λ∫
−E
η((t + E)−1; a, b)(t + E) dt.
(9.9)
Note that η( · ; a, b) is integrable and η(λ; a, b) vanishes for large λ and therefore
the integral in (9.9) converges. Moreover, this definition ensures that η̃(λ; A,B) =
0 for λ < inf(σ(A) ∪ σ(B)). Thus, it is natural to define
η̃(λ;A,B) = 0 for λ ≤ −E. (9.10)
The above calculations prove the following result:
Theorem 9.1. Let A, B, a, b, x be as above. Then there exists a function
η̃( · ; A,B) such that ∫
R
η̃(λ;A,B)(λ + E)−4dλ < ∞ (9.11)
and η̃(λ; A,B) = 0 for λ < inf(σ(A)∪σ(B)) and for all f ∈ C∞
0 (R) the following
trace formula holds:
Tr
(
f(A)− f(B)− d
dα
f((b + αx)−1 − E)
∣∣∣∣
α=0
)
=
∫
R
η̃(λ; A,B)f ′′(λ) dλ. (9.12)
88 Journal of Mathematical Physics, Analysis, Geometry, 2009, vol. 4, No. 1
On the Koplienko Spectral Shift Function. I. Basics
We note that condition (9.11) does not fix the linear term in the definition of
η̃ but (9.10) does.
In [49], a pair of selfadjoint operators A,B was considered under the as-
sumption (9.1) alone (without the lower semiboundedness assumption). Another
regularization of η( · ; A,B) was suggested in this case. The construction of [49] is
more intricate than the above calculation and uses KoSSF for unitary operators.
In [36], the assumption
(A−B)|A− iI|−1/2 ∈ I2
was used. This assumption is intermediate between (A−B) ∈ I2 and (9.1). Under
this assumption, the trace formula (5.7) was proven with 0 ≤ η ∈ L1(R; (1 +
λ)−γdλ) for any γ > 1
2 .
Finally, in [49], the assumption
(A−B)(A− iI)−1 ∈ I2
was used and formula (5.7) was proven with η ∈ L1(R; (1 + λ2)−2dλ).
Note that the difference between the last two results and Th. 9.1 is that in
Th. 9.1, a modified trace formula (9.12) is proven rather than the original formula
(5.7). Theorem 9.1 is nothing but a change of variables in the trace formula for
resolvents, whereas the abovementioned results of [36] and [49] require some work.
10. The Case of Unitary Operators
In this section, we want to briefly discuss a definition of η for a pair of uni-
taries. Once again, there is an issue of interpolation. If A and B are the unitaries,
A(θ) = (1− θ)B + θA, θ ∈ [0, 1], (10.1)
is not unitary, so we cannot define f(A(θ)) for arbitrary C∞-functions on ∂D =
{z ∈ C | |z| = 1}. Neidhardt [49] (see also [52]) discussed one way of interpolating
by writing A = eC , B = eD for suitable C and D and interpolating, but there
is considerable ambiguity in how to choose C, D as well as whether to look at
eθC+(1−θ)D or e(1−θ)DeθC , etc. We also note that Rybkin [57] considered the case
of unitary operators differing by a Hilbert–Schmidt perturbation in the context
of Lax–Phillips scattering theory.
Here, with Szegő’s theorem as background [25], we want to discuss an alter-
native to Neidhardt’s approach.
Lemma 10.1. Let A, B be unitary with (A−B) ∈ I2. Then for any
n = 0, 1, 2, . . . , (
An −Bn − d
dθ
A(θ)n
∣∣∣∣
θ=0
)
∈ I1. (10.2)
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F. Gesztesy, A. Pushnitski, and B. Simon
We have ∥∥∥∥An −Bn − d
dθ
A(θ)n
∣∣∣∣
θ=0
∥∥∥∥
I1
≤ n(n− 1)
2
‖A−B‖2
I2
. (10.3)
In fact,
LHS of (10.3) = o(n2). (10.4)
P r o o f. Let X = A−B. Then, by telescoping,
A(θ)n −Bn =
n−1∑
j=0
A(θ)j(θX)Bn−1−j . (10.5)
Thus, since ‖A(θ)‖ ≤ 1, ‖B‖ = 1,
‖A(θ)n −Bn‖I2 ≤ n|θ| ‖X‖I2 (10.6)
and, of course,
‖A(θ)n −Bn‖ ≤ 2. (10.7)
Dividing (10.5) by θ and taking θ to zero yields
d
dθ
A(θ)n =
n−1∑
j=0
Bj X Bn−1−j ,
so
LHS of (10.2) =
n−1∑
j=0
(Aj −Bj)X Bn−1−j . (10.8)
(10.3) is immediate since (10.8) and (10.6) implies
LHS of (10.3) ≤ ‖X‖2
2
( n−1∑
j=0
j
)
. (10.9)
To get (10.4), we write X = X
(1)
ε + X
(2)
ε where ‖X(2)
ε ‖I2 ≤ ε and ‖X(1)
ε ‖I1 < ∞.
Thus, (10.8) implies
LHS of (10.3) ≤ ε‖X‖I1
n(n− 1)
2
+ 2n‖X(1)
ε ‖I1
using (10.7) instead of (10.6). Dividing by n2, taking n → ∞, and then ε ↓ 0,
show
lim sup
n→∞
n−2 LHS of (10.3) = 0.
90 Journal of Mathematical Physics, Analysis, Geometry, 2009, vol. 4, No. 1
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Theorem 10.2. Let A and B be unitary so (A−B) ∈ I2. Then there exists
a real distribution η(λ;A,B) on ∂D so that for any polynomial P (z),
[
P (A) −
P (B)− d
dθP (A(θ))
]∣∣
θ=0
∈ I2 and
Tr
(
P (A)− P (B)− d
dθ
P (A(θ))
∣∣∣∣
θ=0
)
=
2π∫
0
P ′′(eiθ)η(eiθ; A,B)
dθ
2π
. (10.10)
Moreover, the moments of η satisfy
2π∫
0
einθη(eiθ)
dθ
2π
=
|n|→∞
o(1). (10.11)
R e m a r k s 1. As usual, we use
∫ 2π
0 f(eiθ)η(eiθ) dθ
2π as shorthand for the
distribution η acting on the function f .
2. As we will discuss, η is determined by (10.10) up to three real constants
in an affine term.
3. For a sharp condition on the class of functions for which Neidhardt’s version
of Koplienko’s trace formula for unitary operators holds, we refer to Peller [52].
P r o o f. Let cn, n ∈ Z, be defined by
cn =
0, n = 0, 1,
[n(n− 1)]−1 Tr
(
An −Bn − d
dθ A(θ)n
∣∣
θ=0
)
, n ≥ 2,
c−n, n ≤ −1.
(10.12)
By Lemma 10.1, cn = o(1) as n → ∞, so there is a distribution η = η( · ;A, B)
satisfying
cn =
2π∫
0
ei(n−2)θη(eiθ)
dθ
2π
, n ≥ 2. (10.13)
By (10.12), we have (10.10) for P (z) = zn for n ≥ 2 and both sides are zero
for P (z) = zm, m = 0, 1. Thus, (10.10) holds for all polynomials.
For any c0 ∈ R, c1 ∈ C, we can add c0 + c1e
iθ + c̄1e
−iθ to η without changing
the right-hand side of (10.10). We wonder if η is always in L1(∂D) with η ≥ 0
for some choice of c0 and c1. The condition cn → 0 is, of course, consistent with
η ∈ L1(∂D).
Journal of Mathematical Physics, Analysis, Geometry, 2009, vol. 4, No. 1 91
F. Gesztesy, A. Pushnitski, and B. Simon
11. Open Problems and Conjectures
While we have found some new aspects of η here and summarized much of the
prior literature, there are many open issues. The most important one concerns
properties of η and the invariance of the a.c. spectrum:
Conjecture 11.1. Suppose A, B are selfadjoint with (A − B) ∈ I2 and that
on some interval (a, b) ⊂ σ(A) ∩ σ(B), we have η( · ; A,B) and η( · ; B, A) are
of bounded variation with distributional derivatives in Lp((a, b); dλ) on (a, b) for
some p > 1. Then σac(A) ∩ (a, b) = σac(B) ∩ (a, b).
In the Appendix, we prove the invariance for I1-perturbations using boundary
values of det((A − z)(B − z)−1). When η has the properties in the conjecture,
det2((A−z)(B−z)−1) has boundary values and we hope those can be used to get
the invariance of a.c. spectrum. While we made the conjecture assuming control
of η( · ; A,B) and η( · ; B, A), we wonder if only one suffices. Similarly, we wonder
if Lp, p > 1, can be replaced by the weaker condition that the derivative is a sum
of an L1-piece and the Hilbert transform of an L1-piece.
Open Question 11.2. Is the η we constructed in Sect. 10 for the unitary
case an L1(∂D) function?
Open Question 11.3. Is the class η(I2) introduced in Sect. 6 all of L1(R; dλ)
(of compact support ), or only the Riemann integrable functions, or something in
between?
Open Question 11.4. Is the class η(I1) all functions of bounded variation
or a subset, and if so, what subset?
Appendix: On the KrSSF ξ( · ; A,B)
Both for comparison and because the Krein spectral shift (KrSSF) is needed
in our construction of the KoSSF, we present the basics of the KrSSF here. Most
of the results in this Appendix are known (see, e.g., [7, Sect. 19.1.4], [14], [16],
[38–40], [62], [65], [69, Ch. 8], [70] and the references therein) so this Appendix
is largely pedagogical, but our argument proving the invariance of a.c. spectrum
under trace class perturbations at the end of this Appendix is new. Moreover, we
fill in the details of an approach sketched in [61, Ch. 11] exploiting the method
Gesztesy–Simon [26] used to construct the rank-one KrSSF. Most approaches
define ξ via perturbation determinants.
We will need the following strengthening of Th. 2.2:
Theorem A.1. Let f be a function of compact support whose Fourier trans-
form f̂ satisfies (2.6) for n = 1 (in particular, f can be C2+ε(R)). Then,
92 Journal of Mathematical Physics, Analysis, Geometry, 2009, vol. 4, No. 1
On the Koplienko Spectral Shift Function. I. Basics
(a) For any bounded selfadjoint operators A,B with (A − B) ∈ I1, (f(A) −
f(B)) ∈ I1. Moreover,
‖f(A)− f(B)‖I1 ≤ ‖kf̂‖1‖A−B‖I1 , (A.1)
where
‖kf̂‖1 ≤
∫
R
k|f̂(k)| dk. (A.2)
(b) Let Bn, B, n ∈ N, be uniformly bounded selfadjoint operators such that
Bn −→
n→∞ B strongly. Let Xn, X, n ∈ N, be a sequence of selfadjoint trace
class operators such that ‖X −Xn‖I1 −→n→∞ 0. Then,
Tr(f(Bn + Xn)− f(Bn)) −→
n→∞ Tr(f(B + X)− f(B)). (A.3)
P r o o f. (a) is immediate from Prop. 2.1 which implies
f(A)− f(B)
= (2π)−1/2
∫
R
ikf̂(k)
[ 1∫
0
eiβkA(A−B)ei(1−β)kB dβ
]
dk.
(A.4)
This also implies (b) via the dominated convergence theorem, continuity of the
functional calculus (so Cn −→
n→∞ C strongly implies eitCn −→
n→∞ eitC strongly), and
the fact that if Xn −→
n→∞ X in I1 and Cn −→
n→∞ C strongly (with Cn, C uniformly
bounded), then Tr(CnXn) −→
n→∞ Tr(CX). This latter fact comes from
|Tr(CnXn − CX)| ≤ |Tr(Cn(Xn −X)− (C − Cn)X)|
≤ ‖Cn‖ ‖Xn −X1‖I1 + |Tr((C − Cn)X)|
and if X =
∑
m∈N µm(X)〈ϕm, · 〉ψm, then
|Tr((Cn − C)X)| ≤
∑
m∈N
µm(X)|〈ϕm, (Cn − C)ψm〉| −→
n→∞ 0
by the dominated convergence theorem.
Part (a) in Th. A.1, in a slightly more general form, is stated and proved in
[40, p. 141].
Now let B be a bounded selfadjoint operator and ϕ a unit vector. For α ∈ R,
define
Aα = B + α(ϕ, · )ϕ (A.5)
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F. Gesztesy, A. Pushnitski, and B. Simon
and for z ∈ C\R,
Fα(z) = (ϕ, (Aα − z)−1ϕ), (A.6)
Gα(z) = 1 + αF0(z). (A.7)
The resolvent formula implies (see [61, Sect. 11.2])
Fα(z) =
F0(z)
1 + αF0(z)
, z ∈ C\R, (A.8)
and that
(Aα − z)−1 − (B − z)−1
= − α
1 + αF0(z)
((B − z̄)−1ϕ, · )(B − z)−1ϕ, z ∈ C\R,
(A.9)
implying
Tr((B − z)−1 − (Aα − z)−1) =
α
1 + αF0(z)
(ϕ, (B − z)−2ϕ)
=
d
dz
log(Gα(z)), z ∈ C\R.
(A.10)
Theorem A.2. Let B be a bounded selfadjoint operator and Aα given by (A.5)
for α ∈ R and ϕ with ‖ϕ‖ = 1. Then for a.e. λ ∈ R,
ξα(λ) =
1
π
lim
ε↓0
arg(Gα(λ + iε)) (A.11)
exists and satisfies
(i)
0 ≤ ±ξα( · ) ≤ 1 if 0 < ±α. (A.12)
(ii) ξα(λ) = 0 if λ ≤ min(σ(Aα) ∪ σ(B)) or λ ≥ max(σ(Aα) ∪ σ(B)).
(iii) ∫
|ξα(λ)| dλ = |α| (A.13)
(iv) For any z ∈ C\R,
Gα(z) = exp
(∫
R
(λ− z)−1ξα(λ) dλ
)
. (A.14)
(v)
det((Aα − z)(B − z)−1) = Gα(z). (A.15)
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On the Koplienko Spectral Shift Function. I. Basics
(vi) For any z ∈ C\R,
Tr((B − z)−1 − (Aα − z)−1) =
∫
R
(λ− z)−2ξα(λ) dλ. (A.16)
(vii) For any f satisfying the hypotheses of Th. A.1,
Tr(f(Aα)− f(B)) =
∫
R
f ′(λ)ξα(λ) dλ. (A.17)
R e m a r k s. 1. This theorem and its proof are essentially the same as the
starting point of Krein’s construction in [38] (see also [40, pp. 134–136] or [16,
Sect. 3]).
2. In (A.11), arg(Gα(z)) is defined uniquely for Im(z) > 0 by demanding
continuity in z and
lim
y↑∞
arg(Gα(iy)) = 0. (A.18)
For Im(z) < 0 one has Gα(z) = Gα(z).
3. By (A.9), (Aα − z)(B − z)−1 is of the form I+ rank one, and so lies in
I + I1. The det(·) in (A.15) is the Fredholm determinant (see [61, Ch. 3]). This
is the same as the finite-dimensional determinant det(C) for I + D with D finite
rank and C = (I + D) ¹ K where K is any finite-dimensional space containing
ran(D) and (ker(D))⊥.
4. The exponential Herglotz representation basic to this proof goes back to
Aronszajn and Donoghue [5].
5. Comparing (A.17) and (1.6), one concludes
ξα( · ) = ξ( · ; A,B).
P r o o f. By the spectral theorem, there is a probability measure dµα(λ)
such that
Fα(z) =
∫
R
dµα(λ)
λ− z
. (A.19)
In particular,
Im(F0(z)) > 0 if Im(z) > 0, (A.20)
so on C+ = {z ∈ C | Im(z) > 0},
±Im(Gα(z)) > 0 if ± α > 0. (A.21)
Since Gα(iy) → 1, as y ↑ ∞, we can define
log(Gα(z)) = Hα(z)
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F. Gesztesy, A. Pushnitski, and B. Simon
on C+ uniquely if we require Hα(iy) −→
y↑∞
0. By (A.21),
0 < ±Im(Hα( · )) ≤ π. (A.22)
By the general theory of Herglotz functions (see, e.g., [4, 5]), the limit in
(A.11) exists and (A.12) holds by (A.22). (A.22) also implies that the limiting
measure w− limε↓0± 1
π Im(Hα(λ+ iε)) dλ in the Herglotz representation theorem
is purely absolutely continuous, hence (A.14) holds.
(A.16) then follows from (A.14) and (A.10).
Since
Fα(z) =
z→∞ −z−1 + O(z−2), (A.23)
(A.17) implies
Gα(z) =
z→∞ 1− αz−1 + O(z−2), (A.24)
and thus (A.14) implies ∫
R
ξα(λ) dλ = α, (A.25)
which, given (A.12), implies (A.13). This proves everything except the parts (ii),
(v), and (vii).
To prove (A.15), we note that with Pϕ = (ϕ, · )ϕ, we have
(Aα − z)(B − z)−1 = I + αPϕ(B − z)−1,
which, since Pϕ is rank one, implies
det((Aα − z)(B − z)−1) = 1 + Tr(αPϕ(B − z))
= 1 + αF0(z)
= Gα(z).
Let us prove (ii) for α > 0. The proof of α < 0 is similar. Let a = min(σ(B)),
b = max(σ(B)). Then, by (A.19),
F ′
0(x) =
∫
R
dµ0(λ)
(λ− x)2
> 0
on (−∞, a) ∪ (b,∞) and F0 −→
x→±∞ 0. Thus, F > 0 on (−∞, a) and F < 0 on
(b,∞). Let f = limx↓b F (x) which may be −∞. If 1+αf < 0, there is a unique c
with 1+αF0(c) = 0, and then Gα is positive on (c,∞). By (A.8), Fα(z) is analytic
away from (a, b) ∪ {c}. Thus, σ(Aα) ∈ (a, b) ∪ {c} and c = max(σ(Aα), σ(B)),
so (ii) says that ξα(λ) = 0 on (−∞, b) and (c,∞). Since Gα(x) > 0 there and
0 < arg(Gα(z + iε)) < π, we see that ξα(x) = 0 on these intervals.
96 Journal of Mathematical Physics, Analysis, Geometry, 2009, vol. 4, No. 1
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Finally, we turn to (vii). Since Bn−An
α can be written as a telescoping series,
it is trace class and
‖Bn −An
α‖I1 ≤ n [sup(‖Aα‖, ‖B‖)]n−1‖B −A‖I1 . (A.26)
Thus, both sides of (A.16) are analytic about z = ∞, so identifying Taylor
coefficients,
Tr(Bn −An
α) =
∫
R
nλn−1ξα(λ) dλ. (A.27)
Summing Taylor series for ezλ, using (A.26) and (A.27) proves (ezB − ezAα) ∈ I1
and
Tr(ezB − ezAα) = z
∫
R
ezλξα(λ) dλ. (A.28)
This leads to (A.17) by using (A.4).
In extending this, the following uniqueness result will be useful:
Proposition A.3. Suppose A and B are bounded selfadjoint operators and
(A−B) ∈ I1. Suppose ξj ∈ L1(R; dλ) for j = 1, 2, and for all f ∈ C∞
0 (R),
Tr(f(A)− f(B)) =
∫
R
f ′(λ)ξj(λ) dλ. (A.29)
Then ξ1 = ξ2. Moreover, if (a, b) ⊂ R\σ(A) ∪ σ(B), ξj(·) is an integer on (a, b),
and if a = −∞ or b = ∞, it is zero on (a, b), and so ξj has a compact support.
P r o o f. By (A.29), the distribution ξ1 − ξ2 has vanishing distributional
derivative, so is constant. Since it lies in L1(R; dλ), it must be zero.
If f ∈ C∞
0 ((a, b)), f(A) = f(B) = 0, so ξ′j has zero derivative on (a, b) and so
is constant. If a = −∞ or b = ∞, the constant must be zero since ξj ∈ L1(R; dλ).
Now pick f which is supported on (c, (a+b)/2) for some c < d < min(σ(A)∪ σ(B))
with f = 1 on (d, (3a+ b)/4). Thus, the right-hand side of (A.29) is the negative
of the constant value of ξj on (a, b), while the left-hand side is the trace of a trace
class difference of projections which is always an integer (see [6, 23]).
Theorem A.4. For any pair of bounded selfadjoint operators A, B with (A−
B) of finite rank, there exists a function, ξ( · ; A,B) such that the following holds:
(i) (A.17) holds for any f satisfying the hypotheses of Th. A.1.
(ii)
|ξ( · ;A, B)| ≤ rank(A−B). (A.30)
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F. Gesztesy, A. Pushnitski, and B. Simon
(iii) ∫
R
|ξ(λ; A,B)| dλ ≤ ‖A−B‖I1 . (A.31)
(iv) For z ∈ C\R, one has
det((A− z)(B − z)−1) = exp
( ∫
R
(λ− z)−1ξ(λ) dλ
)
. (A.32)
(v) ξ(λ) = 0 for λ ≤ min(σ(A) ∪ σ(B)) or λ ≥ max(σ(A) ∪ σ(B)).
(vi) If (A−B) and (B − C) are both finite rank,
ξ( · ;A, C) = ξ( · ;A, B) + ξ( · ; B, C). (A.33)
P r o o f. If (A − B) has rank n, we can find A0 = A,A1, . . . , An = B so
(Aj+1 −Aj) has rank one, and
n−1∑
j=0
‖Aj+1 −Aj‖I1 = ‖B −A‖I1 . (A.34)
We define
ξ( · ; A,B) =
n−1∑
j=0
ξ( · ;Aj , Aj+1), (A.35)
where ξ( · ; Aj , Aj+1) is constructed via Th. A.2. (A.17) holds by telescoping and
the rank-one case. (A.30) and (A.31) follow from (A.12), (A.13), and (A.34).
(A.32) follows from
(A− z)(B − z)−1 = [(A0 − z)(A1 − z)−1][(A1 − z)(A2 − z)−1] . . .
using
det((1 + X1)(1 + X2)) = det(1 + X1) det(1 + X2)
for X1, X2 ∈ I1.
Item (v) is proven in Prop. A.3. Item (vi) follows from the uniqueness in
Prop. A.3.
Th. A.4 is essentially the same as Th. 3 in [38] (see also [40] and [16]).
Corollary A.5. If A, A′ are both finite rank perturbations of B with all three
operators selfadjoint, we have
∫
R
|ξ(λ;A,B)− ξ(λ;A′, B)| dλ ≤ ‖A−A′‖I1 . (A.36)
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On the Koplienko Spectral Shift Function. I. Basics
P r o o f. By (A.33),
ξ( · ; A,B)− ξ( · ; A′, B) = ξ( · ; A,A′).
Thus, (A.36) follows from (A.31).
This yields the principal result on existence and properties of the KrSSF (see
[38] or [40]).
Theorem A.6. Let A,B be bounded selfadjoint operators with (A−B) ∈ I1.
Then,
(i) There exists a unique function ξ( · ; A,B) ∈ L1(R; dλ) such that (A.17)
holds for any f satisfying the hypotheses of Th. A.1.
(ii) ∫
R
|ξ(λ; A,B)| dλ ≤ ‖A−B‖I1 . (A.37)
(iii) (A.32) holds.
(iv) ξ(λ) = 0 if λ ≤ min(σ(A) ∪ σ(B)) or λ ≥ max(σ(A) ∪ σ(B)).
(v) If (A−B) and (B − C) are both trace class, (A.33) holds.
(vi) If (A−B) and (A′ −B) are trace class, (A.36) holds.
P r o o f. Find An so (An − B) −→
n→∞ (A − B) in I1 and (An − B) is
finite rank. By (A.36), ξ( · ; An, B) is Cauchy in L1(R) so converges to an L1(R)
function by (A.36). Thus, items (i), (ii), (iii), (v), and (vi) hold by taking limits
(using ‖ · ‖I1-continuity of the mapping C → det(I + C). Uniqueness and (iv)
follow from Prop. A.3.
We refer to [50] (see also [51]) for a description of a class of functions f for
which this theorem holds.
We note that there are interesting extensions of the trace formula (A.17) to
classes of operators A,B different from selfadjoint or unitary operators. While we
cannot possibly list all such extensions here, we refer, for instance, to Adamjan
and Neidhardt [1], Adamjan and Pavlov [2], Jonas [29, 30], Krein [41], Langer
[44], Neidhardt [47, 48], Rybkin [56], Sakhnovich [58], and the literature cited
therein.
Theorem A.7. Let Bn, B, n ∈ N, be uniformly bounded selfadjoint operators
such that Bn −→
n→∞ B strongly. Let Xn, X, n ∈ N, be a sequence of selfadjoint
trace class operators such that ‖X − Xn‖I1 −→
n→∞ 0. Then for any continuous
function, g,
∫
R
g(λ)ξ(λ; Bn + Xn, Bn) dλ −→
n→∞
∫
R
g(λ)ξ(λ; B + X, B) dλ. (A.38)
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F. Gesztesy, A. Pushnitski, and B. Simon
P r o o f. By Th. A.1, we have (A.38) for g ∈ C∞
0 (R). Note that the
‖ · ‖∞-norm closure of C∞
0 includes the continuous functions of compact support.
Thus, by an approximation argument using uniform L1(R; dλ)-bounds on ξ, we
get (A.38) for continuous functions of compact support. Since ξ(λ; A,B) = 0
for λ ∈ [−max(‖A‖, ‖B‖),max(‖A‖, ‖B‖)], the result for continuous functions of
compact support extends to any continuous function.
We want to note the following. Define
ξ(I1) = {ξ( · ;A,B) |A,B bounded and selfadjoint, (A−B) ∈ I1}.
Proposition A.8. ξ(I1) is the set of L1(R; dλ)-elements of compact support.
P r o o f. Since A, B are bounded and selfadjoint, any ξ( · ;A, B) ∈ ξ(I1)
necessarily lies in L1(R; dλ) and has compact support (cf. Th. A.6 (i) and (iv)).
Next, let g ∈ L1(R; dλ) satisfy 0 ≤ g(λ) ≤ 1 and supp(g) ⊂ (a, b) for some
−∞ < a < b < ∞. Define
G(z) = exp
(
1
π
b∫
a
g(λ) dλ
λ− z
)
, Im(z) > 0. (A.39)
Then G satisfies the following items (i)–(iii):
(i) Im(G(z)) > 0 (for Im(z) > 0) since
0 ≤ Im
( b∫
a
g(λ) dλ
λ− z
)
≤ Im
( b∫
a
dλ
λ− z
)
≤ π
on account of 0 ≤ g ≤ 1.
(ii) Im(G(λ + i0)) = 0 if λ < a or λ > b.
(iii) G(z) → 1 as Im(z) → ∞ since g ∈ L1(R; dλ). It follows that there is
α > 0 and a probability measure dµ on [a, b] with
G(z) = 1 + α
b∫
a
dµ(λ)
λ− z
. (A.40)
Let B be multiplication by λ on L2((a, b); dµ), ϕ is the function 1 in L2((a, b); dµ)
and A = B + α(ϕ, · )ϕ. Then, by (A.5), (A.6), (A.7), and (A.14), ξ(λ; A,B) =
π−1g(λ) for a.e. λ ∈ (a, b), and α = π−1
∫ b
a g(λ) dλ. Thus, we have the theorem
if 0 ≤ g ≤ 1 or (by interchanging A and B) if 0 ≥ g ≥ −1. Since any L1(R; dλ)-
function is a sum of such g’s converging in L1(R; dλ) (simple functions are dense
in L1(R; dλ)), we obtain the general result.
100 Journal of Mathematical Physics, Analysis, Geometry, 2009, vol. 4, No. 1
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We note that a similar result for the finite rank case can be found in [42].
Finally, we prove invariance of the absolutely continuous spectrum under
trace class perturbations using the KrSSF and perturbation determinants, that
is, without directly relying on elements from scattering theory.
We start with the following observations:
Lemma A.9. Let A, B be bounded selfadjoint operators with X = (A − B)
of rank one. Then,
σac(A) = σac(B)
and
ξ(λ; A,B) ∈ {−1, 0, 1}
for a.e. λ ∈ R\σac(B).
P r o o f. ξ(λ; A,B) ∈ {−1, 0, 1} follows from (A.5)–(A.7), (A.11), and
(A.12). σac(A) = σac(B) follows in the usual manner by computing the normal
boundary values to the real axis of the imaginary part of Fα in terms of that of
F0 using (A.8).
Lemma A.10. Let A, B be bounded selfadjoint operators with X = (A−B)
∈ I1. Then for a.e. λ ∈ R\σac(B) one has
lim
ε↓0
det(I + X(B − λ− iε)−1) ∈ R.
P r o o f. By (A.32), it suffices to prove that
ξ( · ;A, B) ∈ Z a.e. on R\σac(B). (A.41)
Introducing
X =
∞∑
n=1
xn(φn, · )φn, X0 = 0, XN =
N∑
n=1
xn(φn, · )φn, N ∈ N,
the rank-by-rank construction of ξ( · ;A,B) alluded to in the proof of Th. A.6
yields the L1(R; dλ)-convergent series
ξ( · ; A,B) =
∞∑
n=1
ξ( · ; B + Xn, B + Xn−1). (A.42)
By Lemma A.9, each term in the above series is integer-valued a.e. on R\σac(B)
and hence so is the left-hand side of (A.42), which yields (A.41).
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F. Gesztesy, A. Pushnitski, and B. Simon
Lemma A.11. Let A, B be bounded selfadjoint operators in the Hilbert space
H with X = (A−B) ∈ I1 and ϕ ∈ H, ‖ϕ‖ = 1. Denote Pϕ = (ϕ, · )ϕ. Then,
1− (ϕ, (B − z)−1ϕ) =
det(I − (X + Pϕ)(A− z)−1)
det(I −X(A− z)−1)
, z ∈ C+. (A.43)
P r o o f. One computes
I − Pϕ(B − z)−1 = (B − Pϕ − z)(A− z)−1(A− z)(B − z)−1
= (B − Pϕ − z)(A− z)−1[(B − z)(A− z)−1]−1
= [I − (X + Pϕ)(A− z)−1][I −X(A− z)−1]−1. (A.44)
Taking determinants in (A.44) then yields
det(I − (X + Pϕ)(A− z)−1)
det(I −X(A− z)−1)
= det(I − Pϕ(B − z)−1)
= 1− (ϕ, (B − z)−1ϕ).
Theorem A.12. Let A,B be bounded selfadjoint operators in the Hilbert space
H with (A−B) ∈ I1. Then,
σac(A) = σac(B).
P r o o f. By symmetry between A and B, it suffices to prove σac(B) ⊆ σac(A).
Suppose to the contrary that there exists a set E ⊆ σac(B) such that |E| > 0 and
E ∩ σac(A) = ∅. Choose an element ϕ ∈ H such that limε↓0 Im((ϕ, (B − λ −
iε)−1)ϕ) > 0 for a.e. λ ∈ E . Thus, for a.e. λ ∈ E , the imaginary part of the limit
z → λ + i0 of the left-hand side of (A.43) is nonzero. On the other hand, by
Lemma A.10, the right-hand side of (A.43) is real for a.e. λ ∈ E , a contradiction.
R e m a r k. Employing det(I − A) = det2(I − A)eTr(A) for A ∈ I1, and
using an approximation of Hilbert–Schmidt operators by trace class operators in
the norm ‖ · ‖I2 , one rewrites (A.43) in the case where X = (A−B) ∈ I2 as
1− (ϕ, (B − z)−1ϕ) =
det2(I − (X + Pϕ)(A− z)−1)
det2(I −X(A− z)−1)
e(ϕ,(A−z)−1ϕ),
z ∈ C+. (A.45)
Since in the proof of Th. A.12 one assumes E ⊆ σac(B), |E| > 0, and E ∩
σac(A) = ∅, one concludes that
(ϕ, (A− λ− i0)−1ϕ) is real-valued for a.e. λ ∈ E .
102 Journal of Mathematical Physics, Analysis, Geometry, 2009, vol. 4, No. 1
On the Koplienko Spectral Shift Function. I. Basics
Moreover, if the boundary values of det2(I − X(A − λ − i0)−1) exist for a.e.
λ ∈ σac(B), by (A.45), so do those of det2(I − (X + Pϕ)(A− λ− i0)−1). Hence,
if one can assert real-valuedness of
det2(I − (X + Pϕ)(A− λ− i0)−1)
det2(I −X(A− λ− i0)−1)
for a.e. λ ∈ σac(B), (A.46)
using input from some other sources, one can follow the proof of Th. A.12 step
by step to obtain invariance of the a.c. spectrum.
In the special case of Schrödinger (and similarly for Jacobi) operators with
real-valued potentials V ∈ Lp([0,∞)), p ∈ [1, 2], the existence of the boundary
values of det2(I −X(A − λ − i0)−1) is indeed known due to Christ–Kiselev [17]
(for p ∈ [1, 2) using some heavy machinery) and Killip–Simon [34] (for p = 2).
We will return to this circle of ideas in [25].
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