# The Cotlar-Stein Lemma, Grothendieck’s Inequality and All That

Brian Jefferies  School of Mathematics
The University of New South Wales
NSW 2052 AUSTRALIA<EMAIL_ADDRESS>
###### Abstract.

The purpose of this paper is point out connections between scattering theory,
double operator integrals, Krein’s spectral shift function, integration
theory, bimeasures, Feynman path integrals, harmonic and functional analysis
and many other applications to quantum physics made since the last 50 ears or
so.

The starting point is Kluvanek’s Integration Structures which he hoped to
apply to quantum physics and is now bearing fruit from the contributions of
many authors, especially former Soviet mathematical physicists in the
intervening years. Soon, a practical quantum field theory in four space-time
dimensions satisfying the Wightman axioms may be proved to exist. This is the
aim of one of the Clay Prizes. At the moment, only toy models exist in fewer
that four space-time dimensions.

###### 1991 Mathematics Subject Classification:

Primary 81S40, 58D30; Secondary 46G10, 28B05

###### Contents

  1. 1 Integration Structures
  2. 2 Bimeasures
  3. 3 Spectral Measures
  4. 4 $L^{1}(QP)$ and the Cotlar-Stein Lemma
  5. 5 The Weyl Functional Calculus
  6. 6 Double Operator Integrals
  7. 7 Peller’s First and Second Theorems
  8. 8 Fundamental Results for Double Operator Integrals and Traces
  9. 9 An elementary proof of Krein’s spectral trace formula

##  1\. Integration Structures

Let $(\Omega,{\mathcal{S}},\mu)$ be a measure space, or more generally, let
$\mu:{\mathcal{S}}\to{\mathbb{C}}$ be a $\sigma$-additive set function defined
on a $\delta$-ring of subset ${\mathcal{S}}$ of $\Omega$. In either case, we
have the bound

$\left|\int_{\Omega}f\,d\mu\right|\leq\|f\|_{L^{1}(|\mu|)}$

for all $\mu$-integrable functions $f$. In case $\mu$ is scalar valued,
$|\mu|$ is the variation of $\mu$. On the other hand, if
$\mu(E)\in\\{0,\infty\\}$ for all $E\in{\mathcal{S}}$, then it is well known
that $L^{1}(\mu)=\\{0\\}$. We say that the norm $\|\cdot\|_{L^{1}(\mu)}$ is
integrating for $\mu$.

I. Kluvanek generalised this concept in the following direction [22]. Let
${\mathcal{K}}$ be some family scalar valued functions defined on a set
nonempty set $\Omega$ with the zero function $0\in{\mathcal{K}}$. A gauge
$\rho$ on ${\mathcal{K}}$ is a function $\rho:{\mathcal{K}}\to[0,\infty)$ such
that $\rho(0)=0$. Let ${\mathcal{R}}$ be a family of gauges on
${\mathcal{K}}$. Then ${\mathcal{R}}$ is said to be integrating for a map
$\mu:{\mathcal{K}}\to{\mathbb{C}}$ if the following holds:

  * a)

$f\in{\mathcal{K}}$ and there exists $c_{j}\in{\mathbb{C}}$,
$f_{j}\in{\mathcal{K}}$, $j=1,2,\dots$,
$f(\omega)=\sum_{j=1}^{\infty}c_{j}f_{j}(\omega)$ for all $\omega\in\Omega$
such that $\sum_{j=1}^{\infty}|c_{j}|.|f_{j}(\omega)|<\infty$

  * b)

$\sum_{j=1}^{\infty}|c_{j}|.\rho(f_{j})<\infty$ for all
$\rho\in{\mathcal{R}}$,

  * c)

for every $\rho\in{\mathcal{K}}$, there exists $C_{\rho}>0$ such that
$|\mu(f)|\leq C_{\rho}\sum_{j=1}^{\infty}|c_{j}|.\rho(f_{j})$.

######  Definition 1.

If ${\mathcal{R}}$ satisfies just a) and b) on ${\mathcal{K}}$, then the
quasicomplete lcs $L^{1}({\mathcal{K}},{\mathcal{R}})$ consists of all
${\mathcal{R}}$-equivalence classes of functions $f:\Omega\to{\mathbb{C}}$
such that a) and b) hold. The collection sim$({\mathcal{K}})$ of all finite
linear combinations of functions from ${\mathcal{K}}$ is densely embedded in
$L^{1}({\mathcal{K}},{\mathcal{R}})$ [22]. The theory mimics Lebesgue, Daniell
and Stone integration [34] quite closely but is richer, as the following
examples show.

######  Example 2.

i) Let $\mu$ be Lebesgue measure on ${\mathbb{R}}$ and
${\mathcal{K}}=\\{\smash{\raise
2.15277pt\hbox{$\chi$}}_{I}:I=[a,b),a,b\in{\mathbb{R}},\ a\leq b\\}$. The
length of an interval $I$ is denoted by $|I|$. Then $\rho=|\cdot|$ defines the
gauge $|\cdot|:{\mathcal{K}}\to[0,\infty)$ by $|\smash{\raise
2.15277pt\hbox{$\chi$}}_{I}|=|I|$ that is integrating for $\mu$ on
${\mathcal{K}}$. For any Lebesgue measurable function $f:\Omega\to[0,\infty]$,
we have

$\int_{\Omega}f\,d\mu=\inf\left\\{\sum_{j=1}^{\infty}|c_{j}|.|I_{j}|:f(\omega)=\sum_{j=1}^{\infty}c_{j}\smash{\raise
2.15277pt\hbox{$\chi$}}_{I_{j}}(\omega)\ \forall\omega\in\Omega\text{ with
}\sum_{j=1}^{\infty}|c_{j}|.|f_{j}(\omega)|<\infty\right\\},$

If the $c_{j}$, $j=1,2,\dots$ are elements of a Banach space ${\mathcal{X}}$,
then we obtain the Bochner integral [23].

ii) Let $E$ be a quasicomplete lcs (bounded sets are complete in the
associated uniformity), $(\Omega,{\mathcal{S}})$ a measurable space and
$m:{\mathcal{S}}\to E$ a countably additive measure on the $\sigma$-algebra
${\mathcal{S}}$. Then $m$ has a family $\Delta(m)$ of finite nonnegative
measures that is integrating for every scalar measure $\langle m,\xi\rangle$,
$\xi\in E^{\prime}$ (Bartle-Dunford-Schwartz Theorem).

Furthermore, for every continuous seminorm $p$ on $E$, there exists $C_{p}>0$
and $\mu_{p}\in\Delta$, such that $p(m)(S)\leq C_{p}\mu_{p}(S)$ for all
$S\in{\mathcal{S}}$. Here

$p(m)(S)=\sup_{\xi\in U_{p}^{\circ}}|\langle m,\xi\rangle|(S),\quad
S\in{\mathcal{S}},$

for $U_{p}=\\{x\in E:p(x)<1\\}$ is the $p$-semivariation of $m$ and for any
$m$-integrable function $f:\Omega\to{\mathbb{C}}$ the inequalities

$p\left(\int_{S}f\,dm\right)\leq C_{p}\mu_{p}(|f|)$

obtain so that the collection $\Delta(m)$ is integrating for $m$.

The lcs $L^{1}(m)$ of $m$ integrable functions is endowed with the topology
induced by the continuous seminorms $f\mapsto p(m)(f)$ as $p$ ranges over the
continuous seminorms of $E$. Unsurprisingly, if $E$ is a Fréchet space or
Banach space, then so is $L^{1}(m)$. The space $L^{1}(m)$ is quasicomplete
exactly when the $\Delta(m)$-measure algebra of ${\mathcal{S}}$ is Dedekind
complete. It has only recently been verified by R. Becker, R. Ricker and S.
Okada, see [24], that this occurs precisely when $m$ is totally absolutely
continuous with respect to a single localisable measure $\lambda$, that is,
one for which $L^{1}(\lambda)$ is itself Dedekind complete. The construction
is due to pioneering work of I. Kluvanek on conical measures. There was a gap
in the proof for ${\mathbb{R}}^{I}$ filled by R. Becker.

iii) For a spectral measure $P$ acting on a Banach space ${\mathcal{X}}$, the
gauge $\rho:f\mapsto\|P(|f|)\|_{{\mathcal{L}}({\mathcal{X}})}$, is an
integrating norm for $P$ so that $L^{1}(\rho)=L^{1}(P)$ as vector spaces [22].
However, the usual topology of $L^{1}(P)$ as in ii) above is not a norm
topology if ${\mathcal{X}}$ is infinite dimensional, because $P$ is usually
only $\sigma$-additive in the strong operator topology, except in matrix
theory.

######  Example 3.

(One Dimensional Brownian Motion) Let $\Omega=C([0,1])$, the continuous
functions on the interval $[0,1]$. Let $X_{t}:\Omega\to{\mathbb{R}}$ be the
evaluation functions $X_{t}(\omega)=\omega(t)$ for $0\leq t\leq 1$ and
$\omega\in\Omega$. Let $m([a,b))=X_{b}-X_{a}$ for all $0\leq a\leq b\leq t$.

We seek a Borel probability measure $P$ on $\Omega$ with expectation operator
${\mathbb{E}}$ for which

${\mathbb{E}}|m([a,b))|^{2}=b-a.$

Then $m:{\mathcal{B}}([0,1])\to L^{2}(P)$ is an orthogonally scattered vector
measure in the sense that

$\lambda(A\cap B)=\int_{\Omega}m(A)m(B)\,dP=0\text{ if }A\cap B=\emptyset,\
A,B\in{\mathcal{B}}([0,1]).$

Moreover $\|m(A)\|^{2}\leq\lambda(A)$, the Lebesgue measure of
$A\in{\mathcal{S}}$ and

$\left\|\int_{A}f\,dm\right\|\leq\lambda(|f|.\smash{\raise
2.15277pt\hbox{$\chi$}}_{A})^{\frac{1}{2}},\quad A\in{\mathcal{S}}$

for $f\in L^{1}(m)$, so that $\lambda^{\frac{1}{2}}$ is integrating for $m$ on
subintervals of $[0,1]$.

The random variable $\int_{0}^{1}f\,dm\in L^{2}(P)$ is called the Itô integral
of $f\in L^{2}([0,1])$ and the total integation map is the Itô isometry. The
random variable $\int_{0}^{1}f\,dm$ is usually written as $\int_{0}^{1}f\,dX$
despite the observation that, with probability one, the continuous process $X$
has unbounded variation on every nonempty interval so that $dX$ cannot be a
pointwise $\sigma$-additive with values in the random variables $L^{0}(P)$.

A. Kolmogorov demonstrated the existence of a Baire probability measure
$\tilde{P}$ on ${\mathbb{R}}^{[0.1]}$ with these properties and N. Wiener
proved that a probability measure $P$ with $P(E\cap\Omega)=\tilde{P}(E)$
exists for cylinder sets $E$ contained in ${\mathbb{R}}^{[0.1]}$. It follows
that the quadratic variation process

$[X]_{t}=\lim_{{\mathcal{P}}_{t}}\sum_{[a,b)\in{\mathcal{P}}_{t}}|X_{a}-X_{b}|^{2}$

converges in probability $P$ over all partitions ${\mathcal{P}}_{t}$ of
$[0,t)$ and ${\mathbb{E}}[X]_{t}=t$. The vector measure $m$ does however have
a density $\Phi:[0,1]\to{\mathcal{S}}^{\prime}$ with respect to $\lambda$,
with values in distributions ${\mathcal{S}}^{\prime}$. The process $\Phi$ is
usually called White Noise.

##  2\. Bimeasures

Let ${\mathcal{X}},\ {\mathcal{Y}}$ be lcs and $(\Lambda,{\mathcal{E}})$,
$(\Gamma,{\mathcal{F}})$ measurable spaces. We write the semi-algebra of
product sets $E\times F$, $E\in{\mathcal{E}}$ and $F\in{\mathcal{F}}$ as
${\mathcal{E}}\times{\mathcal{F}}$ and the algebra it generates as
${\mathcal{E}}\otimes{\mathcal{F}}$. The corresponding $\sigma$-algebra is
${\mathcal{E}}\widehat{\otimes}{\mathcal{F}}$. The space of continuous linear
operators $u:{\mathcal{X}}\to{\mathcal{Y}}$ is denoted by
${\mathcal{L}}({\mathcal{X}},{\mathcal{Y}})$. We always give the vector space
${\mathcal{L}}({\mathcal{X}},{\mathcal{Y}})$ or
${\mathcal{L}}({\mathcal{X}}):={\mathcal{L}}({\mathcal{X}},{\mathcal{X}})$ the
topology of strong convergence generated by the seminorms $u\mapsto p(ux)$ for
any $x\in{\mathcal{X}}$ and continuous seminorm
$p:{\mathcal{Y}}\to{\mathbb{R}}_{+}$. Then ${\mathcal{X}}$ is identical to the
lcs ${\mathcal{L}}({\mathbb{C}},{\mathcal{X}})$.

A separately $\sigma$-additive set function
$m:{\mathcal{E}}\times{\mathcal{F}}\to{\mathcal{L}}({\mathcal{X}})$ is called
a bimeasure, that is,

$m(E\times F)=\sum_{j=1}^{\infty}m(G_{j})$

for $G_{j}\in{\mathcal{E}}\otimes{\mathcal{F}}$, $j=1,2,\dots$, pairwise
disjoint and $E\times F=\bigcup_{j=1}^{\infty}G_{j}$. This is the same as
requiring the two set functions

$E\longmapsto m(E\times F)\text{ and }F\longmapsto m(E\times F),\quad
E\in{\mathcal{E}},\ F\in{\mathcal{F}}$

to be $\sigma$-additive [22]. It is well known that a bimeasure on
${\mathcal{E}}\times{\mathcal{F}}$ may not be the restriction of a measure on
the $\sigma$-algebra ${\mathcal{E}}\widehat{\otimes}{\mathcal{F}}$.

######  Example 4.

Let $\varphi\in\ell^{2}$ and $m(E\times F)=\sum_{k\in F}\left(\sum_{j\in
E}\varphi(j)e^{i\pi j}\right)\varphi(k)e^{-i\pi k}$, the convergence being in
$\ell^{2}$ as Fourier series,

(2.1) $|m(E\times F)|\leq\|\varphi\|_{2}^{2},\quad E,F\subseteq{\mathbb{N}}.$

and by linearity, $|m(f\otimes
g)|\leq\|\varphi\|_{2}^{2}\|f\|_{\infty}\|g\|_{\infty}$ for simple functions
$f,g$ on ${\mathbb{N}}$.

But any additive extension of $m$ is unbounded on the $\sigma$-algebra
${\mathcal{E}}\widehat{\otimes}{\mathcal{F}}$.

Extending to functions, we see that

$\int_{\Lambda\times\Gamma}f\otimes g\,dm=m(f\otimes g),\quad f\in
L^{\infty}(\Lambda),\ g\in L^{\infty}(\Gamma),$

in the limit. There exist $C>0$ such that $|m(f\otimes g)|\leq
C\|f\|_{\infty}\|g\|_{\infty}$ in the scalar case or for any continuous
seminorm $p$ in the vector case., there exists $C_{p}>0$ such that

$p\big{(}m(f\otimes g)\big{)}\leq C_{p}\|f\|_{\infty}\|g\|_{\infty}.$

Bimeasures were examined in a series of papers by M. Morse and Transue.

A function $\varphi$ belonging to the projective tensor product
$L^{\infty}(\Lambda)\widehat{\otimes}_{\pi}L^{\infty}(\Gamma)$ has a uniform
expansion $\varphi=\sum_{j=1}^{\infty}f_{j}\otimes g_{j}$ with
$\sum_{j=1}^{\infty}\|f_{j}\|_{\infty}\|g_{j}\|_{\infty}<\infty,$ so

$\int_{\Lambda\times\Gamma}\varphi\,dm=\sum_{j=1}^{\infty}m(f_{j}\otimes
g_{j}).$

Not much more can be said without additional assumptions.

On the other hand, in Example 4, the representation

(2.2) $\psi(j,k)=\sum_{t=1}^{\infty}\alpha(j,t)\beta(k,t),\quad
j,k=1,2,,\dots,$

enables us to write

$\int_{{\mathbb{N}}\times{\mathbb{N}}}\psi\,dm=\sum_{t=1}^{\infty}m\big{(}\alpha(\cdot,t)\otimes\beta(\cdot,t)\big{)}.$

It can happen that $\int_{{\mathbb{N}}\times{\mathbb{N}}}|\psi|\,d|m|=\infty$
because $|m|(E\times F)=\sum_{j\in E,k\in F}|\varphi(j)||\varphi(k)|$.

Applying inequality (2.1), we have

$\left|\sum_{t=1}^{\infty}m\big{(}\alpha(\cdot,t)\otimes\beta(\cdot,t)\big{)}\right|\leq\|\varphi\|_{2}^{2}\sum_{t\in
T}\|\alpha(\cdot,t)\|_{\ell^{\infty}}\|\beta(\cdot,t)\|_{\ell^{\infty}},$

so $\sum_{t\in
T}\|\alpha(\cdot,t)\|_{\ell^{\infty}}\|\beta(\cdot,t)\|_{\ell^{\infty}}<\infty$
ensures that $\psi$ is integrable.

For $\psi$ represented by formula (2.2), let us define

(2.3)
$\displaystyle\|[\psi]\|_{L^{1}_{G}(m)}=\left\|\left(\sum_{t\in{\mathbb{N}}}|\alpha(\cdot,t)|^{2}\right)^{\frac{1}{2}}\right\|_{L^{\infty}({\mathbb{N}})}\left\|\left(\sum_{t\in{\mathbb{N}}}|\beta(\cdot,t)|^{2}\right)^{\frac{1}{2}}\right\|_{L^{\infty}({\mathbb{N}})}.$

We see that the inequality

$\left|\int_{{\mathbb{N}}\times{\mathbb{N}}}\psi\,dm\right|\leq\|[\psi]\|_{L^{1}_{G}(m)}$

characterises $m$-integrability, even though $m$ is not a genuine
$\sigma$-additive measure. This is essentially Grothendieck’s inequality
established in 1958 (see [29]).

######  Definition 5.

The smallest positive number $K_{G}$ for which

$\sum_{t\in
T}\|\alpha(\cdot,t)\|_{\ell^{\infty}}\|\beta(\cdot,t)\|_{\ell^{\infty}}\leq
K_{G}\left\|\left(\sum_{t\in{\mathbb{N}}}|\alpha(\cdot,t)|^{2}\right)^{\frac{1}{2}}\right\|_{\ell^{\infty}}\left\|\left(\sum_{t\in{\mathbb{N}}}|\beta(\cdot,t)|^{2}\right)^{\frac{1}{2}}\right\|_{L^{\infty}({\mathbb{N}})}$

is called Grothendieck’s constant.

Consequently, $\|[\psi]\|_{L^{1}_{G}(m)}<\infty$ ensures that $\psi$ is
$m$-integrable and the Banach function space norm
$[\psi]\mapsto\|[\psi]\|_{L^{1}(m)}$ is integrating for the simple scalar
bimeasure $m$, as is required for a decent theory of integration in the sense
of Lebesgue and modern Harmonic Analysis.

It turns out that the essential property of the simple scalar bimeasure $m$
above is that it is a well defined measure on compact sets, that is to say, on
finite sets.

######  Definition 6.

Let $\Lambda$, $\Gamma$ be Hausdorff topological spaces with Borel
$\sigma$-algebras ${\mathcal{E}}={\mathcal{B}}(\Lambda)$,
${\mathcal{F}}={\mathcal{B}}(\Gamma)$.

A separately $\sigma$-additive set function
$m:{\mathcal{E}}\times{\mathcal{F}}\to{\mathcal{L}}({\mathcal{X}})$ is called
a regular (Radon) bimeasure if

  * a)

$m(\cdot,F)$ and $m(E,\cdot)$ are compact inner regular for every
$E\in{\mathcal{E}}$ and $F\in{\mathcal{F}}$, and

  * b)

$E\times F\mapsto m\big{(}(E\cap K_{1})\times(F\cap K_{2})\big{)}$
$E\in{\mathcal{E}}$, $F\in{\mathcal{F}}$ is the restriction to
${\mathcal{E}}\times{\mathcal{F}}$ of a finite Radon measure on
$\Lambda\times\Gamma$ for each compact set $K_{1},\ K_{2}$.

By virtue of Bourbakist principles, the integral $E\times
F\mapsto\int_{E\times F}\varphi\,dm$ is defined to be the unique regular
bimeasure $\varphi.m$ equal to $\varphi.m$ on compact sets, that is, on

$({\mathcal{E}}\cap K_{1})\otimes({\mathcal{F}}\cap K_{2})$

for all compact sets $K_{1},K_{2}$. The product $\varphi.m$ of $\varphi$ with
$m$, the indefinite integral

$\int\varphi\,dm:E\times F\mapsto\int_{E\times
F}\varphi\,dm:=(\varphi.m)(E\times F),\quad E\in{\mathcal{E}},\
F\in{\mathcal{F}},$

is itself a regular bimeasure as one would hope.

With no topology, we can just use a compact class ${\mathcal{K}}_{j}$ of sets
$K_{j}$, $j=1,2$. [22]. Even if ${\mathcal{X}}$ is a Banach space, the Banach
function space $\\{[\psi]:\|[\psi]\|_{L^{1}_{G}(m)}<\infty\\}$ need not be the
optimal space of $m$-integrable functions because we deal with the strong
operator topology of ${\mathcal{L}}({\mathcal{X}})$ rather then the uniform
operator topology of uniform convergence on the closed unit ball of
${\mathcal{X}}$. In this setup, the Cotlar-Stein Lemma to be mentioned later
characterises $L^{1}(m)$ and its integrating norm topology when $m=QP$ for
spectral measures $Q$ qnd $P$.

Let ${\mathcal{M}}({\mathcal{F}})$ denote the scalar valued measures on the
$\sigma$-algebra ${\mathcal{F}}$ with the total variation norm. Then for a
scalar bimeasure $m$ the set function $E\mapsto m(E,\cdot)$,
$E\in{\mathcal{E}}$, is a measure-valued measure that is $\sigma$-additive for
the topology of setwise convergence, equivalent to the weak topology
$\sigma({\mathcal{M}}({\mathcal{F}}),{\mathcal{L}}^{\infty}(G))$ on
${\mathcal{M}}({\mathcal{F}})$ by uniform convergence. It follows that the
semivariation norm

$\|m\|=\sup\\{|m(f\otimes g)|:\|f\|_{\infty}\leq 1,\ \|g\|_{\infty}\leq 1\\}$

is a norm on the space of scalar bimeasures as well as for
${\mathcal{L}}(X)$-valued bimeasures. The corresponding space is
$L^{1}(\|m\|(\,\cdot\,))=L^{\infty}(Q)\widehat{\otimes}_{\pi}L^{\infty}(P)$.

For a regular bimeasure
$m:{\mathcal{E}}\times{\mathcal{F}}\to{\mathcal{L}}({\mathcal{X}})$, we shall
determine $L^{1}(m)$ and its locally convex topology $\tau_{1}(m)$. The
semivariation norm $\|\cdot\|$ determines a norm topology on $L^{1}(m)$ that
is strictly stronger than $\tau_{1}(m)$ if ${\mathcal{K}}$ is infinite
dimensional. The situation is a standard feature of vector measure theory and
generalises to polymeasures.

The natural choice for $\tau_{1}(m)$ is to take fundamental compact classes
${\mathcal{K}}_{j}$, $j=1,2$ and take the topology generated by
$L^{1}$-topology for the strong operator valued measures

$\\{(\smash{\raise 2.15277pt\hbox{$\chi$}}_{K_{1}}\otimes\smash{\raise
2.15277pt\hbox{$\chi$}}_{K_{2}}).m:K_{j}\in{\mathcal{K}}_{j},\ j=1,2\\}$

using the inner regularity of $m$ together with the semivariation norms
$\|mx\|$ for $x\in{\mathcal{X}}$. The topology first described in [19] was
motivated by quantum mechanics and the bimeasure $QP$ described below.

##  3\. Spectral Measures

We now assume that $Q,P$ are regular spectral measures and $m=QP$, that is,

$m(E\times F)=Q(E)P(F),\quad E\in{\mathcal{E}},\ F\in{\mathcal{F}}.$

A special role is played by the position operator
$Q(E):\psi\mapsto\smash{\raise 2.15277pt\hbox{$\chi$}}_{E}.\psi$,
$E\in{\mathcal{B}}({\mathbb{R}}^{d})$, $\psi\in L^{2}({\mathbb{R}}^{d})$ in
quantum mechanics in dimension $d=1,2,\dots$ . For the unitary Fourier
transform

${\mathfrak{F}}:L^{2}({\mathbb{R}}^{d})\to L^{2}({\mathbb{R}}^{d})$

given by

${\mathfrak{F}}f(\xi)=\hat{f}(\xi):={(2\pi)^{-d/2}}\int_{{\mathbb{R}}^{d}}e^{-i\langle\xi,x\rangle}f(x)\,dx,\quad\xi\in{\mathbb{R}}^{d},$

for $f\in L^{1}({\mathbb{R}}^{d})$, we let
$P(E)={\mathfrak{F}}^{*}Q(E){\mathfrak{F}}$,
$E\in{\mathcal{B}}({\mathbb{R}}^{d})$, be the corresponding momentum operator
(“Questions”) and
$P:={\mathfrak{F}}^{*}Q{\mathfrak{F}}:E\mapsto{\mathfrak{F}}^{*}Q(E){\mathfrak{F}}$,
$E\in{\mathcal{B}}({\mathbb{R}}^{d})$, the spectral measure of the position
operator in quantum mechanics over $L^{2}({\mathbb{R}}^{d})$. Then the
definite integral

$\int_{{\mathbb{R}}^{d}\times{\mathbb{R}}^{d}}\sigma\,d(QP):L^{2}({\mathbb{R}}^{d})\to
L^{2}({\mathbb{R}}^{d})$

is the pseudodifferential operator with symbol
$\sigma:{\mathbb{R}}^{d}\times{\mathbb{R}}^{d}\to{\mathbb{C}}$ given by

$\left(\int_{{\mathbb{R}}^{d}\times{\mathbb{R}}^{d}}\sigma\,d(QP)\psi\right)(x)={(2\pi)^{-d/2}}\int_{{\mathbb{R}}^{d}}e^{i\langle
x,\xi\rangle}\sigma(x,\xi)\hat{\psi}(\xi)\,d\xi,\quad\psi\in
L^{2}({\mathbb{R}}^{d}),$

when $\sigma$ is rapidly decreasing. The Cotlar-Stein result characterises the
class $L^{1}(QP)$ of symbols $\sigma$ determining a bounded pseudodifferential
operator $\int_{{\mathbb{R}}^{d}\times{\mathbb{R}}^{d}}\sigma\,d(QP)$ on
$L^{2}({\mathbb{R}}^{d})$, although it is usually not formulated in this
manner. It is easy to check the basic operator equality

$Q(E)\left(\int_{{\mathbb{R}}^{d}\times{\mathbb{R}}^{d}}\sigma\,d(QP)\right)P(F)=\int_{E\times
F}\sigma\,d(QP),\quad E,\,F\in{\mathcal{B}}({\mathbb{R}}^{d}).$

A similar treatment applies to the space $L^{1}(PQ)$. Below we shall see that
the classical Weyl quantisation procedure is not a bimeasure $W$, but it is
finitely additive. This is actually a difficult problem in harmonic analysis
and singular integral operators. The question is resolved by checking whether
or not $\|W(g,E)\|=\|W(E,g)\|$ is bounded uniformly for all finite unions $E$
of intervals and all uniformly bounded sets of measurable functions $g$.

##  4\. $L^{1}(QP)$ and the Cotlar-Stein Lemma

The argument in the last section suggests that to compute the indefinite
integral $\varphi.(QP)$, if it exists at all, it suffices to show that the
operator $(QP)(\varphi)\in{\mathcal{L}}({\mathcal{H}})$ exists, a standard
feature of spectral theory in its many guises. The Cotlar-Stein Lemma allows
us to compute $(QP)(\varphi)$ and hence, $L^{1}(QP)$ with its topology.

The Cotlar-Stein Lemma concerning matrices tells us that

$\displaystyle\sum_{j}\|Q(|f_{k}|^{2})P(|g_{j}|^{2}\|^{\frac{1}{2}}\leq M$
$\displaystyle,\ \sum_{k}\|Q(|f_{k}|^{2})P(|g_{j}|^{2})\|^{\frac{1}{2}}\leq
M\implies$ $\displaystyle\left\|\sum_{k}(QP)(f_{k}\otimes g_{k})\right\|\leq
M.$

The best reference is Terry Tao’s Blog111Lemma 1,
https://terrytao.wordpress.com/2011/05/25/the-cotlar-stein-lemma and the
Wikipedia article222https://en.wikipedia.org/wiki/Cotlar%E2%80%93Stein_lemma].

Now define the Banach function space norm

$\displaystyle\|\varphi\|_{L^{1}(QP)}$ $\displaystyle=$
$\displaystyle\sup\left\\{\sum_{j}\|Q(|f_{k}|^{2})P(|g_{j}|^{2}\|^{\frac{1}{2}},\
\sum_{k}\|Q(|f_{k}|^{2})P(|g_{j}|^{2})\|^{\frac{1}{2}}:\sum_{j,\,k}|f_{k}|\otimes|g_{j}|\leq|\varphi|\right\\}.$

It follows that $L^{1}(QP)$ is a Banach function space in which
$L^{1}_{G}(QP)$ is strictly embedded, because

$\max\left\\{\sum_{j}(\sup|f_{k}|^{2}))^{\frac{1}{2}}(\sup|g_{j}|^{2})^{\frac{1}{2}},\
\sum_{k}(\sup|f_{k}|^{2}))^{\frac{1}{2}}(\sup|g_{j}|^{2})^{\frac{1}{2}}:j,k\right\\}$
$=\max\left\\{\|f_{k}\|_{\infty}\sum_{j}\|g_{j}\|_{\infty},\|g_{j}\|_{\infty}\sum_{k}\|f_{k}\|_{\infty}:j,k\right\\}$

for $\sum_{j,\,k}|f_{k}|\otimes|g_{j}|\leq|\varphi|$ and this obviously
differs from the $L^{1}_{G}(QP)$-norm which is stronger.

Write this set as ${\mathfrak{C}}(|\varphi|)$. Then

$\left\|(QP)(|\varphi|)\right\|=\sup_{u\in{\mathfrak{C}}(|\varphi|)}\left\|(QP)(u)\right\|.$

Standard Banach lattice arguments ensure that $L^{1}(QP)$ is a Dedekind
complete Banach function space and
$(QP):L^{1}(QP)\to{\mathcal{L}}({\mathcal{H}})$ is a contimuous linear map for
the uniform operator topology. In fact, for regular spectral measures $Q,P$,
the lcs topology $\tau_{1}(QP)$ mentioned above possesses the remarkable
property that the topology defined by the norm
$\varphi\mapsto\left\|(QP)(|\varphi|)\right\|$, which is an integrating norm
for $QP$ on the linear space of simple functions on product sets, has the same
bounded subsets of $L^{1}(QP)$ determined by $\tau_{1}(QP)$. The identity

$\varphi.(QP)=\varphi_{+}\cdot(QP)-\varphi_{-}\cdot(QP)$

analogous to the Lebesgue theory does not rely on $Q$ and $P$ being regular
and

$\left\|(QP)(|\varphi|)\right\|\leq\inf\left\\{\sum_{t\in
T}\|\alpha(\cdot,t)\|_{L^{\infty}(Q)}\|\beta(\cdot,t)\|_{L^{\infty}(P)}:\varphi(\lambda,\gamma)=\sum_{t\in
T}\alpha(\lambda,t)\beta(\gamma,t)\right\\}$

because, clearly

$\left\|\sum_{t\in
T}Q(\alpha(\lambda,t))P(\beta(\gamma,t))\right\|_{{\mathcal{L}}({\mathcal{H}})}\leq\sum_{t\in
T}\|\alpha(\cdot,t)\|_{L^{\infty}(Q)}\|\beta(\cdot,t)\|_{L^{\infty}(P)}$

and the inequality may be strict. The integrals $\varphi.(QP)$ determined by
$L^{1}(QP)$ and the locally convex topology $\tau_{1}(QP)$ are the same and
are identical as vector spaces—the norm topology of $L^{1}(QP)$ and the
locally convex topology $\tau_{1}(QP)$ have the same bounded sets by the
uniform boundedness principle.

Although $QP$ is not a vector valued measure, the Cotlar-Stein lemma ensures
that it possesses properties remarkably similar to a Banach space valued
measure. The same analysis applies to the polymeaures considered in Remark 8
below arising in quantum theory.

##  5\. The Weyl Functional Calculus

Let ${\mathcal{M}},{\mathcal{D}}$ be a $(2n)$-system of
$S_{\omega}({\mathbb{C}})$-sectorial operators and
${\mathcal{W}}_{{\mathcal{M}},{\mathcal{D}}}$ the associated bilnear
$H^{\infty}$-Weyl functional calculus, when square function estimates obtain
[17].

We extend the linear map
$f\mapsto{\mathcal{W}}_{{\mathcal{M}},{\mathcal{D}}}(f)$ to a class larger
than holomorphic functions by the Cotlar-Stein Lemma, so that

${\mathcal{D}}{om}({\mathcal{M}},{\mathcal{D}})=\left[(H^{\infty}\otimes
L^{\infty})\bigcup(L^{\infty}\otimes H^{\infty})\right]$

is optimal or equivalently, the linear space

$\left[({\bf sim}(S_{\omega}({\mathbb{C}}^{n}))\otimes
L^{\infty}({\mathbb{C}}^{n}))\bigcup(L^{\infty}(({\mathbb{C}}^{n}))\otimes{\bf
sim}(S_{\omega}(({\mathbb{C}}^{n}))))\right]$

is optimal by extendibility. The convergence of the associated bilinear
singular integral was a longstanding conjecture in Harmonic Analysis solved
eventually be Michael Lacey, but optimality is actually a simple application
of Cotlar’s matrix bounds and McIntosh methods. Moreover, holomorphic
functions with decay in $S_{\omega}({\mathbb{C}}^{n})$ suffice if there are no
square function estimates.

The map ${\mathcal{W}}_{{\mathcal{M}},{\mathcal{D}}}$ is not a bimeasure or
separately $\sigma$-additive, but
$f\mapsto{\mathcal{W}}_{{\mathcal{M}},{\mathcal{D}}}(f\circ diag)$ is an
$H^{\infty}(S_{\omega})$-functional calculus corresponding to the $n$-system
${\mathcal{W}}_{\hbox{ diag(${\mathcal{M}},{\mathcal{D}}$)}}$ projected onto
the diagonal, so giving the finitely additive “spectral measure” on
$S_{\omega}({\mathbb{C}})$ of the sectorial operator
$\frac{1}{2}(\langle{\mathcal{M}},{\mathcal{D}}\rangle+\langle{\mathcal{D}},{\mathcal{M}}\rangle)$.
For the classical system, this directly defines a genuine selfadloint spectral
measure by the classical vector measure extension theory.

Although
$(E,F)\mapsto{\mathcal{W}}_{{\mathcal{M}},{\mathcal{D}}}(\smash{\raise
2.15277pt\hbox{$\chi$}}_{E\times F})$ is finitely additive on intervals as a
bilinear singular integral operator, it is not regular on product sets and
therefore not separately $\sigma$-additive, so it is not a genuine bimeasure
and the domain ${\mathcal{D}}{om}({\mathcal{M}},{\mathcal{D}})$ is genuinely
optimal for the continuous linear map
${\mathcal{W}}_{{\mathcal{M}},{\mathcal{D}}}:\langle{\bf
sim}({\mathbb{R}}^{2n}),\|\cdot\|_{\infty}\rangle\to{\mathcal{L}}({\mathbb{R}}^{2n})$
in the selfadjoint case. The beast reared its ugly head in 1994 [19] and is
laid to rest in 2023 here.

However, for the classical Weyl system and generalisations, results on Fourier
integral operators provide an integration structure for
${\mathcal{W}}_{{\mathcal{M}},{\mathcal{D}}}$ and an optimal domain via deep
Harmonic Analysis methods, as would be expected [38].

##  6\. Double Operator Integrals

Let ${\mathcal{H}}$ be a separable Hilbert space. Two bounded linear operators
$S,T\in{\mathcal{L}}({\mathcal{H}})$ determine a transformer
$({S\overline{\otimes}T})_{{\mathcal{C}}_{p}({\mathcal{H}})}:u\mapsto SuT$ for
every element of the Schatten class ${\mathcal{C}}_{p}({\mathcal{H}})$, $1\leq
p\leq\infty$, an old idea going back to M. Krein and Yu Daletskii in the 50’s
[6]. By duality

$\|({S\overline{\otimes}T})_{{\mathcal{C}}_{p}({\mathcal{H}})}\|_{{\mathcal{L}}({\mathcal{C}}_{p}({\mathcal{H}}))}=\|({S\overline{\otimes}T})_{{\mathcal{C}}_{q}({\mathcal{H}})}\|_{{\mathcal{L}}({\mathcal{C}}_{q}({\mathcal{H}}))}$

for the dual index $q$ satisfying $1/p+1/q=1$ with $1\leq p,q\leq\infty$. In
particular, for the trace class operators ${\mathcal{C}}_{1}({\mathcal{H}})$
on ${\mathcal{H}}$,
$\langle{\mathcal{C}}_{1}({\mathcal{H}}),{\mathcal{C}}_{\infty}({\mathcal{H}})\rangle$
is a dual pair for the compact linear operators
${\mathcal{C}}_{\infty}({\mathcal{H}})={\mathcal{C}}_{1}({\mathcal{H}})^{\prime}$
[6]. The crude inequality
$\|ST\|_{{\mathcal{L}}({\mathcal{H}})}\leq\|({S\overline{\otimes}T})_{{\mathcal{C}}_{1}({\mathcal{H}})}\|_{{\mathcal{L}}({\mathcal{C}}_{1}({\mathcal{H}}))}$
helps us estimate a proper subspace of $L^{1}(QP)$ because $QP=QIP$ for the
identity operator $I$ on ${\mathcal{H}}$, see M. Birman and M. Solomyak [6,
Section 3.1] and V.Peller.

The following result was proved by M. Birman and M. Solomyak [6, Section 3.1]
who initiated this program in scattering theory

######  Theorem 7.

Let $(\Lambda,{\mathcal{E}})$ and $(\Gamma,{\mathcal{F}})$ be measurable
spaces and ${\mathcal{H}}$ a separable Hilbert space. Let
$P:{\mathcal{E}}\to{\mathcal{L}}_{s}({\mathcal{H}})$ and
$Q:{\mathcal{F}}\to{\mathcal{L}}_{s}({\mathcal{H}})$ be spectral measures.

Then there exists a unique spectral measure
$({P\overline{\otimes}Q})_{{\mathcal{C}}_{2}({\mathcal{H}})}:{\mathcal{E}}\otimes{\mathcal{F}}\to{\mathcal{L}}({\mathcal{C}}_{2}({\mathcal{H}}))$
such that

$({P\overline{\otimes}Q})_{{\mathcal{C}}_{2}({\mathcal{H}})}(A)=({P\otimes
Q})_{{\mathcal{C}}_{2}({\mathcal{H}})}(A)$

for every set $A$ belonging to the algebra ${\mathcal{A}}$ of all finite
unions of product sets $E\times F$ for $E\in{\mathcal{E}}$,
$F\in{\mathcal{F}}$, and

$\int_{A}\varphi\,d(P\otimes
Q)_{{\mathcal{C}}_{2}({\mathcal{H}})}=\int_{A}\varphi\,d({P\overline{\otimes}Q})_{{\mathcal{C}}_{2}({\mathcal{H}})}\in{\mathcal{L}}({\mathcal{C}}_{2}({\mathcal{H}})),\quad
A\in{\mathcal{E}}\otimes{\mathcal{F}},$

for every bounded $({\mathcal{E}}\otimes{\mathcal{F}})$-measurable function
$\varphi:{\mathcal{L}}\times M\to{\mathbb{C}}$. Moreover,

$\|({P\overline{\otimes}Q})_{{\mathcal{C}}_{2}({\mathcal{H}})}(\varphi)\|_{{\mathcal{L}}({\mathcal{C}}_{2}({\mathcal{H}}))}=\|\varphi\|_{\infty}.$

For spectral measures $P$ and $Q$, the formula

$\left(\int_{E\times F}\varphi\,d(P\otimes
Q)_{{\mathfrak{S}}}\right)T=\left(\int_{\Lambda\times\Gamma}\varphi\,d(P\otimes
Q)_{{\mathfrak{S}}}\right)P(E)TQ(F)$

holds for each $E\in{\mathcal{E}}$, $F\in{\mathcal{F}}$ and
$T\in{\mathfrak{S}}$, so it is only necessary to verify that

$\int_{\Lambda\times\Gamma}\varphi\,d(P\otimes
Q)_{{\mathfrak{S}}}\in{\mathcal{L}}({\mathfrak{S}})$

in order to show that $\varphi$ is $(P\otimes Q)_{{\mathfrak{S}}}$-integrable.

The following observation is just a multiliear extension of the concepts
above. Now for simplicity, we take the Fourier transform of $f\in
L^{1}({\mathbb{R}})$ to be the function $\hat{f}:{\mathbb{R}}\to{\mathbb{C}}$
defined by $\hat{f}(\xi)=\int_{\mathbb{R}}e^{-i\xi x}f(x)\,dx$ for
$\xi\in{\mathbb{R}}$.

######  Remark 8.

Although we shall not consider multiple operator integrals, as mentioned
above, transformers like $(Q\otimes\cdots\otimes
Q)_{{\mathcal{C}}_{2}({\mathcal{H}})}$ are relevant to Feynman path integrals
in theoretical physics [18]. Suppose that $S$ is a C0-semigroup of Hilbert-
Schmidt operators or operators in the Schatten class
${\mathcal{C}}_{p}({\mathcal{H}})$ for some $1<p<\infty$. Then for every
$0<t_{1}<\cdots<t_{n}<t$ the expression

$(Q\otimes\cdots\otimes
Q)_{{\mathcal{C}}_{2}({\mathcal{H}})}(S(t_{n}-t_{n-1}),\dots,S(t_{1}))=QS(t_{n}-t_{n-1})\cdots
QS(t_{1})Q$

defines a separately $\sigma$-additive set function

$F_{0}\times F_{1}\times\cdots\times F_{n}\longmapsto
Q(F_{n})S(t_{n}-t_{n-1})\cdots Q(F_{1})S(t_{1})Q(F_{0}),\quad
F_{j}\in{\mathcal{F}},\ j=1,\dots,n.$

that is the resriction of an operator valued measure. It is possible to make
sense of the inclusions

$L^{\infty}(M\times\cdots\times M)=L^{1}\big{(}(Q\otimes\cdots\otimes
Q)_{{\mathcal{C}}_{2}({\mathcal{H}}))}\big{)}\subset
L^{1}\big{(}QS(t_{n}-t_{n-1})\cdots QS(t_{1})Q\big{)}$

and

$L^{1}\big{(}(Q\otimes\cdots\otimes
Q)_{{\mathcal{C}}_{p}({\mathcal{H}}))}\big{)}\subset
L^{1}\big{(}QS(t_{n}-t_{n-1})\cdots QS(t_{1})Q\big{)}.$

In the case $p\neq 2$, the inclusions provide a criterion for integration with
respect to operator valued set functions that may only be separately
$\sigma$-additive and not fully $\sigma$-additive on the algebra of product
sets (polymeasures) giving elementary Feynman-type integrals. Multiple
operator integrals are considered in greater detail in [35].

The group $S(t)=e^{-it\Delta/2}$, $t\in{\mathbb{R}}$, of bounded linear
operators on $L^{2}({\mathbb{R}}^{d})$ belongs to no $p$-Schatten class for
$p\neq\infty$ and $L^{1}\big{(}QS(t_{n}-t_{n-1})\cdots QS(t_{1})Q\big{)}$ is
best described in terms of symbol classes of certain oscillatory pseudo-
differential operators as mention above for the bimeasure $QP$. If the
operator valued Feynman set function $M_{-i}^{t}$ described in [18] is
restricted to cylinder sets

${\mathcal{E}}_{t_{1},\dots,t_{n}}=\\{X_{t_{1}}\in B_{1},\dots,X_{t_{n}}\in
B_{n}\\}$

with $0<t_{1}<\cdots<t_{n}\leq t$ , then

$L^{1}(M_{-i}^{t}\restriction{\mathcal{E}}_{t_{1},\dots,t_{n}})=L^{1}\big{(}QS(t_{n}-t_{n-1})\cdots
QS(t_{1})Q\big{)}$

is completely understood by the Cotlar-Stein Lemma as a Banach function space
similar to the Lebesgue space $L^{1}(\mu)$ for an abstract measure $\mu$. This
is a small step towards treating Feynman’s mathematical and emotional
inadequacies, which seems to have been missed until now.

On the other hand, product functions $\varPhi=f_{0}\otimes
f_{1}\otimes\cdots\otimes f_{n}$, $f_{j}\in L^{\infty}((M,{\mathcal{F}},Q))$,
$j=1,\dots,n$, always belong to

$L^{1}\big{(}QS(t_{n}-t_{n-1})\cdots QS(t_{1})Q\big{)}$

so that

$\displaystyle\int_{M\times\cdots\times
M}\varPhi\,d\big{(}QS(t_{n}-t_{n-1})\cdots QS(t_{1})Q\big{)}$
$\displaystyle=Q(f_{n})S(t_{n}-t_{n-1})\cdots Q(f_{1})S(t_{1})Q(f_{0})$
$\displaystyle=\int_{\Omega}\varPhi\circ X\,dM_{-i}^{t}$

as expected. The space of paths $\Omega$ consists of continuous functions
$\omega$ on $[0,\infty)$ with $X_{t}(\omega)=\omega(t)$ for $t\geq 0$ as for
Brownian motion.

Subspaces of the symbol classes $L^{1}((QP)\stackrel{{\scriptstyle
n}}{{\cdots}}(QP))$ and $L^{1}((PQ)\stackrel{{\scriptstyle n}}{{\cdots}}(PQ))$
are also described in terms of the Cotlar-Stein lemma in an analogous
procedure to that above, in the case that $Q$ and $P$ are regular spectral
measures. We use the integrating seminorms

$\rho_{h}^{(n)}(f)=\|\left(QP\cdots QP\right)(|f|)h\|,\quad
h\in{\mathcal{H}},\quad\rho_{h}=\rho_{h}^{(1)}$

on product functions $f$. Note that the spectral measure property implies the
useful characterisation

$f\in L^{1}(QP)\iff\smash{\raise 2.15277pt\hbox{$\chi$}}_{E}f\smash{\raise
2.15277pt\hbox{$\chi$}}_{F}\in L^{1}(QP),\quad\forall E,F,$

$Q(E)(QP)(f)Q(F)=(QP)(\smash{\raise 2.15277pt\hbox{$\chi$}}_{E}f\smash{\raise
2.15277pt\hbox{$\chi$}}_{F})$ and

$L^{\infty}(Q)\widehat{\otimes}_{\pi}L^{\infty}(P)\subset
L^{1}((QP)_{{\mathfrak{S}}_{1}({\mathcal{H}})})\subset L^{1}(QP).$

It is proved above that by the Cotlar-Stein lemma, the bornological
isomorphism

$L^{1}(QP)=L^{1}({\mathcal{R}})$

holds for ${\mathcal{R}}=\\{\rho_{h}:h\in{\mathcal{H}}\\}$ in the sense the
the bounded sets are the same but not the topologies. Here $f\in L^{1}(QP)$
implies $\smash{\raise 2.15277pt\hbox{$\chi$}}_{K}f$ is $(QP)$-Lebesgue
integrable for $K$ a compact product set. A similar argument holds for the
locally convex topology $\tau_{1}(QP)$. Here we have a type of uniform
boundedness principle for the bimeasure $QP$.

It is remarkable that the Lebesgue norm topology of the Banach function space
$L^{1}(QP)$ is bornologically equivalent to the much weaker locally convex
${\mathcal{R}}$-topology and the locally convex topology $\tau_{1}(QP)$ for
the spectral bimeasure $QP$. The Banach function space $L^{1}(QP)$ is an
optimal domain for the spectral bimeasure $QP$ defined on simple functions.

##  7\. Peller’s First and Second Theorems

Let ${\mathcal{H}}$ be a separable Hilbert space. Let
$P:{\mathcal{E}}\to{\mathcal{L}}_{s}({\mathcal{H}})$ and
$Q:{\mathcal{F}}\to{\mathcal{L}}_{s}({\mathcal{H}})$ be spectral measures on
the measurable spaces $(\Lambda,{\mathcal{E}})$ and $(\Gamma,{\mathcal{F}})$.
Let ${\mathfrak{S}}={\mathcal{C}}_{1}({\mathcal{H}})$ the trace class
operators on ${\mathcal{H}}$. V. Peller obtained a Grothendieck-style
characterisation of the function space $L^{1}((P\otimes
Q)_{\mathfrak{S}})=L^{1}((P\otimes Q)_{{\mathcal{L}}({\mathcal{H}})})$. One
direction is easy to see. If $\varphi$ has the decomposition

$\varphi=\int_{T}\alpha(\cdot,t)\beta(\cdot,t)\,d\nu(t)$

with

$\int_{T}\|\alpha(\cdot,t)\|_{L^{\infty}(Q)}\|\beta(\cdot,t)\|_{L^{\infty}(P)}\,d\nu(t)<\infty,$

then

$\big{(}(P\otimes
Q)_{\mathfrak{S}}(\varphi)\big{)}u=\int_{T}P(\alpha(\cdot,t))uQ\beta((\cdot,t))\,d\nu(t)$

is trace class for $u\in{\mathfrak{S}}$.

In terms of Schur mulipliers of matrices, equation (2.3) shows that the
$L^{1}_{G}(m)$-norm of the function (2.2) is given by the expression (2.3).
The appropriate extension of Grothendieck’s inequality to measure spaces gives
Peller’s representation
$\varphi=\int_{T}\alpha(\cdot,t)\beta(\cdot,t)\,d\nu(t)$ for

$\varphi\in L^{1}((P\otimes Q)_{\mathfrak{S}}),$

an earler conjecture of M. Birman and M. Solomyak [6], illustrating how
Grothendieck’s deep studies eventually solved a problem in scattering theory.
The characterisation of $L^{1}((P\otimes Q)_{\mathfrak{S}})$ may be acheived
via martingale convergence [18] and a direct application of Grothendieck’s
fundamental metric theory [29, 18]

This, in turn, leads to the strict inclusion of the space $L^{1}((P\otimes
Q)_{\mathfrak{S}})$ in $L^{1}((PQ))$. A better handle on membership in
$L^{1}((P\otimes Q)_{\mathfrak{S}})$ is achieved with Peller’s study of Besov
space [25, 26]. Membership of $L^{1}((PQ))$ and $L^{1}((QP))$ on abelian
groups can also be achieved with Hörmander’s general study of symbol classes.

##  8\. Fundamental Results for Double Operator Integrals and Traces

Here we sketch a few results that can be found in greater depth in [35].

######  Theorem 9.

Let ${\mathcal{H}}$ be a separable Hilbert space. Let
$P:{\mathcal{B}}({\mathbb{R}})\to{\mathcal{L}}_{s}({\mathcal{H}})$ and
$Q:{\mathcal{B}}({\mathbb{R}})\to{\mathcal{L}}_{s}({\mathcal{H}})$ be spectral
measures on ${\mathbb{R}}$. Let
${\mathfrak{S}}={\mathcal{C}}_{p}({\mathcal{H}})$ for some $1\leq p<\infty$ or
${\mathfrak{S}}={\mathcal{L}}({\mathcal{H}})$. Suppose that $f\in
L^{1}({\mathbb{R}})$ and $\varphi(\lambda,\mu)=\hat{f}(\lambda-\mu)$ for all
$\lambda,\mu\in{\mathbb{R}}$. Then
$\int_{{\mathbb{R}}\times{\mathbb{R}}}\varphi\,d(P\otimes
Q)_{\mathfrak{S}}\in{\mathcal{L}}({\mathfrak{S}})$ and

$\left\|\int_{{\mathbb{R}}\times{\mathbb{R}}}\varphi\,d(P\otimes
Q)_{\mathfrak{S}}\right\|_{{\mathcal{L}}({\mathfrak{S}})}\leq\|f\|_{1}.$

###### Proof.

For each $T\in{\mathcal{C}}_{1}({\mathcal{H}})$, the set function $E\times
F\longmapsto P(E)TQ(F)$, $E,F\in{\mathcal{B}}({\mathbb{R}})$, is the
restriction to all measurable rectangles of an
${\mathcal{L}}({\mathcal{H}})$-valued measure $\sigma$-additive for the strong
operator topology and the integral

(8.1) $\displaystyle\int_{{\mathbb{R}}\times{\mathbb{R}}}\varphi\,d(PTQ)$
$\displaystyle=\int_{{\mathbb{R}}\times{\mathbb{R}}}\left(\int_{\mathbb{R}}e^{-it(\lambda-\mu)t}f(t)\,dt\right)\,d(PTQ)(\lambda,\mu)$
(8.2)
$\displaystyle=\int_{\mathbb{R}}\left(\int_{{\mathbb{R}}\times{\mathbb{R}}}e^{-it(\lambda-\mu)t}\,d(PTQ)(\lambda,\mu)\right)f(t)\,dt$
(8.3) $\displaystyle=\int_{\mathbb{R}}e^{-itA}Te^{itB}f(t)\,dt$

converges as a Bochner integral in the strong operator topology to an element
of the operator ideal ${\mathcal{C}}_{1}({\mathcal{H}})$ of trace class
operators. The interchange of integrals is verified scalarly.

It follows that $\varphi$ is a $(P\otimes
Q)_{{\mathcal{C}}_{1}({\mathcal{H}})}$-integrable function and

$\left\|\int_{{\mathbb{R}}\times{\mathbb{R}}}\varphi\,d(P\otimes
Q)_{{\mathcal{C}}_{1}({\mathcal{H}})}\right\|_{{\mathcal{L}}({\mathcal{C}}_{1}({\mathcal{H}}))}\leq\|f\|_{1}.$

The corresponding bound for ${\mathfrak{S}}={\mathcal{C}}_{p}({\mathcal{H}})$
for $1\leq p<\infty$ and ${\mathfrak{S}}={\mathcal{L}}({\mathcal{H}})$ follows
by duality and interpolation, or directly from formula (8.3). ∎

######  Proposition 10.

Let ${\mathcal{H}}$ be a separable Hilbert space and let $A,B$ be selfadjoint
operators with spectral measures
$P_{A}:{\mathcal{B}}(\sigma(A))\to{\mathcal{L}}_{s}({\mathcal{H}})$ and
$P_{B}:{\mathcal{B}}(\sigma(B))\to{\mathcal{L}}_{s}({\mathcal{H}})$,
respectively. Let ${\mathfrak{S}}={\mathcal{C}}_{p}({\mathcal{H}})$ for some
$1\leq p<\infty$ or ${\mathfrak{S}}={\mathcal{L}}({\mathcal{H}})$. If the
spectra of $A$ and $B$ are separated by a distance
$d(\sigma(A),\sigma(B))=\delta>0$, then
$\int_{\sigma(A)\times\sigma(B)}(\lambda-\mu)^{-1}(P_{A}\otimes
P_{B})_{\mathfrak{S}}(d\lambda,d\mu)\in{\mathcal{L}}({\mathfrak{S}})$ and

$\left\|\int_{\sigma(A)\times\sigma(B)}\frac{(P_{A}\otimes
P_{B})_{\mathfrak{S}}(d\lambda,d\mu)}{\lambda-\mu}\right\|_{{\mathcal{L}}({\mathfrak{S}})}\leq\frac{\pi}{2\delta}.$

In particular, $AX-XB=Y$ has a unique strong solution for $Y\in{\mathfrak{S}}$
given by the double operator integral

$X=\int_{\sigma(A)\times\sigma(B)}\frac{dP_{A}(\lambda)YdP_{B}(\mu)}{\lambda-\mu}:=\left(\int_{\sigma(A)\times\sigma(B)}\frac{(P_{A}\otimes
P_{B})_{\mathfrak{S}}(d\lambda,d\mu)}{\lambda-\mu}\right)Y,$

so that $\|X\|_{\mathfrak{S}}\leq\frac{\pi}{2\delta}\|Y\|_{\mathfrak{S}}$ [4].

Although the Heaviside function $\smash{\raise
2.15277pt\hbox{$\chi$}}_{(0,\infty)}$ is not the Fourier transform of an
$L^{1}$-function, the following result of I. Gohberg and M. Krein [15, Section
III.6] holds, in case $P=Q$.

######  Theorem 11.

Let ${\mathcal{H}}$ be a separable Hilbert space. Let
$P:{\mathcal{B}}({\mathbb{R}})\to{\mathcal{L}}_{s}({\mathcal{H}})$ and
$Q:{\mathcal{B}}({\mathbb{R}})\to{\mathcal{L}}_{s}({\mathcal{H}})$ be spectral
measures on ${\mathbb{R}}$. Then

$\int_{{\mathbb{R}}\times{\mathbb{R}}}\smash{\raise
2.15277pt\hbox{$\chi$}}_{\\{\lambda>\mu\\}}\,d(P\otimes
Q)_{{\mathcal{C}}_{p}({\mathcal{H}})}\in{\mathcal{L}}({\mathcal{C}}_{p}({\mathcal{H}}))$

for every $1<p<\infty$.

The following recent result of F. Sukochev and D. Potapov (see [37]) settled a
long outstanding conjecture of M. Krein for the index $p$ in the range
$1<p<\infty$.

######  Theorem 12.

Let ${\mathcal{H}}$ be a separable Hilbert space. Let
$P:{\mathcal{B}}({\mathbb{R}})\to{\mathcal{L}}_{s}({\mathcal{H}})$ and
$Q:{\mathcal{B}}({\mathbb{R}})\to{\mathcal{L}}_{s}({\mathcal{H}})$ be spectral
measures on ${\mathbb{R}}$. Suppose that $f:{\mathbb{R}}\to{\mathbb{R}}$ is a
continuous function for which the difference quotient

$\varphi_{f}(\lambda,\mu)=\left\\{\begin{array}[]{ll}\frac{f(\lambda)-f(\mu)}{\lambda-\mu}&,\quad\lambda\neq\mu,\\\
0&,\quad\lambda=\mu,\end{array}\right.$

is uniformly bounded. Then for every $1<p<\infty$,

$\int_{{\mathbb{R}}\times{\mathbb{R}}}\varphi_{f}\,d(P\otimes
Q)_{{\mathcal{C}}_{p}({\mathcal{H}})}\in{\mathcal{L}}({\mathcal{C}}_{p}({\mathcal{H}}))$

and there exists $C_{p}>0$ such that

$\left\|\int_{{\mathbb{R}}\times{\mathbb{R}}}\varphi_{f}\,d(P\otimes
Q)_{{\mathcal{C}}_{p}({\mathcal{H}})}\right\|_{{\mathcal{C}}_{p}({\mathcal{H}})}\leq
C_{p}\|\varphi_{f}\|_{\infty}.$

Such a function $f$ is said to be uniformly Lipschitz on ${\mathbb{R}}$ and
$\|f\|_{{\rm Lip}_{1}}:=\|\varphi_{f}\|_{\infty}$.

######  Corollary 13.

Suppose that $f:{\mathbb{R}}\to{\mathbb{R}}$ is a uniformly Lipschitz
function. Then for every $1<p<\infty$, there exists $C_{p}>0$ such that

$\|f(A)-f(B)\|_{{\mathcal{C}}_{p}({\mathcal{H}})}\leq C_{p}\|f\|_{{\rm
Lip}_{1}}\|A-B\|_{{\mathcal{C}}_{p}({\mathcal{H}})}$

for any selfadjoint operators $A$ and $B$ on a separable Hilbert space
${\mathcal{H}}$.

###### Proof.

Let $P_{A}$ and $P_{B}$ be the spectral meaures of $A$ and $B$, respectively,
and suppose that $\|A-B\|_{{\mathcal{C}}_{p}({\mathcal{H}})}<\infty$. Then
according to [6, Theorem 8.1] (see also [11, Corollary 7.2]), the equality

$f(A)-f(B)=\left(\int_{{\mathbb{R}}\times{\mathbb{R}}}\varphi_{f}\,d(P_{A}\otimes
P_{B})_{{\mathcal{C}}_{p}({\mathcal{H}})}\right)(A-B)$

holds and the norm estimate follows from Theorem 12. ∎

##  9\. An elementary proof of Krein’s spectral trace formula

We refer to the survey paper of M Birman and M. Solomyak [6] . According to
their history of the subject, they hoped to prove Krein’s formula just with
general double operator integral arguments, which go back to old work of Yu.
Daletskii and M. Krein in the 1950’s, see [6]. They were aware of the
arguments of Feynman’s Operational Calculus which has received a new boost in
the monograph [20] of M. Lapidus, L. Nielsen and the late G.W. Johnson.

The proof below is adapted from [8] which has an attractive and elementary
Fourier Theory argument in the upper half-plane. Other proofs use elementary
Complex Analysis but all finally appeal to a Bootstrap estimate on eigenvalues
of a positive trace class operator. The original argument was simply by
analogy from the case of matrices and determinants, which is often a good
testing ground for infinite dimensional operator perturbation theory, see
[21].

######  Theorem 14.

Let ${\mathcal{H}}$ be a separable Hilbert space and let $A$ and $B$ be
selfadjoint operators with the same domain such that
$A-B\in{\mathcal{C}}_{1}({\mathcal{H}})$. Then there exists a spectral shift
function $\xi\in L^{1}({\mathbb{R}})$ such that

(9.1) $\hbox{\rm
tr}(f(A)-f(B))=\int_{\mathbb{R}}f^{\prime}(\lambda)\xi(\lambda)\,d\lambda$

for every function $f:{\mathbb{R}}\to{\mathbb{C}}$ for which there exists a
finite positive Borel measure $\mu$ on ${\mathbb{R}}$ such that

$f(x)=i\int_{\mathbb{R}}\frac{e^{-isx}-1}{s}\,d\mu(s),\quad x\in{\mathbb{R}}.$

Furthermore, $\xi$ possesses the following properties.

  1. a)

$\hbox{\rm tr}(A-B)=\int_{\mathbb{R}}\xi(\lambda)\,d\lambda$.

  2. b)

$\|\xi\|_{1}\leq\|A-B\|_{{\mathcal{C}}_{1}({\mathcal{H}})}$.

  3. c)

If $B\leq A$, then $\xi\geq 0$ a.e.

  4. d)

$\xi$ is zero a.e. outside the interval
$(\inf(\sigma(A)\cup\sigma(B)),\sup(\sigma(A)\cup\sigma(B)))$.

###### Proof.

It suffices to assume that $A$ and $B$ are hermitian matrices. The estimate

(9.2)
$\|f(A)-f(B)\|_{{\mathcal{C}}_{1}({\mathcal{H}})}\leq\mu({\mathbb{R}})\|A-B\|_{{\mathcal{C}}_{1}({\mathcal{H}})}$

follows from the bound of Corollary 13 and the calculation

$f(A)-f(B)=\frac{i}{2\pi}\int_{\mathbb{R}}\frac{e^{-isA}-e^{-isB}}{s}\,d\mu(s)$

obtained from an application of Fubini’s theorem with respect to
$P_{A}\otimes\mu$ and $P_{B}\otimes\mu$ on
${\mathbb{R}}\times[\epsilon,\infty)$ for $\epsilon>0$. Then

$\hbox{\rm tr}(f(A)-f(B))=\frac{1}{2\pi}\int_{\mathbb{R}}\Phi\,d\mu.$

An expression for the spectral shift function $\xi$ may be obtained from
Fatou’s Theorem [30, Theorem 11.24]. Suppose that $\nu$ is a finite measure on
${\mathbb{R}}$

$\phi_{\nu}(z)=\frac{1}{2\pi
i}\int_{\mathbb{R}}\frac{d\nu(\lambda)}{\lambda-z},\quad
z\in{\mathbb{C}}\setminus{\mathbb{R}},$

is the Cauchy transform of $\nu$. Then $\nu$ is absolutely continuous if

$\hat{\nu}(\xi)=\int_{\mathbb{R}}e^{-i\xi
x}(\phi_{\nu}(x+i0+)-\phi_{\nu}(x+i0-))\,dx,\quad\xi\in{\mathbb{R}}.$

The function $x\longmapsto\phi_{\nu}(x+i0+)-\phi_{\nu}(x+i0-)$ defined for
almost all $x\in{\mathbb{R}}$ is then the density of $\nu$ with respect to
Lebesgue measure. For $\nu=\Xi$, if the representation

$\displaystyle\Phi(s)$ $\displaystyle=i\frac{\hbox{tr}(e^{-isA}-e^{-isB})}{s}$
$\displaystyle=\frac{1}{2\pi
i}\int_{\mathbb{R}}e^{-isx}\bigg{(}\lim_{\epsilon\to
0+}\int_{0}^{1}\hbox{tr}(V(A+tV-x-i\epsilon)^{-1}-$ $\displaystyle\qquad
V(A+tV-x+i\epsilon)^{-1})\,dt\bigg{)}dx$

were valid, we would expect that $\xi=\check{\Phi}$ has the representation

$\displaystyle\xi(s)$ $\displaystyle=\frac{1}{2\pi i}\lim_{\epsilon\to
0+}\int_{\mathbb{R}}e^{isx-\epsilon|x|}\frac{\hbox{tr}(e^{-ixA}-e^{-ixB})}{x}dx,\quad
s\in{\mathbb{R}},$ (9.3) $\displaystyle=\lim_{\epsilon\to
0+}\frac{1}{\pi}\hbox{tr}\left[\arctan\left(\frac{A-sI}{\epsilon}\right)-\arctan\left(\frac{B-sI}{\epsilon}\right)\right],$

where the arctan function may be expressed as

(9.4) $\arctan
t=\frac{1}{2i}\int_{\mathbb{R}}\frac{e^{ist}-1}{s}e^{-|s|}\,ds,\quad
t\in{\mathbb{R}}.$

For the function defined by

$\displaystyle h(x,y)$
$\displaystyle=\frac{1}{\pi}\hbox{tr}\left[\arctan\left(\frac{A-xI}{y}\right)-\arctan\left(\frac{B-xI}{y}\right)\right]$

we have the bounds

$\displaystyle\pi|h(x,y)|$
$\displaystyle\leq\left\|\arctan\left(\frac{A-xI}{y}\right)-\arctan\left(\frac{B-xI}{y}\right)\right\|_{{\mathcal{C}}_{1}({\mathcal{H}})}$
$\displaystyle\leq\frac{1}{y}\|A-B\|_{{\mathcal{C}}_{1}({\mathcal{H}})},$

from the bound (9.2) and the representation (9.4). Rewriting

$h(x,y)=\frac{1}{2\pi
i}\int_{\mathbb{R}}e^{-ixs-y|s|}\hbox{tr}\left[\frac{e^{isA}-e^{isB}}{s}\right]ds$

using (9.4), it follows that $h(x,y)$ is harmonic in the upper half-plane

$\\{(x,y):x\in{\mathbb{R}},\ y>0\\}.$

We first look at the case that $A-B=\alpha(\cdot,w)w$ for $\alpha>0$ and
$w\in{\mathcal{H}}$, $\|w\|=1$, so that $A$ is a rank one perturbation of the
bounded selfadjoint operator $B$.

If we set

$X=2\arctan\frac{A-x}{y},Y=2\arctan\frac{B-x}{y},$

then $2\pi h=\hbox{tr}(X-Y)$. The formula tr
$\log(e^{iX}e^{-iY})=i\hbox{tr}(X-Y)$ follows from the Baker-Campbell-
Hausdorff formula for large $y>0$, see [8, Lemma 1.1]. Let $T_{A}=e^{-iX}$,
$T_{B}=e^{-iY}$. Then for $z=x+iy$, elementary spectral theory gives

$\begin{array}[]{ll}T_{A}&=(A-\overline{z}I)(A-zI)^{-1}=I+2iy(A-zI)^{-1},\\\
T_{B}&=(B-\overline{z}I)(B-zI)^{-1}=I+2iy(B-zI)^{-1}.\end{array}$

Our aim is to compute tr $\log(U)$ for the unitary operator
$U=T_{A}^{*}T_{B}$. Because

$\displaystyle U-I$ $\displaystyle=T_{A}^{*}T_{B}-T_{B}^{*}T_{B}$
$\displaystyle=(T_{A}^{*}-T_{B}^{*})T_{B}$
$\displaystyle=-i2y[(A-\overline{z}I)^{-1}-(B-\overline{z}I)^{-1}]T_{B},$

we obtain

$U=I+i2y(A-\overline{z}I)^{-1}(A-B)(B-zI)^{-1}.$

Substituting $A-B=\alpha(\cdot,w)w$ gives

$U=I+i2y\alpha(\cdot,(B-\overline{z}I)^{-1}w)(A-\overline{z}I)^{-1}w.$

The vector $(A-\overline{z}I)^{-1}w$ is an eigenvector for the unitary
operator $U$ with eigenvalue

$1+i2y\alpha((A-\overline{z}I)^{-1}w,(B-\overline{z}I)^{-1}w)$

which can be expressed as $e^{i2\pi\theta(x,y)}$ for a continuous function
$\theta$ in the upper half plane such that $0<\theta<1$. Consequently, for
large $y>0$,

$i2\pi\theta=\hbox{tr}\log(U)=i\hbox{tr}(X-Y)=i2\pi h.$

Then $\theta$ is harmonic for large $y>0$ so it is harmonic on the upper half
plane and it is equal to $h$ there, so $0<h<1$. For matrices, the formula for
$h$ is quite explicit and we can proceed by direct calculation.

By Fatou’s Theorem, the boundary values $\xi(x)=\lim_{y\to 0+}h(x,y)$ are
defined for almost all $x\in{\mathbb{R}}$ and satisfy

$\lim_{y\to\infty}\pi
yh(x,y)=\int_{\mathbb{R}}\xi(t)\,dt=\|\xi\|_{1}\leq\|A-B\|_{{\mathcal{C}}_{1}({\mathcal{H}})}$

for every $x\in{\mathbb{R}}$, so in the case that $A-B$ has rank one, formula
(9.3) is valid.

For an arbitrary selfadjoint perturbation
$V=\sum_{j=1}^{\infty}\alpha_{j}(\cdot,w_{j})w_{j}$ with

$\sum_{j=1}^{\infty}|\alpha_{j}|=\|A-B\|_{{\mathcal{C}}_{1}({\mathcal{H}})}<\infty,$

the function $\xi_{n}\in L^{1}({\mathbb{R}})$ may be defined in a similar
fashion for $A_{n}=B+\sum_{j=1}^{n}\alpha_{j}(\cdot,w_{j})w_{j}$,
$n=1,2,\dots$, so that $\xi_{n}\to\xi$ in $L^{1}({\mathbb{R}})$ as
$n\to\infty$ from which it verified that $\xi=\check{\Phi}$. ∎

The representation $\xi=\check{\Phi}$ obtained above may be viewed as the
Fourier transform approach. In the case of a rank one perturbation
$V=\alpha(\cdot,w)w$, the Cauchy transform approach is developed by B. Simon
[33] with the formula

$\hbox{\rm
tr}((A-zI)^{-1}-(B-zI)^{-1})=-\int_{\mathbb{R}}\frac{\xi(\lambda)}{(\lambda-z)^{2}}\,d\lambda$

for $z\in{\mathbb{C}}\setminus[a,\infty)$ for some $a\in{\mathbb{R}}$,
established in [33, Theorem 1.9] by computing a contour integral. Here the
boundary value $\xi(x)=\lim_{y\to 0+}h(x,y)$ is expressed as

$\xi(x)=\frac{1}{\pi}\hbox{Arg}(1+\alpha F(\lambda+i0+))$

for almost all $x\in{\mathbb{R}}$ with respect to the Cauchy transform

$F(z)=\int_{\mathbb{R}}\frac{d(P_{B}w,w)(\lambda)}{\lambda-z},\quad
z\in{\mathbb{C}}\setminus(-\infty,a).$

The Cauchy transform approach is generalised to type II von Neumann algebras
in [3].

Many different proofs of Krein’s formula (9.1) are available for a wide class
of functions $f$, especially in a form that translates into the setting of
noncommutative integration [3, 26]. As remarked in [6, p. 163], an ingredient
additional to double operator integrals (such as complex function theory) is
needed to show that the measure $\Xi$ is absolutely continuous with respect to
Lebesgue measure on ${\mathbb{R}}$. Krein’s original argument uses
perturbation determinants from which follows the representation
Det$(S(\lambda))=e^{-2\pi i\xi(\lambda)}$ for the scattering matrix
$S(\lambda)$ for $A$ and $B$ [39, Chapter 8] ,which is still a basic tool of
scattering theory.

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## Index