# Lattice Trace Operators

Authors: Brian Jefferies
Journal: Journal of Operators
 (2014)

---

## Abstract

A bounded linear operatorT on a Hilbert space ℋ is trace class if its singular values are summable. The trace class operators
on ℋ form an operator ideal and in the case that ℋ is finite-dimensional, the trace tr(
T
) of T is given by ∑
j
a
j
j for any matrix representation {
a
i
j
} of T. In applications of trace class operators to scattering theory and representation theory, the subject is complicated by the fact that if k is an integral kernel of the operator T on the Hilbert space L
2
(
μ
) with μ a σ-finite measure, then k
(
x
,
x
) may not be defined, because the diagonal {
(
x
,
x
)
} may be a set of (
μ
⊗
μ
)-measure zero. The present note describes a class of linear operators acting on a Banach function space X which forms a lattice ideal of operators on X, rather than an operator ideal, but coincides with the collection of hermitian positive trace class operators in the case of X
=
L
2
(
μ
).


---

## Body

## 1. Introduction

Atrace class operator
A on a separable Hilbert space ℋ is a compact operator whosesingular values
λ
j
(
A
), j
=
1,2
,
…, satisfy
(1)
∥
A
∥
1
=
∑
j
=
1
∞
λ
j
(
A
)
<
∞
.
The decreasing sequence {
λ
j
(
A
)
}
j
=
1
∞ consists of eigenvalues of (
A
*
A
)
1
/
2. Equivalently, A is trace class if and only if, for any orthonormal basis {
h
j
}
j
=
1
∞ of ℋ, the sum ∑
j
=
1
∞
‍
|
(
A
h
j
,
h
j
)
| is finite. The number
(2)
tr
⁡
(
A
)
=
∑
j
=
1
∞
‍
(
A
h
j
,
h
j
)
is called thetrace of A and is independent of the orthonormal basis {
h
j
}
j
=
1
∞ of ℋ.Lidskii’s equality asserts that tr
⁡
(
A
) is actually the sum of the eigenvalues of the compact operator T [1, Theorem 3.7].We refer to [1] for properties of trace class operators. The collection 𝒞
1
(
ℋ
) of trace class operators on ℋ is anoperator ideal and Banach space with the norm ∥
·
∥
1. The following facts are worth noting in the case of the Hilbert space L
2
(
[
0,1
]
) with respect to Lebesgue measure on the interval [
0,1
].(a)
IfT
:
L
2
(
[
0,1
]
)
→
L
2
(
[
0,1
]
) is a trace class linear operator, then there exist ϕ
j
,
ψ
j
∈
L
2
(
[
0,1
]
), j
=
1,2
,
…, with ∑
j
=
1
∞
‍
∥
ϕ
j
∥
2
∥
ψ
j
∥
2
<
∞ and
(3)
(
T
f
)
(
x
)
=
∫
0
1
‍
k
(
x
,
y
)
f
(
y
)
d
y
,
a
.
e
.
for
f
∈
L
2
(
[
0,1
]
)
,
wherek
=
∑
j
=
1
∞
‍
ϕ
j
⊗
ψ
j a.e.. In particular, T is regular and |
T
| has an integral kernel |
k
|
≤
∑
j
=
1
∞
‍
|
ϕ
j
|
⊗
|
ψ
j
|. Moreover,
(4)
tr
⁡
(
T
)
=
∑
j
=
1
∞
‍
∫
0
1
‍
ϕ
j
(
x
)
ψ
j
(
x
)
d
x
.
(b)
Suppose thatT
:
L
2
(
[
0,1
]
)
→
L
2
(
[
0,1
]
) is a regular linear operator defined by formula (3) for a continuous function k
:
[
0,1
]
×
[
0,1
]
→
ℂ. If T is trace class, then ∫
0
1
‍
|
k
(
x
,
x
)
|
d
x
<
∞, and tr
⁡
(
T
)
=
∫
0
1
‍
k
(
x
,
x
)
d
x [2, Theorem V
.
3.1
.
1].
(c)
Suppose that the functionk
:
[
0,1
]
×
[
0,1
]
→
ℂ is continuous and positive definite; that is, ∑
j
,
ℓ
=
1
n
‍
z
j
z
ℓ
¯
k
(
x
j
,
x
ℓ
)
≥
0 for all z
j
∈
ℂ and x
j
∈
[
0,1
], j
=
1
,
…
,
n, and any n
=
1,2
,
…. Then k
(
x
,
x
)
≥
0 for all x
∈
[
0,1
]. If ∫
0
1
‍
k
(
x
,
x
)
d
x
<
∞, then there exists a unique trace class operator defined by formula (3) [1, Theorem 2.12].Let(
Σ
,
ℬ
,
μ
) be a measure space. Theprojective tensor product
L
2
(
μ
)
⊗
^
π
L
2
(
μ
) is the set of all sums:
(5)
k
=
∑
j
=
1
∞
‍
ϕ
j
⊗
ψ
j
a
.
e
.
,
with
∑
j
=
1
∞
‍
∥
ϕ
j
∥
2
∥
ψ
j
∥
2
<
∞
.
The norm of k
∈
L
2
(
μ
)
⊗
^
π
L
2
(
μ
) is given by ∥
k
∥
π
=
inf
⁡
{
∑
j
=
1
∞
‍
∥
ϕ
j
∥
2
∥
ψ
j
∥
2
} where the infimum is taken over all sums for which the representation (5) holds. The Banach space L
2
(
μ
)
⊗
^
π
L
2
(
μ
) is actually the completion of the algebraic tensor product L
2
(
μ
)
⊗
L
2
(
μ
) with respect to the projective tensor product norm [3, Section 6.1].There is a one-to-one correspondence between the space of trace class operators acting onL
2
(
μ
) and L
2
(
μ
)
⊗
^
π
L
2
(
μ
), so that the trace class operator T
k has an integral kernel k
∈
L
2
(
μ
)
⊗
^
π
L
2
(
μ
). If the integral kernel k given by (5) has the property that
(6)
k
(
x
,
y
)
=
∑
j
=
1
∞
‍
ϕ
j
(
x
)
ψ
j
(
y
)
for all x
,
y
∈
Σ such that the sum ∑
j
=
1
∞
‍
|
ϕ
j
(
x
)
ψ
j
(
y
)
| is finite, then the equality
(7)
tr
⁡
(
T
k
)
=
∑
j
=
1
∞
‍
∫
Σ
‍
ϕ
j
(
x
)
ψ
j
(
x
)
d
μ
(
x
)
=
∫
Σ
‍
k
(
x
,
x
)
d
μ
(
x
)
holds. Because the diagonal {
(
x
,
x
)
:
x
∈
Σ
} may be a set of (
μ
⊗
μ
)-measure zero in Σ
×
Σ, it may be difficult to determine whether or not a given integral kernel k
:
Σ
×
Σ
→
ℂ has such adistinguished representation.The difficulty is addressed by Brislawn [4, 5], [1, Appendix D] who shows that, for a trace class operator T
k
:
L
2
(
μ
)
→
L
2
(
μ
) with integral kernel k, the equality
(8)
tr
⁡
(
T
k
)
=
∫
Σ
‍
k
~
(
x
,
x
)
d
μ
(
x
)
holds. The measure μ is supposed in [5] to be a σ-finite Borel measure on a second countable topological space Σ and the regularised kernel k
~ is defined from k by averaging with respect to the product measure μ
⊗
μ. Extending the result (c) of M. Duflo given above, Brislawn [5, Theorem 4.3] shows that a hermitian positive Hilbert-Schmidt operator T
k is a trace class operator if and only if ∫
Σ
‍
k
~
(
x
,
x
)
d
μ
(
x
)
<
∞.The present paper examines the spaceℭ
1
(
X
) of absolute integral operators T
k
:
X
→
X defined on a Banach function space for which ∫
Σ
‍
|
k
~
(
x
,
x
)
|
d
μ
(
x
)
<
∞. Elements of ℭ
1
(
X
) are calledlattice trace operators because ℭ
1
(
X
) is alattice ideal in the Banach lattice of regular operators on X, whereas the collection 𝒞
1
(
ℋ
) of trace class operators on a Hilbert space ℋ is anoperator ideal in the Banach algebra ℒ
(
ℋ
) of all bounded linear operators on ℋ. The intersections of ℭ
1
(
X
) and 𝒞
1
(
L
2
(
μ
)
) with the hermitian positive operators on L
2
(
μ
) are equal for locally square integrable kernels; see Proposition 4.The regularised kernelk
~
:
Σ
×
Σ
→
ℂ of an absolute integral operator T
k is defined by adapting the method of Brislawn [5] to positive operators with an integral kernel. The generalised trace ∫
Σ
‍
k
~
(
x
,
x
)
d
μ
(
x
) may be viewed alternatively as a bilinear integral ∫
Σ
‍
〈
T
k
,
d
m
〉 with respect to the measure m
:
E
↦
χ
E, E
∈
ℬ. Lattice trace operators are employed in the proof of the Cwikel-Lieb-Rosenblum inequality for dominated semigroups [6].The basic definitions of Banach function spaces and operators with an integral kernel which act upon them are set out in Section2. The martingale regularisation of the integral kernel of an operator between Banach function spaces is set out in Section 3 and the connection with trace class operators on L
2
(
μ
) is set out in Section 4.


## 2. Banach Function Spaces and Regular Operators

LetΣ be a second countable topological space with Borel σ-algebra ℬ. The diagonal diag
⁡
(
Σ
×
Σ
)
=
{
(
x
,
x
)
:
x
∈
Σ
} is a closed subset of the Cartesian product Σ
×
Σ. Because the Borel σ-algebra of Σ
×
Σ is equal to ℬ
⊗
ℬ, the diagonal diag
⁡
(
Σ
×
Σ
) belongs to the σ-algebra ℬ
⊗
ℬ.We suppose that(
Σ
,
ℬ
,
μ
) is a σ-finite measure space. The space of all μ-equivalence classes of Borel measurable scalar functions is denoted by L
0
(
μ
). It is equipped with the topology of convergence in μ-measure over sets of finite measure and vector operations pointwise μ-almost everywhere. Any Banach space X that is a subspace of L
0
(
μ
) with the properties that(i)
X is an order ideal of L
0
(
μ
), that is, if g
∈
X, f
∈
L
0
(
μ
), and |
f
|
≤
|
g
|
μ-a.e., then f
∈
X and
(ii)
iff
,
g
∈
X and |
f
|
≤
|
g
|
μ-a.e., then ∥
f
∥
X
≤
∥
g
∥
X,is called aBanach function space (based on (
Σ
,
ℬ
,
μ
)). The Banach function space X is necessarily Dedekind complete; that is, every order bounded set has a sup and an inf [7, page 116]. The set of f
∈
X with f
≥
0
μ-a.e. is written as X
+.We suppose thatX contains the characteristic functions of sets of finite measure and m
:
S
↦
χ
S, S
∈
𝒮, is σ-additive in X on sets of finite measure; for example, X is σ-order continuous; see [8, Corollary 3.6]. If X is reflexive and μ is finite and nonatomic, then it follows from [8, Corollary 3.23] that the values of the variation V
(
m
) of m are either zero or infinity. In particular, this is the case for X
=
L
p
(
[
0,1
]
) with 1
<
p
<
∞.Following the account of Brislawn [5], we extend the mapping T
↦
∫
Σ
‍
〈
T
,
d
m
〉 from the space 𝒞
1
(
L
2
(
μ
)
) of trace class linear operators to a larger class of regular operators by representing T by a “regularised” kernel, so that the collection of regular operators T for which ∫
Σ
‍
〈
|
T
|
,
d
m
〉
<
∞ is a vector sublattice of the Riesz space of regular operators—a property not necessarily enjoyed by the trace class operators.LetX be a Banach function space based on the σ-finite measure space (
Σ
,
ℬ
,
μ
) as above. A continuous linear operator T
:
X
→
X is calledpositive if T
:
X
+
→
X
+. The collection of all positive continuous linear operators on X is written as ℒ
+
(
X
). If the real and imaginary parts of a continuous linear operator T
:
X
→
X can be written as the difference of two positive operators, it is said to beregular. Themodulus
|
T
| of a regular operator T is defined by
(9)
|
T
|
f
=
sup
⁡
|
g
|
≤
f
|
T
g
|
,
f
∈
X
+
.
The collection of all regular operators is written as ℒ
r
(
X
) and it is given the norm T
↦
∥
|
T
|
∥, T
∈
ℒ
r
(
X
) under which it becomes a Banach lattice [7, Proposition 1.3
.
6].A continuous linear operatorT
:
X
→
X has an integral kernel k if k
:
Σ
×
Σ
→
ℂ is a Borel measurable function such that T
=
T
k for the operator given by
(10)
(
T
k
f
)
(
x
)
=
∫
Σ
‍
k
(
x
,
y
)
f
(
y
)
d
μ
(
y
)
,
μ
-almost  all
x
∈
Σ
,
for each f
∈
X, in the sense that ∫
Σ
‍
|
k
(
x
,
y
)
f
(
y
)
|
d
μ
(
y
)
<
∞ for μ-almost all x
∈
Σ and x
↦
∫
Σ
‍
k
(
x
,
y
)
f
(
y
)
d
μ
(
y
) is an element of X. Then k is (
μ
⊗
μ
)-integrable on any product set A
×
B with finite measure. If T
k
≥
0, then k
≥
0
(
μ
⊗
μ
)-a.e. on Σ
×
Σ [7, Theorem 3.3
.
5].A continuous linear operatorT is anabsolute integral operator if it has an integral kernel k for which T
|
k
| is a bounded linear operator on L
2
(
μ
). Then |
T
k
|
=
T
|
k
| [7, Theorem 3.3
.
5]. The collection of all absolute integral operators is a lattice ideal in ℒ
r
(
X
) [7, Theorem 3.3
.
6].Suppose thatT
∈
ℒ
(
X
) has an integral kernel k
=
∑
j
=
1
n
‍
f
j
χ
A
j, that is, an X-valued simple function with μ
(
A
j
)
<
∞. Then it is natural to view
(11)
∫
Σ
‍
〈
T
,
d
m
〉
=
∑
j
=
1
n
‍
∫
A
j
‍
f
j
d
μ
=
∫
Σ
‍
k
(
x
,
x
)
d
μ
(
x
)
as a bilinear integral. Our aim is to extend the integral to a wider class of absolute integral operators.


## 3. Martingale Regularisation

Let𝒰
=
{
U
1
,
U
2
,
…
} be a countable base for the topology of Σ. An increasing family of countable partitions 𝒫
n, n
=
1,2
,
…, is defined recursively by setting 𝒫
1 equal to a partition of Σ into Borel sets of finite μ-measure and
(12)
𝒫
j
+
1
=
{
P
∩
U
j
,
P
∖
U
j
:
P
∈
𝒫
j
}
for j
=
1,2
,
….. For each n
=
1,2
,
…, let ℰ
n be the σ-algebra for all countable unions of elements of 𝒫
n.Suppose thatk
≥
0 is a Borel measurable function defined on Σ
×
Σ that is integrable on every set of finite (
μ
⊗
μ
)-measure.For eachx
∈
Σ, the set U
n
(
x
) is the unique element of the partition 𝒫
n containing x. For each n
=
1,2
,
…, the conditional expectation k
n
=
𝔼
(
k
∣
ℰ
n
⊗
ℰ
n
) can be represented for μ-almost all x
,
y
∈
Σ as
(13)
𝔼
(
k
∣
ℰ
n
⊗
ℰ
n
)
(
x
,
y
)
=
1
μ
(
U
n
(
x
)
)
μ
(
U
n
(
y
)
)
∫
U
n
(
x
)
‍
∫
U
n
(
y
)
‍
k
(
s
,
t
)
d
μ
(
s
)
d
μ
(
t
)
=
∑
U
,
V
∈
𝒫
n
‍
∫
U
×
V
‍
k
d
(
μ
⊗
μ
)
μ
(
U
)
μ
(
V
)
χ
U
×
V
(
x
,
y
)
.
Let 𝒩 be the set of all x
∈
Σ for which there exists n
=
1,2
,
… such that μ
(
U
n
(
x
)
)
=
0. Then μ
(
U
m
(
x
)
)
=
0 for all m
>
n because 𝒫
m is a refinement of 𝒫
n if m
>
n. Moreover 𝒩 is μ-null because 𝒩
⊂
⋃
n
=
1
∞
‍
⋃
{
U
∈
𝒫
n
:
μ
(
U
)
=
0
}. If 0
≤
k
1
≤
k
2
(
μ
⊗
μ
)-a.e., then
(14)
𝔼
(
k
1
∣
ℰ
n
⊗
ℰ
n
)
(
x
,
y
)
≤
𝔼
(
k
2
∣
ℰ
n
⊗
ℰ
n
)
(
x
,
y
)
,
n
=
1,2
,
…
,
for all (
x
,
y
)
∈
𝒩
c
×
𝒩
c. In particular,
(15)
𝔼
(
k
1
∣
ℰ
n
⊗
ℰ
n
)
(
x
,
x
)
≤
𝔼
(
k
2
∣
ℰ
n
⊗
ℰ
n
)
(
x
,
x
)
,
n
=
1,2
,
…
,
for all x
∈
𝒩
c. Although diag(
Σ
×
Σ
) may be a set of (
μ
⊗
μ
)-measure zero, the application of the conditional expectation operators k
↦
𝔼
(
k
∣
ℰ
n
⊗
ℰ
n
), n
=
1,2
,
…, has the effect of regularising k. By an appeal to the martingale convergence theorem, k
n converges (
μ
⊗
μ
)-a.e. to k as n
→
∞.Letk
~
(
x
,
y
)
=
limsup
⁡
n
→
∞
𝔼
(
k
∣
ℰ
n
⊗
ℰ
n
)
(
x
,
y
) for all x
,
y
∈
Σ and we set
(16)
∫
Σ
‍
〈
T
,
d
m
〉
=
∫
Σ
‍
k
~
(
x
,
x
)
d
μ
(
x
)
∈
[
0
,
∞
]
.
If ∫
Σ
‍
〈
T
,
d
m
〉
<
∞, then A
↦
∫
A
‍
〈
T
,
d
m
〉
=
∫
A
‍
k
~
(
x
,
x
)
d
μ
(
x
), A
∈
ℬ, is a finite measure. For a regular operator T
=
T
+
-
T
- with positive and negative parts T
±, we set
(17)
∫
Σ
‍
〈
T
,
d
m
〉
=
∫
Σ
‍
〈
T
+
,
d
m
〉
-
∫
Σ
‍
〈
T
-
,
d
m
〉
if one of the integrals on the right-hand side of the equation is finite. The integral ∫
Σ
‍
〈
T
,
d
m
〉 is defined by linearity for each regular operator T
:
X
→
X. It is clear from the construction that the collection of absolute integral operators T such that ∫
Σ
‍
〈
|
T
|
,
d
m
〉
<
∞ is a vector sublattice ℭ
1
(
X
) of the space of regular operators on L
2
(
μ
). We call elements of ℭ
1
(
X
)
lattice trace operators.Theorem 1.
The spaceℭ
1
(
X
) is a lattice ideal in ℒ
r
(
X
); that is, if  S
,
T
∈
ℒ
r
(
X
), |
S
|
≤
|
T
| and T
∈
ℭ
1
(
X
), then S
∈
ℭ
1
(
X
). Moreover, ℭ
1
(
X
) is a Dedekind complete Banach lattice with the norm
(18)
T
⟼
∥
|
T
|
∥
+
∫
Σ
‍
〈
|
T
|
,
d
m
〉
,
T
∈
ℭ
1
(
X
)
.
The map T
↦
∫
Σ
‍
〈
T
,
d
m
〉 is a positive continuous linear function on ℭ
1
(
X
).Proof.
IfS
,
T
∈
ℒ
r
(
X
) and |
S
|
≤
|
T
|, then S is an absolute integral operator by [7, Theorem 3.3
.
6]. If k
1 is the integral kernel of S and k
2 is the integral kernel of T, then by [7, Theorem 3.3
.
5], the inequality |
k
1
|
≤
|
k
2
| holds (
μ
⊗
μ
)-a.e.. Then |
k
~
1
(
x
,
x
)
|
≤
|
k
~
2
(
x
,
x
)
| for μ-almost all x
∈
Σ, so that
(19)
∫
Σ
‍
〈
|
S
|
,
d
m
〉
≤
∫
Σ
‍
〈
|
T
|
,
d
m
〉
<
∞
.
Hence, S
∈
ℭ
1
(
X
).
To show thatℭ
1
(
X
) is complete in its norm, suppose that
(20)
∑
j
=
1
∞
‍
(
∥
|
T
j
|
∥
+
∫
Σ
‍
〈
|
T
j
|
,
d
m
〉
)
<
∞
for T
j
∈
ℭ
1
(
X
). Then T
=
∑
j
=
1
∞
‍
T
j in the space of regular operators on X. The inequality |
T
|
≤
∑
j
=
1
∞
‍
|
T
j
| ensures that T is an absolute integral operator with kernel k by [7, Theorem 3.3
.
6] and |
k
|
≤
∑
j
=
1
∞
‍
|
k
j
|
(
μ
⊗
μ
)-a.e..
Suppose first thatX is a real Banach function space. Each positive part T
j
+ of T
j, j
=
1,2
,
…, has an integral kernel k
j
+ such that
(21)
∫
Σ
〈
T
j
+
,
d
m
〉
=
∫
Σ
k
j
+
~
(
x
,
x
)
d
μ
(
x
)
.
By monotone convergence, there exists a set of full μ-measures on which
(22)
𝔼
(
k
+
∣
ℰ
n
⊗
ℰ
n
)
(
x
,
x
)
≤
∑
j
=
1
∞
‍
𝔼
(
k
j
+
∣
ℰ
n
⊗
ℰ
n
)
(
x
,
x
)
for each n
=
1,2
,
….. Taking the limsup and applying the monotone convergence theorem pointwise and under the sum show that
(23)
k
+
~
(
x
,
x
)
≤
∑
j
=
1
∞
‍
k
j
+
~
(
x
,
x
)
for μ-almost all x
∈
Σ and ∫
Σ
‍
k
+
~
(
x
,
x
)
<
∞. Applying the same argument to T
- and then the real and imaginary parts of T ensures that T
∈
ℭ
1
(
X
) and
(24)
∥
|
T
|
∥
+
∫
Σ
‍
〈
|
T
|
,
d
m
〉
≤
∑
j
=
1
∞
‍
(
∥
|
T
j
|
∥
+
∫
Σ
‍
〈
|
T
j
|
,
d
m
〉
)
.
Dedekind completeness is inherited from ℒ
r
(
X
) [7, Theorem 1.3
.
2] and L
1
(
μ
) [7, Example v, page 9]. The bound
(25)
|
∫
Σ
‍
〈
T
,
d
m
〉
|
≤
∥
|
T
|
∥
+
∫
Σ
‍
〈
|
T
|
,
d
m
〉
defines a positive continuous linear function on ℭ
1
(
X
).Example 2 (see [4, Example 3.2]).
There exist lattice-positive, compact linear operatorsT
:
L
2
(
[
0,1
]
)
→
L
2
(
[
0,1
]
) such that ∫
0
1
‍
〈
T
,
d
m
〉 is finite but T is not a trace class linear operator on the Hilbert space L
2
(
[
0,1
]
).
In particular, the Volterra operatorT is defined by
(26)
(
T
f
)
(
x
)
=
∫
0
x
‍
f
(
y
)
d
y
,
x
∈
[
0,1
]
,
for
f
∈
L
2
(
[
0,1
]
)
.
The (lattice) positive linear map T
:
L
2
(
[
0,1
]
)
→
L
2
(
[
0,1
]
) is a Hilbert-Schmidt operator but not trace class. Nevertheless,∫
0
1
‍
〈
T
,
d
m
〉
=
1
/
2.


## 4. Trace Class Operators

Proposition 3 (see [5, Theorem 3.1]).
IfT
:
L
2
(
μ
)
→
L
2
(
μ
) is a trace class linear operator, then, for any function k
:
Σ
×
Σ
→
ℂ such that  T
=
T
k, where
(27)
k
=
∑
j
=
1
∞
‍
ϕ
j
⊗
ψ
j
,
(
μ
⊗
μ
)
-
a
.
e
.
,
with  ∑
j
=
1
∞
‍
∥
ϕ
j
∥
2
∥
ψ
j
∥
2
<
∞, the equalities
(28)
tr
⁡
(
T
)
=
∫
Σ
‍
〈
T
,
d
m
〉
=
∫
Σ
‍
k
~
(
x
,
x
)
d
μ
(
x
)
hold.Ifk is continuous almost everywhere along the diagonal diag
⁡
(
Σ
×
Σ
), then k
~
(
x
,
x
)
=
k
(
x
,
x
) for μ-almost all x
∈
Σ [5, Theorem 2.4].For positive operators in the Hilbert space sense, we have the following sufficient condition for traceability. The operatoru
↦
χ
B
u, u
∈
L
2
(
μ
), for a Borel set B, is denoted by Q
(
B
).Proposition 4.
LetT
:
L
2
(
μ
)
→
L
2
(
μ
) be an absolute integral operator whose integral kernel is square integrable on any set of finite (
μ
⊗
μ
)-measure. If  (
T
u
,
u
)
≥
0 for all u
∈
L
2
(
μ
), then T is trace class if and only if  ∫
Σ
‍
〈
T
,
d
m
〉 is finite, and in this case
(29)
tr
⁡
(
T
)
=
∫
Σ
‍
〈
T
,
d
m
〉
.Proof.
The case whereT is assumed to be trace class is covered by Proposition 3 above. Suppose that T
:
L
2
(
μ
)
→
L
2
(
μ
) is an absolute integral operator such that (
T
u
,
u
)
≥
0 for all u
∈
L
2
(
μ
) and ∫
Σ
‍
〈
T
,
d
m
〉 is finite.
If the integral kernelk of T is square integrable on any set of finite (
μ
⊗
μ
)-measure, then for any Borel set B with μ
(
B
)
<
∞, the operator Q
(
B
)
T
Q
(
B
) is a positive Hilbert-Schmidt operator. If B
j
↑
Σ as j
→
∞, then
(30)
sup
⁡
j
sup
⁡
n
≥
m
𝔼
(
(
χ
B
j
⊗
χ
B
j
)
k
∣
ℰ
n
⊗
ℰ
n
)
(
x
,
x
)
=
sup
⁡
n
≥
m
𝔼
(
k
∣
ℰ
n
⊗
ℰ
n
)
(
x
,
x
)
by monotone convergence, so
(31)
sup
⁡
j
∫
Σ
‍
〈
Q
(
B
j
)
T
Q
(
B
j
)
,
d
m
〉
≤
∫
Σ
‍
〈
T
,
d
m
〉
.
By choosing B
j
=
∪
m
=
1
j
Σ
m for Σ
m
∈
𝒫
1 for m
,
j
=
1,2
…, we have
(32)
𝔼
(
(
χ
B
j
⊗
χ
B
j
)
k
∣
ℰ
n
⊗
ℰ
n
)
(
x
,
x
)
=
𝔼
(
k
∣
ℰ
n
⊗
ℰ
n
)
(
x
,
x
)
χ
B
j
(
x
)
for all n
,
j
=
1,2
…, so
(33)
∫
Σ
‍
〈
Q
(
B
j
)
T
Q
(
B
j
)
,
d
m
〉
=
∫
B
j
‍
〈
T
,
d
m
〉
⟶
∫
Σ
‍
〈
T
,
d
m
〉
as j
→
∞. According to [5, Theorem 4.3], Q
(
B
j
)
T
Q
(
B
j
) is trace class and
(34)
tr
⁡
(
Q
(
B
j
)
T
Q
(
B
j
)
)
=
∫
B
j
‍
〈
T
,
d
m
〉
.
For every u
∈
L
2
(
μ
), the inequality
(35)
(
Q
(
B
j
)
T
Q
(
B
j
)
u
,
u
)
≤
∥
u
∥
2
tr
⁡
(
Q
(
B
j
)
T
Q
(
B
j
)
)
≤
∥
u
∥
2
∫
Σ
‍
〈
T
,
d
m
〉
.
By polarisation, Q
(
B
j
)
T
Q
(
B
j
)
→
T in the weak operator topology as j
→
∞, so
(36)
|
tr
⁡
(
T
C
)
|
≤
∥
C
∥
lim
⁡
j
→
∞
|
tr
⁡
(
Q
(
B
j
)
T
Q
(
B
j
)
)
|
≤
∥
C
∥
∫
Σ
‍
〈
T
,
d
m
〉
for very finite rank operator C. By [1, Theorem 2.14], T is a trace class operator and an appeal to Proposition 3 gives (28).Proposition 5.
If(
Σ
,
ℬ
,
μ
) is an atomic measure space with countably many atoms, then ℭ
1
(
L
2
(
μ
)
)
=
𝒞
1
(
L
2
(
μ
)
) and
(37)
tr
⁡
(
T
)
=
∫
Σ
‍
〈
T
,
d
m
〉
,
T
∈
𝒞
1
(
L
2
(
μ
)
)
.


## 5. Lattice Properties

LetJ
:
Σ
→
diag
⁡
(
Σ
×
Σ
) be the diagonal embedding J
(
x
)
=
(
x
,
x
), x
∈
Σ. Let ν
=
(
μ
⊗
μ
)
+
μ
∘
J
-
1. If limsup
⁡
n
→
∞
𝔼
(
k
∣
ℰ
n
⊗
ℰ
n
) converges pointwise ν-a.e. and in L
1
(
ν
), then there exist scalars c
j and Borel sets C
j
,
D
j such that
(38)
∑
j
=
1
∞
‍
|
c
j
|
ν
(
C
j
×
D
j
)
<
∞
,
and we can write
(39)
k
(
x
,
y
)
=
∑
j
=
1
∞
‍
c
j
χ
C
j
×
D
j
(
x
,
y
)
for every x
,
y
∈
Σ such that ∑
j
=
1
∞
‍
|
c
j
|
χ
C
j
×
D
j
(
x
,
y
)
<
∞ and k
(
x
,
x
)
=
k
~
(
x
,
x
) for μ-almost all x
∈
Σ; see [9].Proposition 6.
LetT
:
X
→
X be a positive kernel operator. For any nonnegative μ-measurable functions V
1,  V
2, the equalities
(40)
∫
Σ
‍
〈
Q
(
V
2
)
T
Q
(
V
1
)
,
d
m
〉
=
∫
Σ
‍
〈
Q
(
V
1
V
2
)
T
,
d
m
〉
=
∫
Σ
‍
〈
T
Q
(
V
1
V
2
)
,
d
m
〉
of extended real numbers hold.
For any essentially boundedμ-measurable function V,
(41)
|
∫
Σ
‍
〈
Q
(
V
)
T
,
d
m
〉
|
≤
∥
V
∥
∞
∫
Σ
‍
〈
T
,
d
m
〉
∈
[
0
,
∞
]
.Proof.
If the kernelk of T has the representation (39), then for any sets W
1
,
W
2
∈
ℬ, we have
(42)
(
Q
(
W
1
)
k
Q
(
W
2
)
)
~
(
x
,
x
)
=
(
Q
(
W
1
)
Q
(
W
2
)
k
)
~
(
x
,
x
)
=
(
k
Q
(
W
1
)
Q
(
W
2
)
)
~
(
x
,
x
)
is equal to
(43)
∑
j
=
1
∞
‍
c
j
χ
C
j
∩
D
j
∩
W
1
∩
W
2
(
x
)
for μ-almost all x
∈
Σ. The result follows by linearity and approximating V
1 and V
2 by simple functions.It is well known that ifT is a trace class operator on a Hilbert space ℋ and B is any bounded linear operator on ℋ then B
T and T
B are also trace class operators (i.e., 𝒞
1
(
ℋ
) is anoperator ideal) and [1, Corollary 3.8]
(44)
tr
⁡
(
B
T
)
=
tr
⁡
(
T
B
)
.
By contrast, the space ℭ
1
(
L
2
(
μ
)
) is alattice ideal in ℒ
r
(
L
2
(
μ
)
). For T
∈
ℭ
1
(
L
2
(
μ
)
)
) and B
∈
ℒ
(
L
2
(
μ
)
), the operator B
T may not even be a kernel operator, but we have the following trace property.Proposition 7.
LetT
j
:
X
→
X, j
=
1,2, be positive kernel operators. Then the equalities
(45)
∫
Σ
‍
〈
T
1
T
2
,
d
m
〉
=
∫
Σ
‍
〈
T
2
T
1
,
d
m
〉
of extended real numbers hold.Proof.
Suppose that the kernelsk
j of T
j, j
=
1,2, have the representation (39).
Ifℰ
n, n
=
1,2
…, is an increasing sequence of sub-σ-algebras of ℬ such that the σ-algebra σ
(
k
j
) generated by k
j is contained in ∨
n
ℰ
n
⊗
ℰ
n for j
=
1,2, then
(46)
∫
Σ
‍
〈
T
1
T
2
,
d
m
〉
=
∫
Σ
‍
(
∫
Σ
‍
k
~
1
(
x
,
y
)
k
~
2
(
y
,
x
)
d
μ
(
y
)
)
d
μ
(
x
)
.
By the Fubini-Tonelli Theorem this is equal to
(47)
∫
Σ
‍
(
∫
Σ
‍
k
~
2
(
y
,
x
)
k
~
1
(
x
,
y
)
d
μ
(
x
)
)
d
μ
(
y
)
=
∫
Σ
‍
〈
T
2
T
1
,
d
m
〉
.We also note that a bilinear version of the Fubini-Tonelli Theorem holds.Let(
Ξ
,
ℰ
,
ν
) be a σ-finite measure space. For any function f
:
Ξ
→
ℒ
+
(
X
) such that ∫
Ξ
‍
∫
Σ
‍
〈
f
(
ξ
)
,
d
m
〉
d
ν
(
ξ
)
<
∞, we say that f is (
m
⊗
ν
)-integrable if for each u
∈
X, v
∈
X
′, the scalar function 〈
f
u
,
v
〉
:
ξ
↦
〈
f
(
ξ
)
u
,
v
〉 is ν-integrable and there exists T
∈
ℭ
1
(
X
) such that
(48)
∫
Ξ
‍
∫
Σ
‍
〈
f
(
ξ
)
,
d
m
〉
d
ν
(
ξ
)
=
∫
Σ
‍
〈
T
,
d
m
〉
,
(49)
∫
Ξ
‍
〈
f
(
ξ
)
u
,
v
〉
d
ν
=
〈
T
u
,
v
〉
for all u
∈
X, v
∈
X
′. Then we set
(50)
∫
Σ
×
Ξ
‍
〈
f
,
d
(
m
⊗
ν
)
〉
=
∫
Σ
‍
〈
T
,
d
m
〉
.
Because ℭ
1
(
X
) is a lattice ideal, for each A
∈
ℬ, there exists a positive operator ∫
A
‍
f
d
ν
∈
ℭ
1
(
X
) such that
(51)
〈
(
∫
A
‍
f
d
ν
)
u
,
v
〉
=
∫
A
‍
〈
f
(
ξ
)
u
,
v
〉
d
ν
≤
〈
T
u
,
v
〉
,
for all u
∈
X
+, v
∈
X
+
′.Remark 8.
For eachu
∈
X, v
∈
X
′, the tensor product u
⊗
v and T
↦
∫
Σ
‍
〈
T
,
d
m
〉 are continuous linear functionals on ℭ
1
(
X
), so it is natural to assume that both (48) and (49) hold.The following statement is a consequence of the definitions.Proposition 9.
Letf
:
Ξ
→
ℒ
+
(
X
) be a positive operator valued function such that f is (
m
⊗
ν
)-integrable.
Thenf
(
ξ
)
∈
ℭ
1
(
X
) for ν-almost all ξ
∈
Ξ, the scalar valued function ξ
↦
∫
Σ
‍
〈
f
(
ξ
)
,
d
m
〉 is ν-integrable, and the equalities
(52)
∫
Σ
×
Ξ
‍
〈
f
,
d
(
m
⊗
ν
)
〉
=
∫
Σ
‍
〈
∫
Ξ
‍
f
d
ν
,
d
m
〉
(53)
=
∫
Ξ
‍
∫
Σ
‍
〈
f
(
ξ
)
,
d
m
〉
d
ν
(
ξ
)
hold. Moreover, ∫
Σ
‍
〈
∫
A
‍
f
d
ν
,
d
m
〉
=
∫
A
‍
∫
Σ
‍
〈
f
(
ξ
)
,
d
m
〉
d
ν
(
ξ
) for every A
∈
ℬ.Proof.
Equation (52) is the definition of ∫
Σ
×
Ξ
‍
〈
f
,
d
(
m
⊗
ν
)
〉 and (53) is a reformulation of assumption (48). For ν-almost all ξ
∈
Ξ, we can find a martingale ℱ
ξ and a regularisation k
ξ
(
x
,
y
), x
,
y
∈
Σ, of the kernel associated with f
(
ξ
) such that
(54)
〈
(
∫
A
‍
f
d
ν
)
u
,
v
〉
=
∫
A
‍
∫
Σ
‍
∫
Σ
‍

k
ξ

(
x
,
y
)

u
(
x
)

v
¯

(
y
)

d
μ

(
x
)

d
μ

(
y
)

d
ν

(
ξ
)
for all A
∈
ℬ and u
∈
X, v
∈
X
′. Then, for each A
∈
ℬ, we have
(55)
∫
Σ
‍
〈
∫
A
‍
f
d
ν
,
d
m
〉
=
∫
A
‍
∫
Σ
‍
k
ξ
(
x
,
x
)
d
μ
(
x
)
d
ν
(
ξ
)
=
∫
A
‍
∫
Σ
‍
〈
f
(
ξ
)
,
d
m
〉
d
ν
(
ξ
)
by the scalar Fubini-Tonelli Theorem.The following result follows from the observation in Theorem1 that ℭ
1
(
X
) is a lattice ideal and an application of monotone convergence.Proposition 10.
LetM
:
ℬ
→
ℒ
+
(
X
) be a positive operator valued measure on a measurable space (
Ξ
,
ℬ
). If  ∫
Σ
‍
〈
M
(
Ξ
)
,
d
m
〉
<
∞, then the set function 〈
M
,
m
〉
:
A
↦
∫
Σ
‍
〈
M
(
A
)
,
d
m
〉, A
∈
ℬ, is a finite measure such that
(56)
∫
Σ
‍
〈
M
(
A
)
,
d
m
〉
≤
∫
Σ
‍
〈
M
(
Ξ
)
,
d
m
〉
,
A
∈
ℬ
,
∫
Σ
‍
〈
M
(
f
)
,
d
m
〉
=
∫
Ξ
‍
f
d
〈
M
,
m
〉
≤
∥
f
∥
∞
∫
Σ
‍
〈
M
(
Ξ
)
,
d
m
〉
for all ℬ-measurable f
:
Ξ
→
[
0
,
∞
].



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*Source: 629502-2014-04-14.xml*