Research Article Lattice Trace Operatorsdownloads.hindawi.com/journals/joper/2014/629502.pdf · Research Article Lattice Trace Operators BrianJefferies School of Mathematics,e University

Research ArticleLattice Trace Operators

Brian Jefferies

School of Mathematics The University of New South Wales NSW 2052 Australia

Correspondence should be addressed to Brian Jefferies bjefferiesunsweduau

Received 31 October 2013 Accepted 19 March 2014 Published 14 April 2014

Academic Editor Antun Milas

Copyright copy 2014 Brian JefferiesThis is an open access article distributed under the Creative CommonsAttribution License whichpermits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

A bounded linear operator 119879 on a Hilbert space H is trace class if its singular values are summable The trace class operatorson H form an operator ideal and in the case that H is finite-dimensional the trace tr(119879) of 119879 is given by sum119895 119886119895119895 for any matrixrepresentation 119886119894119895 of 119879 In applications of trace class operators to scattering theory and representation theory the subject iscomplicated by the fact that if 119896 is an integral kernel of the operator 119879 on the Hilbert space 1198712(120583) with 120583 a 120590-finite measure then119896(119909 119909)may not be defined because the diagonal (119909 119909)may be a set of (120583 otimes 120583)-measure zeroThe present note describes a class oflinear operators acting on a Banach function space 119883 which forms a lattice ideal of operators on 119883 rather than an operator idealbut coincides with the collection of hermitian positive trace class operators in the case of119883 = 119871

2(120583)

1 Introduction

A trace class operator 119860 on a separable Hilbert space H is acompact operator whose singular values 120582119895(119860) 119895 = 1 2 satisfy

1198601 =

infin

sum

119895=1

120582119895 (119860) lt infin (1)

The decreasing sequence 120582119895(119860)infin119895=1 consists of eigenvalues of

(119860lowast119860)

12 Equivalently 119860 is trace class if and only if for anyorthonormal basis ℎ119895

infin119895=1 of H the sum sum

infin119895=1 |(119860ℎ119895 ℎ119895)| is

finite The number

tr (119860) =infin

sum

119895=1

(119860ℎ119895 ℎ119895) (2)

is called the trace of119860 and is independent of the orthonormalbasis ℎ119895

infin119895=1 of H Lidskiirsquos equality asserts that tr (119860) is

actually the sum of the eigenvalues of the compact operator119879 [1 Theorem 37]

We refer to [1] for properties of trace class operatorsThe collection C1(H) of trace class operators on H is anoperator ideal and Banach space with the norm sdot 1 Thefollowing facts are worth noting in the case of the Hilbertspace 1198712([0 1]) with respect to Lebesgue measure on theinterval [0 1]

(a) If 119879 1198712([0 1]) rarr 119871

2([0 1]) is a trace class linear

operator then there exist 120601119895 120595119895 isin 1198712([0 1]) 119895 =

1 2 with suminfin119895=1 1206011198952

1205951198952lt infin and

(119879119891) (119909) = int

1

0

119896 (119909 119910) 119891 (119910) 119889119910 ae for 119891 isin 1198712([0 1])

(3)

where 119896 = suminfin119895=1 120601119895 otimes 120595119895 ae In particular 119879 is regular

and |119879| has an integral kernel |119896| le suminfin119895=1 |120601119895| otimes |120595119895|

Moreover

tr (119879) =infin

sum

119895=1

int

1

0

120601119895 (119909) 120595119895 (119909) 119889119909 (4)

(b) Suppose that 119879 1198712([0 1]) rarr 119871

2([0 1]) is a

regular linear operator defined by formula (3) for acontinuous function 119896 [0 1] times [0 1] rarr C If 119879 istrace class then int

1

0|119896(119909 119909)| 119889119909 lt infin and tr (119879) =

int1

0119896(119909 119909) 119889119909 [2 Theorem V311]

(c) Suppose that the function 119896 [0 1] times [0 1] rarr

C is continuous and positive definite that issum119899119895ℓ=1 119911119895119911ℓ119896(119909119895 119909ℓ) ge 0 for all 119911119895 isin C and 119909119895 isin [0 1]

119895 = 1 119899 and any 119899 = 1 2 Then 119896(119909 119909) ge 0 for

Hindawi Publishing CorporationJournal of OperatorsVolume 2014 Article ID 629502 6 pageshttpdxdoiorg1011552014629502

2 Journal of Operators

all 119909 isin [0 1] If int10119896(119909 119909) 119889119909 lt infin then there exists a

unique trace class operator defined by formula (3) [1Theorem 212]

Let (ΣB 120583) be a measure space The projective tensorproduct 1198712(120583)otimes1205871198712(120583) is the set of all sums

119896 =

infin

sum

119895=1

120601119895 otimes 120595119895 ae withinfin

sum

119895=1

10038171003817100381710038171003817120601119895

100381710038171003817100381710038172

10038171003817100381710038171003817120595119895

100381710038171003817100381710038172lt infin (5)

The norm of 119896 isin 1198712(120583)otimes120587119871

2(120583) is given by 119896120587 =

infsuminfin119895=1 1206011198952

1205951198952 where the infimum is taken over all

sums for which the representation (5) holds The Banachspace1198712(120583)otimes120587119871

2(120583) is actually the completion of the algebraic

tensor product 1198712(120583) otimes 1198712(120583) with respect to the projective

tensor product norm [3 Section 61]There is a one-to-one correspondence between the space

of trace class operators acting on 1198712(120583) and 1198712(120583)otimes1205871198712(120583) so

that the trace class operator 119879119896 has an integral kernel 119896 isin

1198712(120583)otimes120587119871

2(120583) If the integral kernel 119896 given by (5) has the

property that

119896 (119909 119910) =

infin

sum

119895=1

120601119895 (119909) 120595119895 (119910) (6)

for all 119909 119910 isin Σ such that the sum suminfin119895=1 |120601119895(119909)120595119895(119910)| is finite

then the equality

tr (119879119896) =infin

sum

119895=1

intΣ

120601119895 (119909) 120595119895 (119909) 119889120583 (119909) = intΣ

119896 (119909 119909) 119889120583 (119909)

(7)

holds Because the diagonal (119909 119909) 119909 isin Σ may be a set of(120583otimes120583)-measure zero in ΣtimesΣ it may be difficult to determinewhether or not a given integral kernel 119896 Σ times Σ rarr C hassuch a distinguished representation

The difficulty is addressed by Brislawn [4 5] [1 AppendixD] who shows that for a trace class operator 119879119896 119871

2(120583) rarr

1198712(120583) with integral kernel 119896 the equality

tr (119879119896) = intΣ

(119909 119909) 119889120583 (119909) (8)

holds The measure 120583 is supposed in [5] to be a 120590-finiteBorel measure on a second countable topological space Σ andthe regularised kernel is defined from 119896 by averaging withrespect to the product measure 120583otimes120583 Extending the result (c)ofMDuflo given above Brislawn [5Theorem43] shows thata hermitian positive Hilbert-Schmidt operator 119879119896 is a traceclass operator if and only if int

Σ(119909 119909) 119889120583(119909) lt infin

The present paper examines the space C1(119883) of absoluteintegral operators119879119896 119883 rarr 119883 defined on a Banach functionspace for which int

Σ|(119909 119909)| 119889120583(119909) lt infin Elements of C1(119883)

are called lattice trace operators because C1(119883) is a latticeideal in the Banach lattice of regular operators on119883 whereasthe collection C1(H) of trace class operators on a Hilbertspace H is an operator ideal in the Banach algebra L(H)

of all bounded linear operators on H The intersections ofC1(119883) and C1(119871

2(120583)) with the hermitian positive operators

on 1198712(120583) are equal for locally square integrable kernels seeProposition 4

The regularised kernel Σ times Σ rarr C of anabsolute integral operator 119879119896 is defined by adapting themethod of Brislawn [5] to positive operators with an integralkernel The generalised trace int

Σ(119909 119909) 119889120583(119909)may be viewed

alternatively as a bilinear integral intΣ⟨119879119896 119889119898⟩ with respect to

the measure 119898 119864 997891rarr 120594119864 119864 isin B Lattice trace operatorsare employed in the proof of the Cwikel-Lieb-Rosenbluminequality for dominated semigroups [6]

The basic definitions of Banach function spaces andoperators with an integral kernel which act upon them are setout in Section 2Themartingale regularisation of the integralkernel of an operator between Banach function spaces is setout in Section 3 and the connection with trace class operatorson 1198712(120583) is set out in Section 4

2 Banach Function Spaces andRegular Operators

Let Σ be a second countable topological space with Borel 120590-algebra B The diagonal diag(Σ times Σ) = (119909 119909) 119909 isin Σ is aclosed subset of theCartesian productΣtimesΣ Because the Borel120590-algebra ofΣtimesΣ is equal toBotimesB the diagonal diag (ΣtimesΣ)belongs to the 120590-algebraB otimesB

We suppose that (ΣB 120583) is a 120590-finite measure spaceThe space of all 120583-equivalence classes of Borel measurablescalar functions is denoted by 119871

0(120583) It is equipped with

the topology of convergence in 120583-measure over sets offinite measure and vector operations pointwise 120583-almosteverywhere Any Banach space 119883 that is a subspace of 1198710(120583)with the properties that

(i) 119883 is an order ideal of 1198710(120583) that is if 119892 isin 119883 119891 isin

1198710(120583) and |119891| le |119892| 120583-ae then 119891 isin 119883 and

(ii) if 119891 119892 isin 119883 and |119891| le |119892| 120583-ae then 119891119883 le 119892119883

is called a Banach function space (based on (ΣB 120583)) TheBanach function space 119883 is necessarily Dedekind completethat is every order bounded set has a sup and an inf [7 page116] The set of 119891 isin 119883 with 119891 ge 0 120583-ae is written as119883+

We suppose that 119883 contains the characteristic functionsof sets of finite measure and 119898 119878 997891rarr 120594119878 119878 isin S is 120590-additive in 119883 on sets of finite measure for example 119883 is 120590-order continuous see [8 Corollary 36] If 119883 is reflexive and120583 is finite and nonatomic then it follows from [8 Corollary323] that the values of the variation119881(119898) of119898 are either zeroor infinity In particular this is the case for119883 = 119871

119901([0 1])with

1 lt 119901 lt infinFollowing the account of Brislawn [5] we extend the

mapping 119879 997891rarr intΣ⟨119879 119889119898⟩ from the space C1(119871

2(120583)) of trace

class linear operators to a larger class of regular operators byrepresenting119879 by a ldquoregularisedrdquo kernel so that the collectionof regular operators 119879 for which int

Σ⟨|119879| 119889119898⟩ lt infin is a vector

sublattice of the Riesz space of regular operatorsmdasha propertynot necessarily enjoyed by the trace class operators

Journal of Operators 3

Let 119883 be a Banach function space based on the 120590-finite measure space (ΣB 120583) as above A continuous linearoperator 119879 119883 rarr 119883 is called positive if 119879 119883+ rarr 119883+The collection of all positive continuous linear operators on119883 is written as L+(119883) If the real and imaginary parts of acontinuous linear operator 119879 119883 rarr 119883 can be written as thedifference of two positive operators it is said to be regularThemodulus |119879| of a regular operator 119879 is defined by

|119879| 119891 = sup|119892|le119891

10038161003816100381610038161198791198921003816100381610038161003816 119891 isin 119883+ (9)

The collection of all regular operators is written asL119903(119883) andit is given the norm 119879 997891rarr |119879| 119879 isin L119903(119883) under which itbecomes a Banach lattice [7 Proposition 136]

A continuous linear operator 119879 119883 rarr 119883 has an integralkernel 119896 if 119896 ΣtimesΣ rarr C is a Borel measurable function suchthat 119879 = 119879119896 for the operator given by

(119879119896119891) (119909) = intΣ

119896 (119909 119910) 119891 (119910) 119889120583 (119910) 120583-almost all 119909 isin Σ

(10)

for each 119891 isin 119883 in the sense that intΣ|119896(119909 119910)119891(119910)|119889120583(119910) lt infin

for 120583-almost all 119909 isin Σ and 119909 997891rarr intΣ119896(119909 119910)119891(119910)119889120583(119910) is an

element of119883 Then 119896 is (120583 otimes 120583)-integrable on any product set119860times119861 with finite measure If 119879119896 ge 0 then 119896 ge 0(120583 otimes120583)-ae onΣ times Σ [7 Theorem 335]

A continuous linear operator 119879 is an absolute integraloperator if it has an integral kernel 119896 for which 119879|119896| isa bounded linear operator on 119871

2(120583) Then |119879119896| = 119879|119896|

[7 Theorem 335] The collection of all absolute integraloperators is a lattice ideal inL119903(119883) [7 Theorem 336]

Suppose that 119879 isin L(119883) has an integral kernel 119896 =

sum119899119895=1 119891119895 120594119860119895

that is an 119883-valued simple function with120583(119860119895) lt infin Then it is natural to view

intΣ⟨119879 119889119898⟩ =

119899

sum

119895=1

int119860119895

119891119895 119889120583 = intΣ

119896 (119909 119909) 119889120583 (119909) (11)

as a bilinear integral Our aim is to extend the integral to awider class of absolute integral operators

3 Martingale Regularisation

LetU = 1198801 1198802 be a countable base for the topology ofΣAn increasing family of countable partitionsP119899 119899 = 1 2 is defined recursively by setting P1 equal to a partition of Σinto Borel sets of finite 120583-measure and

P119895+1 = 119875 cap 119880119895 119875 119880119895 119875 isin P119895 (12)

for 119895 = 1 2 For each 119899 = 1 2 let E119899 be the 120590-algebrafor all countable unions of elements ofP119899

Suppose that 119896 ge 0 is a Borelmeasurable function definedonΣtimesΣ that is integrable on every set of finite (120583otimes120583)-measure

For each 119909 isin Σ the set 119880119899(119909) is the unique elementof the partition P119899 containing 119909 For each 119899 = 1 2

the conditional expectation 119896119899 = E(119896 | E119899 otimes E119899) can berepresented for 120583-almost all 119909 119910 isin Σ as

E (119896 | E119899 otimesE119899) (119909 119910)

=1

120583 (119880119899 (119909)) 120583 (119880119899 (119910))int119880119899(119909)

int119880119899(119910)

119896 (119904 119905) 119889120583 (119904) 119889120583 (119905)

= sum

119880119881isinP119899

int119880times119881

119896 119889 (120583 otimes 120583)

120583 (119880) 120583 (119881)120594119880times119881 (119909 119910)

(13)

LetN be the set of all 119909 isin Σ for which there exists 119899 = 1 2 such that 120583(119880119899(119909)) = 0 Then 120583(119880119898(119909)) = 0 for all 119898 gt 119899

because P119898 is a refinement of P119899 if 119898 gt 119899 Moreover N is120583-null becauseN sub ⋃

infin119899=1⋃119880 isin P119899 120583(119880) = 0 If 0 le 1198961 le

1198962(120583 otimes 120583)-ae then

E (1198961 | E119899 otimesE119899) (119909 119910)

le E (1198962 | E119899 otimesE119899) (119909 119910) 119899 = 1 2

(14)

for all (119909 119910) isin N119888timesN119888 In particular

E (1198961 | E119899 otimesE119899) (119909 119909)

le E (1198962 | E119899 otimesE119899) (119909 119909) 119899 = 1 2

(15)

for all 119909 isin N119888 Although diag (Σ times Σ)may be a set of (120583 otimes 120583)-measure zero the application of the conditional expectationoperators 119896 997891rarr E(119896 | E119899 otimes E119899) 119899 = 1 2 has the effectof regularising 119896 By an appeal to the martingale convergencetheorem 119896119899 converges (120583 otimes 120583)-ae to 119896 as 119899 rarr infin

Let (119909 119910) = lim sup119899rarrinfinE(119896 | E119899 otimes E119899)(119909 119910) for all119909 119910 isin Σ and we set

intΣ⟨119879 119889119898⟩ = int

Σ

(119909 119909) 119889120583 (119909) isin [0infin] (16)

If intΣ⟨119879 119889119898⟩ lt infin then 119860 997891rarr int

119860⟨119879 119889119898⟩ = int

119860(119909 x) 119889120583(119909)

119860 isin B is a finite measure For a regular operator119879 = 119879+minus119879minus

with positive and negative parts 119879plusmn we set

intΣ⟨119879 119889119898⟩ = int

Σ

⟨119879+ 119889119898⟩ minus intΣ

⟨119879minus 119889119898⟩ (17)

if one of the integrals on the right-hand side of the equation isfinite The integral int

Σ⟨119879 119889119898⟩ is defined by linearity for each

regular operator119879 119883 rarr 119883 It is clear from the constructionthat the collection of absolute integral operators 119879 such thatintΣ⟨|119879| 119889119898⟩ lt infin is a vector sublattice C1(119883) of the space of

regular operators on 1198712(120583) We call elements of C1(119883) latticetrace operators

Theorem 1 The space C1(119883) is a lattice ideal in L119903(119883) thatis if 119878 119879 isin L119903(119883) |119878| le |119879| and 119879 isin C1(119883) then 119878 isin C1(119883)Moreover C1(119883) is a Dedekind complete Banach lattice withthe norm

119879 997891997888rarr |119879| + intΣ⟨|119879| 119889119898⟩ 119879 isin C1 (119883) (18)

The map 119879 997891rarr intΣ⟨119879 119889119898⟩ is a positive continuous linear

function on C1(119883)


Proof If 119878 119879 isin L119903(119883) and |119878| le |119879| then 119878 is an absoluteintegral operator by [7 Theorem 336] If 1198961 is the integralkernel of 119878 and 1198962 is the integral kernel of 119879 then by [7Theorem 335] the inequality |1198961| le |1198962| holds (120583 otimes 120583)-aeThen |1(119909 119909)| le |2(119909 119909)| for 120583-almost all 119909 isin Σ so that

intΣ⟨|119878| 119889119898⟩ le int

Σ⟨|119879| 119889119898⟩ lt infin (19)

Hence 119878 isin C1(119883)To show thatC1(119883) is complete in its norm suppose that

infin

sum

119895=1

(10038171003817100381710038171003817

10038161003816100381610038161003816119879119895

10038161003816100381610038161003816

10038171003817100381710038171003817+ int

Σ

⟨10038161003816100381610038161003816119879119895

10038161003816100381610038161003816 119889119898⟩) lt infin (20)

for 119879119895 isin C1(119883) Then 119879 = suminfin119895=1 119879119895 in the space of regular

operators on119883 The inequality |119879| le suminfin119895=1 |119879119895| ensures that 119879

is an absolute integral operator with kernel 119896 by [7 Theorem336] and |119896| le suminfin

119895=1 |119896119895| (120583 otimes 120583)-aeSuppose first that119883 is a real Banach function space Each

positive part 119879+119895 of 119879119895 119895 = 1 2 has an integral kernel 119896+119895such that

intΣ

⟨119879+119895 119889119898⟩ = int

Σ

119896+119895 (119909 119909) 119889120583 (119909) (21)

By monotone convergence there exists a set of full 120583-measures on which

E (119896+| E119899 otimesE119899) (119909 119909) le

infin

sum

119895=1

E (119896+119895 | E119899 otimesE119899) (119909 119909)

(22)

for each 119899 = 1 2 Taking the limsup and applying themonotone convergence theorem pointwise and under thesum show that

119896+ (119909 119909) le

infin

sum

119895=1

119896+119895 (119909 119909) (23)

for 120583-almost all 119909 isin Σ and intΣ119896+(119909 119909) lt infin Applying the

same argument to 119879minus and then the real and imaginary partsof 119879 ensures that 119879 isin C1(119883) and

|119879| + intΣ⟨|119879| 119889119898⟩ le

infin

sum

119895=1

(10038171003817100381710038171003817

10038161003816100381610038161003816119879119895

10038161003816100381610038161003816

10038171003817100381710038171003817+ int

Σ

⟨10038161003816100381610038161003816119879119895

10038161003816100381610038161003816 119889119898⟩) (24)

Dedekind completeness is inherited fromL119903(119883) [7Theorem132] and 1198711(120583) [7 Example v page 9] The bound

1003816100381610038161003816100381610038161003816intΣ⟨119879 119889119898⟩

1003816100381610038161003816100381610038161003816le |119879| + int

Σ⟨|119879| 119889119898⟩ (25)

defines a positive continuous linear function on C1(119883)

Example 2 (see [4 Example 32]) There exist lattice-positivecompact linear operators 119879 119871

2([0 1]) rarr 119871

2([0 1]) such

thatint10⟨119879 119889119898⟩ is finite but119879 is not a trace class linear operator

on the Hilbert space 1198712([0 1])

In particular the Volterra operator 119879 is defined by

(119879119891) (119909) = int

119909

0

119891 (119910) 119889119910 119909 isin [0 1] for 119891 isin 1198712([0 1])

(26)

The (lattice) positive linear map 119879 1198712([0 1]) rarr

1198712([0 1]) is a Hilbert-Schmidt operator but not trace class

Neverthelessint10⟨119879 119889119898⟩ = 12

4 Trace Class Operators

Proposition 3 (see [5Theorem31]) If 119879 1198712(120583) rarr 119871

2(120583) is

a trace class linear operator then for any function 119896 ΣtimesΣ rarr

C such that 119879 = 119879119896 where

119896 =

infin

sum

119895=1

120601119895 otimes 120595119895 (120583 otimes 120583) -ae (27)

with suminfin119895=1 1206011198952

1205951198952lt infin the equalities

tr (119879) = intΣ⟨119879 119889119898⟩ = int

Σ

(119909 119909) 119889120583 (119909) (28)

hold

If 119896 is continuous almost everywhere along the diagonaldiag(Σ times Σ) then (119909 119909) = 119896(119909 119909) for 120583-almost all 119909 isin Σ [5Theorem 24]

For positive operators in the Hilbert space sense wehave the following sufficient condition for traceability Theoperator 119906 997891rarr 120594119861119906 119906 isin 119871

2(120583) for a Borel set 119861 is denoted by

119876(119861)

Proposition 4 Let 119879 1198712(120583) rarr 119871

2(120583) be an absolute

integral operator whose integral kernel is square integrable onany set of finite (120583 otimes 120583)-measure If (119879119906 119906) ge 0 for all 119906 isin

1198712(120583) then 119879 is trace class if and only if int

Σ⟨119879 119889119898⟩ is finite

and in this case

tr (119879) = intΣ⟨119879 119889119898⟩ (29)

Proof Thecasewhere119879 is assumed to be trace class is coveredby Proposition 3 above Suppose that 119879 119871

2(120583) rarr 119871

2(120583)

is an absolute integral operator such that (119879119906 119906) ge 0 for all119906 isin 119871

2(120583) and int

Σ⟨119879 119889119898⟩ is finite

If the integral kernel 119896 of 119879 is square integrable on anyset of finite (120583 otimes 120583)-measure then for any Borel set 119861 with120583(119861) lt infin the operator 119876(119861)119879119876(119861) is a positive Hilbert-Schmidt operator If 119861119895 uarr Σ as 119895 rarr infin then

sup119895

sup119899ge119898

E ((120594119861119895 otimes 120594119861119895) 119896 | E119899 otimesE119899) (119909 119909)

= sup119899ge119898

E (119896 | E119899 otimesE119899) (119909 119909)

(30)

by monotone convergence so

sup119895

intΣ

⟨119876 (119861119895) 119879119876 (119861119895) 119889119898⟩ le intΣ⟨119879 119889119898⟩ (31)


By choosing 119861119895 = cup119895

119898=1Σ119898 for Σ119898 isin P1 for119898 119895 = 1 2 wehave

E (( 120594119861119895 otimes 120594119861119895) 119896 | E119899 otimesE119899) (119909 119909)

= E (119896 | E119899 otimesE119899) (119909 119909) 120594119861119895(119909)

(32)

for all 119899 119895 = 1 2 so

intΣ

⟨119876 (119861119895) 119879119876 (119861119895) 119889119898⟩ = int119861119895

⟨119879 119889119898⟩ 997888rarr intΣ⟨119879 119889119898⟩

(33)

as 119895 rarr infin According to [5 Theorem 43] 119876(119861119895)119879119876(119861119895) istrace class and

tr (119876 (119861119895) 119879119876 (119861119895)) = int119861119895

⟨119879 119889119898⟩ (34)

For every 119906 isin 1198712(120583) the inequality

(119876 (119861119895) 119879119876 (119861119895) 119906 119906) le 1199062 tr (119876 (119861119895) 119879119876 (119861119895))

le 1199062intΣ⟨119879 119889119898⟩

(35)

By polarisation 119876(119861119895)119879119876(119861119895) rarr 119879 in the weak operatortopology as 119895 rarr infin so

|tr (119879119862)| le 119862 lim119895rarrinfin

10038161003816100381610038161003816tr (119876 (119861119895) 119879119876 (119861119895))

10038161003816100381610038161003816

le 119862intΣ⟨119879 119889119898⟩

(36)

for very finite rank operator 119862 By [1 Theorem 214] 119879 is atrace class operator and an appeal to Proposition 3 gives (28)

Proposition 5 If (ΣB 120583) is an atomic measure space withcountably many atoms then C1(119871

2(120583)) = C1(119871

2(120583)) and

tr (119879) = intΣ⟨119879 119889119898⟩ 119879 isin C1 (119871

2(120583)) (37)

5 Lattice Properties

Let 119869 Σ rarr diag(Σ times Σ) be the diagonal embedding 119869(119909) =(119909 119909) 119909 isin Σ Let ] = (120583 otimes 120583) + 120583 ∘ 119869

minus1 If lim sup119899rarrinfinE(119896 |

E119899 otimes E119899) converges pointwise ]-ae and in 1198711(]) then thereexist scalars 119888119895 and Borel sets 119862119895 119863119895 such that

infin

sum

119895=1

10038161003816100381610038161003816119888119895

10038161003816100381610038161003816] (119862119895 times 119863119895) lt infin (38)

and we can write

119896 (119909 119910) =

infin

sum

119895=1

119888119895 120594119862119895times119863119895(119909 119910) (39)

for every 119909 119910 isin Σ such that suminfin119895=1 |119888119895| 120594119862119895times119863119895

(119909 119910) lt infin and119896(119909 119909) = (119909 119909) for 120583-almost all 119909 isin Σ see [9]

Proposition 6 Let 119879 119883 rarr 119883 be a positive kerneloperator For any nonnegative 120583-measurable functions 1198811 1198812the equalities

intΣ

⟨119876 (1198812) 119879119876 (1198811) 119889119898⟩ = intΣ

⟨119876 (11988111198812) 119879 119889119898⟩

= intΣ

⟨119879119876 (11988111198812) 119889119898⟩

(40)

of extended real numbers holdFor any essentially bounded 120583-measurable function 119881

1003816100381610038161003816100381610038161003816intΣ

⟨119876 (119881) 119879 119889119898⟩

1003816100381610038161003816100381610038161003816le 119881infin int

Σ

⟨119879 119889119898⟩ isin [0infin] (41)

Proof If the kernel 119896 of 119879 has the representation (39) thenfor any sets11988211198822 isin B we have

(119876 (1198821) 119896119876 (1198822))sim(119909 119909) = (119876 (1198821) 119876 (1198822) 119896)

sim(119909 119909)

= (119896119876 (1198821) 119876 (1198822))sim(119909 119909)

(42)

is equal to

infin

sum

119895=1

119888119895 120594119862119895cap119863119895cap1198821cap1198822(119909) (43)

for 120583-almost all 119909 isin Σ The result follows by linearity andapproximating 1198811 and 1198812 by simple functions

It is well known that if 119879 is a trace class operator on aHilbert spaceH and 119861 is any bounded linear operator onHthen 119861119879 and 119879119861 are also trace class operators (ie C1(H) isan operator ideal) and [1 Corollary 38]

tr (119861119879) = tr (119879119861) (44)

By contrast the spaceC1(1198712(120583)) is a lattice ideal inL119903(119871

2(120583))

For 119879 isin C1(1198712(120583))) and 119861 isin L(119871

2(120583)) the operator 119861119879may

not even be a kernel operator but we have the following traceproperty

Proposition 7 Let 119879119895 119883 rarr 119883 119895 = 1 2 be positive kerneloperators Then the equalities

intΣ

⟨11987911198792 119889119898⟩ = intΣ

⟨11987921198791 119889119898⟩ (45)

of extended real numbers hold

Proof Suppose that the kernels 119896119895 of 119879119895 119895 = 1 2 have therepresentation (39)

If E119899 119899 = 1 2 is an increasing sequence of sub-120590-algebras ofB such that the 120590-algebra 120590(119896119895) generated by 119896119895 iscontained in or119899E119899 otimesE119899 for 119895 = 1 2 then

intΣ

⟨11987911198792 119889119898⟩ = intΣ

(intΣ

1 (119909 119910) 2 (119910 119909) 119889120583 (119910)) 119889120583 (119909)

(46)


By the Fubini-Tonelli Theorem this is equal to

intΣ

(intΣ

2 (119910 119909) 1 (119909 119910) 119889120583 (119909)) 119889120583 (119910) = intΣ

⟨11987921198791 119889119898⟩

(47)

We also note that a bilinear version of the Fubini-TonelliTheorem holds

Let (ΞE ]) be a 120590-finite measure space For any function119891 Ξ rarr L+(119883) such that int

ΞintΣ⟨119891(120585) 119889119898⟩ 119889](120585) lt infin we

say that 119891 is (119898 otimes ])-integrable if for each 119906 isin 119883 V isin 1198831015840 thescalar function ⟨119891119906 V⟩ 120585 997891rarr ⟨119891(120585)119906 V⟩ is ]-integrable andthere exists 119879 isin C1(119883) such that

intΞ

intΣ

⟨119891 (120585) 119889119898⟩ 119889] (120585) = intΣ⟨119879 119889119898⟩ (48)

intΞ

⟨119891 (120585) 119906 V⟩ 119889] = ⟨119879119906 V⟩ (49)

for all 119906 isin 119883 V isin 1198831015840 Then we set

intΣtimesΞ

⟨119891 119889 (119898 otimes ])⟩ = intΣ⟨119879 119889119898⟩ (50)

Because C1(119883) is a lattice ideal for each 119860 isin B there existsa positive operator int

119860119891 119889] isin C1(119883) such that

⟨(int119860

119891 119889]) 119906 V⟩ = int119860

⟨119891 (120585) 119906 V⟩ 119889] le ⟨119879119906 V⟩ (51)

for all 119906 isin 119883+ V isin 1198831015840+

Remark 8 For each 119906 isin 119883 V isin 1198831015840 the tensor product 119906 otimes

V and 119879 997891rarr intΣ⟨119879 119889119898⟩ are continuous linear functionals on

C1(119883) so it is natural to assume that both (48) and (49) hold

The following statement is a consequence of the defini-tions

Proposition 9 Let 119891 Ξ rarr L+(119883) be a positive operatorvalued function such that 119891 is (119898 otimes ])-integrable

Then119891(120585) isin C1(119883) for ]-almost all 120585 isin Ξ the scalar valuedfunction 120585 997891rarr int

Σ⟨f(120585) 119889119898⟩ is ]-integrable and the equalities

intΣtimesΞ

⟨119891 119889 (119898 otimes ])⟩ = intΣ

⟨intΞ

119891119889] 119889119898⟩ (52)

= intΞ

intΣ

⟨119891 (120585) 119889119898⟩ 119889] (120585) (53)

hold Moreover intΣ⟨int119860119891119889] 119889119898⟩ = int

119860intΣ⟨119891(120585) 119889119898⟩ 119889](120585) for

every 119860 isin B

Proof Equation (52) is the definition of intΣtimesΞ

⟨119891 119889(119898 otimes ])⟩and (53) is a reformulation of assumption (48) For ]-almostall 120585 isin Ξ we can find a martingale F120585 and a regularisation119896120585(119909 119910) 119909 119910 isin Σ of the kernel associated with119891(120585) such that

⟨(int119860

119891 119889]) 119906 V⟩

= int119860

intΣ

intΣ

119896120585(119909 119910)119906 (119909)V(119910)119889120583(119909)119889120583(119910)119889](120585)(54)

for all 119860 isin B and 119906 isin 119883 V isin 1198831015840 Then for each 119860 isin B wehave

intΣ

⟨int119860

119891 119889] 119889119898⟩ = int119860

intΣ

119896120585 (119909 119909) 119889120583 (119909) 119889] (120585)

= int119860

intΣ

⟨119891 (120585) 119889119898⟩ 119889] (120585)(55)

by the scalar Fubini-Tonelli Theorem

The following result follows from the observation inTheorem 1 that C1(119883) is a lattice ideal and an application ofmonotone convergence

Proposition 10 Let 119872 B rarr L+(119883) be a positiveoperator valued measure on a measurable space (ΞB) IfintΣ⟨119872(Ξ) 119889119898⟩ lt infin then the set function ⟨119872119898⟩ 119860 997891rarr

intΣ⟨119872(119860) 119889119898⟩ 119860 isin B is a finite measure such that

intΣ⟨119872 (119860) 119889119898⟩ le int

Σ⟨119872 (Ξ) 119889119898⟩ 119860 isin B

intΣ

⟨119872(119891) 119889119898⟩ = intΞ

119891 119889 ⟨119872119898⟩ le10038171003817100381710038171198911003817100381710038171003817infin int

Σ⟨119872 (Ξ) 119889119898⟩

(56)

for allB-measurable 119891 Ξ rarr [0infin]

Conflict of Interests

The author declares that there is no conflict of interestsregarding the publication of this paper

References

[1] B Simon Trace Ideals and Their Applications vol 120 ofMath-ematical Surveys and Monographs American MathematicalSociety Providence RI USA 2nd edition 2005

[2] M Duflo ldquoGeneralites sur les representations induitesRepresentations des Groupes de Lie ResolublesrdquoMonographiesde la Societe Mathematique de France vol 4 pp 93ndash119 1972

[3] H H Schaefer and M P Wolff Topological Vector Spaces vol3 of Graduate Texts in Mathematics Springer New York NYUSA 2nd edition 1999

[4] C Brislawn ldquoKernels of trace class operatorsrdquo Proceedings of theAmerican Mathematical Society vol 104 no 4 pp 1181ndash11901988

[5] C Brislawn ldquoTraceable integral kernels on countably generatedmeasure spacesrdquo Pacific Journal of Mathematics vol 150 no 2pp 229ndash240 1991

[6] B Jefferies ldquoThe CLR inequality for dominated semigroupsrdquo toappear inMathematical Physics Analysis and Geometry

[7] P Meyer-Nieberg Banach Lattices Universitext SpringerBerlin Germany 1991

[8] S Okada W J Ricker and E A Sanchez Perez OptimalDomain and Integral Extension of Operators Acting in FunctionSpaces vol 180 of Operator Theory Advances and ApplicationsBirkhauser Basel Switzerland 2008

[9] J Mikusinski The Bochner Integral vol 55 of Lehrbucher undMonographien aus dem Gebiete der exakten WissenschaftenMathematische Reihe Birkhauser Basel Switzerland 1978

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Algebra

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Stochastic AnalysisInternational Journal of


all 119909 isin [0 1] If int10119896(119909 119909) 119889119909 lt infin then there exists a

unique trace class operator defined by formula (3) [1Theorem 212]

Let (ΣB 120583) be a measure space The projective tensorproduct 1198712(120583)otimes1205871198712(120583) is the set of all sums

119896 =

infin

sum

119895=1

120601119895 otimes 120595119895 ae withinfin

sum

119895=1

10038171003817100381710038171003817120601119895

100381710038171003817100381710038172

10038171003817100381710038171003817120595119895

100381710038171003817100381710038172lt infin (5)

The norm of 119896 isin 1198712(120583)otimes120587119871

2(120583) is given by 119896120587 =

infsuminfin119895=1 1206011198952

1205951198952 where the infimum is taken over all

sums for which the representation (5) holds The Banachspace1198712(120583)otimes120587119871

2(120583) is actually the completion of the algebraic

tensor product 1198712(120583) otimes 1198712(120583) with respect to the projective

tensor product norm [3 Section 61]There is a one-to-one correspondence between the space

of trace class operators acting on 1198712(120583) and 1198712(120583)otimes1205871198712(120583) so

that the trace class operator 119879119896 has an integral kernel 119896 isin

1198712(120583)otimes120587119871

2(120583) If the integral kernel 119896 given by (5) has the

property that

119896 (119909 119910) =

infin

sum

119895=1

120601119895 (119909) 120595119895 (119910) (6)

for all 119909 119910 isin Σ such that the sum suminfin119895=1 |120601119895(119909)120595119895(119910)| is finite

then the equality

tr (119879119896) =infin

sum

119895=1

intΣ

120601119895 (119909) 120595119895 (119909) 119889120583 (119909) = intΣ

119896 (119909 119909) 119889120583 (119909)

(7)

holds Because the diagonal (119909 119909) 119909 isin Σ may be a set of(120583otimes120583)-measure zero in ΣtimesΣ it may be difficult to determinewhether or not a given integral kernel 119896 Σ times Σ rarr C hassuch a distinguished representation

The difficulty is addressed by Brislawn [4 5] [1 AppendixD] who shows that for a trace class operator 119879119896 119871

2(120583) rarr

1198712(120583) with integral kernel 119896 the equality

tr (119879119896) = intΣ

(119909 119909) 119889120583 (119909) (8)

holds The measure 120583 is supposed in [5] to be a 120590-finiteBorel measure on a second countable topological space Σ andthe regularised kernel is defined from 119896 by averaging withrespect to the product measure 120583otimes120583 Extending the result (c)ofMDuflo given above Brislawn [5Theorem43] shows thata hermitian positive Hilbert-Schmidt operator 119879119896 is a traceclass operator if and only if int

Σ(119909 119909) 119889120583(119909) lt infin

The present paper examines the space C1(119883) of absoluteintegral operators119879119896 119883 rarr 119883 defined on a Banach functionspace for which int

Σ|(119909 119909)| 119889120583(119909) lt infin Elements of C1(119883)

are called lattice trace operators because C1(119883) is a latticeideal in the Banach lattice of regular operators on119883 whereasthe collection C1(H) of trace class operators on a Hilbertspace H is an operator ideal in the Banach algebra L(H)

of all bounded linear operators on H The intersections ofC1(119883) and C1(119871

2(120583)) with the hermitian positive operators

on 1198712(120583) are equal for locally square integrable kernels seeProposition 4

The regularised kernel Σ times Σ rarr C of anabsolute integral operator 119879119896 is defined by adapting themethod of Brislawn [5] to positive operators with an integralkernel The generalised trace int

Σ(119909 119909) 119889120583(119909)may be viewed

alternatively as a bilinear integral intΣ⟨119879119896 119889119898⟩ with respect to

the measure 119898 119864 997891rarr 120594119864 119864 isin B Lattice trace operatorsare employed in the proof of the Cwikel-Lieb-Rosenbluminequality for dominated semigroups [6]

The basic definitions of Banach function spaces andoperators with an integral kernel which act upon them are setout in Section 2Themartingale regularisation of the integralkernel of an operator between Banach function spaces is setout in Section 3 and the connection with trace class operatorson 1198712(120583) is set out in Section 4

2 Banach Function Spaces andRegular Operators

Let Σ be a second countable topological space with Borel 120590-algebra B The diagonal diag(Σ times Σ) = (119909 119909) 119909 isin Σ is aclosed subset of theCartesian productΣtimesΣ Because the Borel120590-algebra ofΣtimesΣ is equal toBotimesB the diagonal diag (ΣtimesΣ)belongs to the 120590-algebraB otimesB

We suppose that (ΣB 120583) is a 120590-finite measure spaceThe space of all 120583-equivalence classes of Borel measurablescalar functions is denoted by 119871

0(120583) It is equipped with

the topology of convergence in 120583-measure over sets offinite measure and vector operations pointwise 120583-almosteverywhere Any Banach space 119883 that is a subspace of 1198710(120583)with the properties that

(i) 119883 is an order ideal of 1198710(120583) that is if 119892 isin 119883 119891 isin

1198710(120583) and |119891| le |119892| 120583-ae then 119891 isin 119883 and

(ii) if 119891 119892 isin 119883 and |119891| le |119892| 120583-ae then 119891119883 le 119892119883

is called a Banach function space (based on (ΣB 120583)) TheBanach function space 119883 is necessarily Dedekind completethat is every order bounded set has a sup and an inf [7 page116] The set of 119891 isin 119883 with 119891 ge 0 120583-ae is written as119883+

We suppose that 119883 contains the characteristic functionsof sets of finite measure and 119898 119878 997891rarr 120594119878 119878 isin S is 120590-additive in 119883 on sets of finite measure for example 119883 is 120590-order continuous see [8 Corollary 36] If 119883 is reflexive and120583 is finite and nonatomic then it follows from [8 Corollary323] that the values of the variation119881(119898) of119898 are either zeroor infinity In particular this is the case for119883 = 119871

119901([0 1])with

1 lt 119901 lt infinFollowing the account of Brislawn [5] we extend the

mapping 119879 997891rarr intΣ⟨119879 119889119898⟩ from the space C1(119871

2(120583)) of trace

class linear operators to a larger class of regular operators byrepresenting119879 by a ldquoregularisedrdquo kernel so that the collectionof regular operators 119879 for which int

Σ⟨|119879| 119889119898⟩ lt infin is a vector

sublattice of the Riesz space of regular operatorsmdasha propertynot necessarily enjoyed by the trace class operators



|119879| 119891 = sup|119892|le119891

10038161003816100381610038161198791198921003816100381610038161003816 119891 isin 119883+ (9)



(119879119896119891) (119909) = intΣ

119896 (119909 119910) 119891 (119910) 119889120583 (119910) 120583-almost all 119909 isin Σ

(10)





2(120583) Then |119879119896| = 119879|119896|



sum119899119895=1 119891119895 120594119860119895


intΣ⟨119879 119889119898⟩ =

119899

sum

119895=1

int119860119895

119891119895 119889120583 = intΣ

119896 (119909 119909) 119889120583 (119909) (11)




P119895+1 = 119875 cap 119880119895 119875 119880119895 119875 isin P119895 (12)





E (119896 | E119899 otimesE119899) (119909 119910)

=1

120583 (119880119899 (119909)) 120583 (119880119899 (119910))int119880119899(119909)

int119880119899(119910)

119896 (119904 119905) 119889120583 (119904) 119889120583 (119905)

= sum

119880119881isinP119899


119896 119889 (120583 otimes 120583)

120583 (119880) 120583 (119881)120594119880times119881 (119909 119910)

(13)




1198962(120583 otimes 120583)-ae then

E (1198961 | E119899 otimesE119899) (119909 119910)

le E (1198962 | E119899 otimesE119899) (119909 119910) 119899 = 1 2

(14)


E (1198961 | E119899 otimesE119899) (119909 119909)

le E (1198962 | E119899 otimesE119899) (119909 119909) 119899 = 1 2

(15)



intΣ⟨119879 119889119898⟩ = int

Σ

(119909 119909) 119889120583 (119909) isin [0infin] (16)


119860⟨119879 119889119898⟩ = int

119860(119909 x) 119889120583(119909)



intΣ⟨119879 119889119898⟩ = int

Σ

⟨119879+ 119889119898⟩ minus intΣ

⟨119879minus 119889119898⟩ (17)






119879 997891997888rarr |119879| + intΣ⟨|119879| 119889119898⟩ 119879 isin C1 (119883) (18)





intΣ⟨|119878| 119889119898⟩ le int

Σ⟨|119879| 119889119898⟩ lt infin (19)


infin

sum

119895=1

(10038171003817100381710038171003817

10038161003816100381610038161003816119879119895

10038161003816100381610038161003816

10038171003817100381710038171003817+ int

Σ

⟨10038161003816100381610038161003816119879119895

10038161003816100381610038161003816 119889119898⟩) lt infin (20)






intΣ

⟨119879+119895 119889119898⟩ = int

Σ

119896+119895 (119909 119909) 119889120583 (119909) (21)


E (119896+| E119899 otimesE119899) (119909 119909) le

infin

sum

119895=1

E (119896+119895 | E119899 otimesE119899) (119909 119909)

(22)


119896+ (119909 119909) le

infin

sum

119895=1

119896+119895 (119909 119909) (23)



|119879| + intΣ⟨|119879| 119889119898⟩ le

infin

sum

119895=1

(10038171003817100381710038171003817

10038161003816100381610038161003816119879119895

10038161003816100381610038161003816

10038171003817100381710038171003817+ int

Σ

⟨10038161003816100381610038161003816119879119895

10038161003816100381610038161003816 119889119898⟩) (24)


1003816100381610038161003816100381610038161003816intΣ⟨119879 119889119898⟩

1003816100381610038161003816100381610038161003816le |119879| + int

Σ⟨|119879| 119889119898⟩ (25)



2([0 1]) rarr 119871

2([0 1]) such




(119879119891) (119909) = int

119909

0

119891 (119910) 119889119910 119909 isin [0 1] for 119891 isin 1198712([0 1])

(26)






2(120583) is


C such that 119879 = 119879119896 where

119896 =

infin

sum

119895=1

120601119895 otimes 120595119895 (120583 otimes 120583) -ae (27)

with suminfin119895=1 1206011198952


tr (119879) = intΣ⟨119879 119889119898⟩ = int

Σ

(119909 119909) 119889120583 (119909) (28)

hold




119876(119861)





Σ⟨119879 119889119898⟩ is finite

and in this case

tr (119879) = intΣ⟨119879 119889119898⟩ (29)


2(120583) rarr 119871

2(120583)


2(120583) and int

Σ⟨119879 119889119898⟩ is finite


sup119895

sup119899ge119898


= sup119899ge119898

E (119896 | E119899 otimesE119899) (119909 119909)

(30)


sup119895

intΣ

⟨119876 (119861119895) 119879119876 (119861119895) 119889119898⟩ le intΣ⟨119879 119889119898⟩ (31)


By choosing 119861119895 = cup119895



= E (119896 | E119899 otimesE119899) (119909 119909) 120594119861119895(119909)

(32)

for all 119899 119895 = 1 2 so

intΣ

⟨119876 (119861119895) 119879119876 (119861119895) 119889119898⟩ = int119861119895

⟨119879 119889119898⟩ 997888rarr intΣ⟨119879 119889119898⟩

(33)


tr (119876 (119861119895) 119879119876 (119861119895)) = int119861119895

⟨119879 119889119898⟩ (34)


(119876 (119861119895) 119879119876 (119861119895) 119906 119906) le 1199062 tr (119876 (119861119895) 119879119876 (119861119895))

le 1199062intΣ⟨119879 119889119898⟩

(35)



10038161003816100381610038161003816tr (119876 (119861119895) 119879119876 (119861119895))

10038161003816100381610038161003816

le 119862intΣ⟨119879 119889119898⟩

(36)



2(120583)) = C1(119871

2(120583)) and

tr (119879) = intΣ⟨119879 119889119898⟩ 119879 isin C1 (119871

2(120583)) (37)





infin

sum

119895=1

10038161003816100381610038161003816119888119895

10038161003816100381610038161003816] (119862119895 times 119863119895) lt infin (38)

and we can write

119896 (119909 119910) =

infin

sum

119895=1

119888119895 120594119862119895times119863119895(119909 119910) (39)




intΣ

⟨119876 (1198812) 119879119876 (1198811) 119889119898⟩ = intΣ

⟨119876 (11988111198812) 119879 119889119898⟩

= intΣ

⟨119879119876 (11988111198812) 119889119898⟩

(40)


1003816100381610038161003816100381610038161003816intΣ

⟨119876 (119881) 119879 119889119898⟩

1003816100381610038161003816100381610038161003816le 119881infin int

Σ

⟨119879 119889119898⟩ isin [0infin] (41)


(119876 (1198821) 119896119876 (1198822))sim(119909 119909) = (119876 (1198821) 119876 (1198822) 119896)

sim(119909 119909)

= (119896119876 (1198821) 119876 (1198822))sim(119909 119909)

(42)

is equal to

infin

sum

119895=1

119888119895 120594119862119895cap119863119895cap1198821cap1198822(119909) (43)



tr (119861119879) = tr (119879119861) (44)


2(120583))


2(120583)) the operator 119861119879may



intΣ

⟨11987911198792 119889119898⟩ = intΣ

⟨11987921198791 119889119898⟩ (45)




intΣ

⟨11987911198792 119889119898⟩ = intΣ

(intΣ

1 (119909 119910) 2 (119910 119909) 119889120583 (119910)) 119889120583 (119909)

(46)



intΣ

(intΣ

2 (119910 119909) 1 (119909 119910) 119889120583 (119909)) 119889120583 (119910) = intΣ

⟨11987921198791 119889119898⟩

(47)



ΞintΣ⟨119891(120585) 119889119898⟩ 119889](120585) lt infin we


intΞ

intΣ

⟨119891 (120585) 119889119898⟩ 119889] (120585) = intΣ⟨119879 119889119898⟩ (48)

intΞ

⟨119891 (120585) 119906 V⟩ 119889] = ⟨119879119906 V⟩ (49)


intΣtimesΞ

⟨119891 119889 (119898 otimes ])⟩ = intΣ⟨119879 119889119898⟩ (50)


119860119891 119889] isin C1(119883) such that

⟨(int119860

119891 119889]) 119906 V⟩ = int119860

⟨119891 (120585) 119906 V⟩ 119889] le ⟨119879119906 V⟩ (51)









intΣtimesΞ

⟨119891 119889 (119898 otimes ])⟩ = intΣ

⟨intΞ

119891119889] 119889119898⟩ (52)

= intΞ

intΣ

⟨119891 (120585) 119889119898⟩ 119889] (120585) (53)


119860intΣ⟨119891(120585) 119889119898⟩ 119889](120585) for

every 119860 isin B



⟨(int119860

119891 119889]) 119906 V⟩

= int119860

intΣ

intΣ

119896120585(119909 119910)119906 (119909)V(119910)119889120583(119909)119889120583(119910)119889](120585)(54)


intΣ

⟨int119860

119891 119889] 119889119898⟩ = int119860

intΣ

119896120585 (119909 119909) 119889120583 (119909) 119889] (120585)

= int119860

intΣ

⟨119891 (120585) 119889119898⟩ 119889] (120585)(55)





intΣ⟨119872 (119860) 119889119898⟩ le int

Σ⟨119872 (Ξ) 119889119898⟩ 119860 isin B

intΣ

⟨119872(119891) 119889119898⟩ = intΞ

119891 119889 ⟨119872119898⟩ le10038171003817100381710038171198911003817100381710038171003817infin int

Σ⟨119872 (Ξ) 119889119898⟩

(56)




References

















Volume 2014




Journal of











Journal of


Function Spaces






Algebra











|119879| 119891 = sup|119892|le119891

10038161003816100381610038161198791198921003816100381610038161003816 119891 isin 119883+ (9)



(119879119896119891) (119909) = intΣ

119896 (119909 119910) 119891 (119910) 119889120583 (119910) 120583-almost all 119909 isin Σ

(10)





2(120583) Then |119879119896| = 119879|119896|



sum119899119895=1 119891119895 120594119860119895


intΣ⟨119879 119889119898⟩ =

119899

sum

119895=1

int119860119895

119891119895 119889120583 = intΣ

119896 (119909 119909) 119889120583 (119909) (11)




P119895+1 = 119875 cap 119880119895 119875 119880119895 119875 isin P119895 (12)





E (119896 | E119899 otimesE119899) (119909 119910)

=1

120583 (119880119899 (119909)) 120583 (119880119899 (119910))int119880119899(119909)

int119880119899(119910)

119896 (119904 119905) 119889120583 (119904) 119889120583 (119905)

= sum

119880119881isinP119899


119896 119889 (120583 otimes 120583)

120583 (119880) 120583 (119881)120594119880times119881 (119909 119910)

(13)




1198962(120583 otimes 120583)-ae then

E (1198961 | E119899 otimesE119899) (119909 119910)

le E (1198962 | E119899 otimesE119899) (119909 119910) 119899 = 1 2

(14)


E (1198961 | E119899 otimesE119899) (119909 119909)

le E (1198962 | E119899 otimesE119899) (119909 119909) 119899 = 1 2

(15)



intΣ⟨119879 119889119898⟩ = int

Σ

(119909 119909) 119889120583 (119909) isin [0infin] (16)


119860⟨119879 119889119898⟩ = int

119860(119909 x) 119889120583(119909)



intΣ⟨119879 119889119898⟩ = int

Σ

⟨119879+ 119889119898⟩ minus intΣ

⟨119879minus 119889119898⟩ (17)






119879 997891997888rarr |119879| + intΣ⟨|119879| 119889119898⟩ 119879 isin C1 (119883) (18)





intΣ⟨|119878| 119889119898⟩ le int

Σ⟨|119879| 119889119898⟩ lt infin (19)


infin

sum

119895=1

(10038171003817100381710038171003817

10038161003816100381610038161003816119879119895

10038161003816100381610038161003816

10038171003817100381710038171003817+ int

Σ

⟨10038161003816100381610038161003816119879119895

10038161003816100381610038161003816 119889119898⟩) lt infin (20)






intΣ

⟨119879+119895 119889119898⟩ = int

Σ

119896+119895 (119909 119909) 119889120583 (119909) (21)


E (119896+| E119899 otimesE119899) (119909 119909) le

infin

sum

119895=1

E (119896+119895 | E119899 otimesE119899) (119909 119909)

(22)


119896+ (119909 119909) le

infin

sum

119895=1

119896+119895 (119909 119909) (23)



|119879| + intΣ⟨|119879| 119889119898⟩ le

infin

sum

119895=1

(10038171003817100381710038171003817

10038161003816100381610038161003816119879119895

10038161003816100381610038161003816

10038171003817100381710038171003817+ int

Σ

⟨10038161003816100381610038161003816119879119895

10038161003816100381610038161003816 119889119898⟩) (24)


1003816100381610038161003816100381610038161003816intΣ⟨119879 119889119898⟩

1003816100381610038161003816100381610038161003816le |119879| + int

Σ⟨|119879| 119889119898⟩ (25)



2([0 1]) rarr 119871

2([0 1]) such




(119879119891) (119909) = int

119909

0

119891 (119910) 119889119910 119909 isin [0 1] for 119891 isin 1198712([0 1])

(26)






2(120583) is


C such that 119879 = 119879119896 where

119896 =

infin

sum

119895=1

120601119895 otimes 120595119895 (120583 otimes 120583) -ae (27)

with suminfin119895=1 1206011198952


tr (119879) = intΣ⟨119879 119889119898⟩ = int

Σ

(119909 119909) 119889120583 (119909) (28)

hold




119876(119861)





Σ⟨119879 119889119898⟩ is finite

and in this case

tr (119879) = intΣ⟨119879 119889119898⟩ (29)


2(120583) rarr 119871

2(120583)


2(120583) and int

Σ⟨119879 119889119898⟩ is finite


sup119895

sup119899ge119898


= sup119899ge119898

E (119896 | E119899 otimesE119899) (119909 119909)

(30)


sup119895

intΣ

⟨119876 (119861119895) 119879119876 (119861119895) 119889119898⟩ le intΣ⟨119879 119889119898⟩ (31)


By choosing 119861119895 = cup119895



= E (119896 | E119899 otimesE119899) (119909 119909) 120594119861119895(119909)

(32)

for all 119899 119895 = 1 2 so

intΣ

⟨119876 (119861119895) 119879119876 (119861119895) 119889119898⟩ = int119861119895

⟨119879 119889119898⟩ 997888rarr intΣ⟨119879 119889119898⟩

(33)


tr (119876 (119861119895) 119879119876 (119861119895)) = int119861119895

⟨119879 119889119898⟩ (34)


(119876 (119861119895) 119879119876 (119861119895) 119906 119906) le 1199062 tr (119876 (119861119895) 119879119876 (119861119895))

le 1199062intΣ⟨119879 119889119898⟩

(35)



10038161003816100381610038161003816tr (119876 (119861119895) 119879119876 (119861119895))

10038161003816100381610038161003816

le 119862intΣ⟨119879 119889119898⟩

(36)



2(120583)) = C1(119871

2(120583)) and

tr (119879) = intΣ⟨119879 119889119898⟩ 119879 isin C1 (119871

2(120583)) (37)





infin

sum

119895=1

10038161003816100381610038161003816119888119895

10038161003816100381610038161003816] (119862119895 times 119863119895) lt infin (38)

and we can write

119896 (119909 119910) =

infin

sum

119895=1

119888119895 120594119862119895times119863119895(119909 119910) (39)




intΣ

⟨119876 (1198812) 119879119876 (1198811) 119889119898⟩ = intΣ

⟨119876 (11988111198812) 119879 119889119898⟩

= intΣ

⟨119879119876 (11988111198812) 119889119898⟩

(40)


1003816100381610038161003816100381610038161003816intΣ

⟨119876 (119881) 119879 119889119898⟩

1003816100381610038161003816100381610038161003816le 119881infin int

Σ

⟨119879 119889119898⟩ isin [0infin] (41)


(119876 (1198821) 119896119876 (1198822))sim(119909 119909) = (119876 (1198821) 119876 (1198822) 119896)

sim(119909 119909)

= (119896119876 (1198821) 119876 (1198822))sim(119909 119909)

(42)

is equal to

infin

sum

119895=1

119888119895 120594119862119895cap119863119895cap1198821cap1198822(119909) (43)



tr (119861119879) = tr (119879119861) (44)


2(120583))


2(120583)) the operator 119861119879may



intΣ

⟨11987911198792 119889119898⟩ = intΣ

⟨11987921198791 119889119898⟩ (45)




intΣ

⟨11987911198792 119889119898⟩ = intΣ

(intΣ

1 (119909 119910) 2 (119910 119909) 119889120583 (119910)) 119889120583 (119909)

(46)



intΣ

(intΣ

2 (119910 119909) 1 (119909 119910) 119889120583 (119909)) 119889120583 (119910) = intΣ

⟨11987921198791 119889119898⟩

(47)



ΞintΣ⟨119891(120585) 119889119898⟩ 119889](120585) lt infin we


intΞ

intΣ

⟨119891 (120585) 119889119898⟩ 119889] (120585) = intΣ⟨119879 119889119898⟩ (48)

intΞ

⟨119891 (120585) 119906 V⟩ 119889] = ⟨119879119906 V⟩ (49)


intΣtimesΞ

⟨119891 119889 (119898 otimes ])⟩ = intΣ⟨119879 119889119898⟩ (50)


119860119891 119889] isin C1(119883) such that

⟨(int119860

119891 119889]) 119906 V⟩ = int119860

⟨119891 (120585) 119906 V⟩ 119889] le ⟨119879119906 V⟩ (51)









intΣtimesΞ

⟨119891 119889 (119898 otimes ])⟩ = intΣ

⟨intΞ

119891119889] 119889119898⟩ (52)

= intΞ

intΣ

⟨119891 (120585) 119889119898⟩ 119889] (120585) (53)


119860intΣ⟨119891(120585) 119889119898⟩ 119889](120585) for

every 119860 isin B



⟨(int119860

119891 119889]) 119906 V⟩

= int119860

intΣ

intΣ

119896120585(119909 119910)119906 (119909)V(119910)119889120583(119909)119889120583(119910)119889](120585)(54)


intΣ

⟨int119860

119891 119889] 119889119898⟩ = int119860

intΣ

119896120585 (119909 119909) 119889120583 (119909) 119889] (120585)

= int119860

intΣ

⟨119891 (120585) 119889119898⟩ 119889] (120585)(55)





intΣ⟨119872 (119860) 119889119898⟩ le int

Σ⟨119872 (Ξ) 119889119898⟩ 119860 isin B

intΣ

⟨119872(119891) 119889119898⟩ = intΞ

119891 119889 ⟨119872119898⟩ le10038171003817100381710038171198911003817100381710038171003817infin int

Σ⟨119872 (Ξ) 119889119898⟩

(56)




References

















Volume 2014




Journal of











Journal of


Function Spaces






Algebra











intΣ⟨|119878| 119889119898⟩ le int

Σ⟨|119879| 119889119898⟩ lt infin (19)


infin

sum

119895=1

(10038171003817100381710038171003817

10038161003816100381610038161003816119879119895

10038161003816100381610038161003816

10038171003817100381710038171003817+ int

Σ

⟨10038161003816100381610038161003816119879119895

10038161003816100381610038161003816 119889119898⟩) lt infin (20)






intΣ

⟨119879+119895 119889119898⟩ = int

Σ

119896+119895 (119909 119909) 119889120583 (119909) (21)


E (119896+| E119899 otimesE119899) (119909 119909) le

infin

sum

119895=1

E (119896+119895 | E119899 otimesE119899) (119909 119909)

(22)


119896+ (119909 119909) le

infin

sum

119895=1

119896+119895 (119909 119909) (23)



|119879| + intΣ⟨|119879| 119889119898⟩ le

infin

sum

119895=1

(10038171003817100381710038171003817

10038161003816100381610038161003816119879119895

10038161003816100381610038161003816

10038171003817100381710038171003817+ int

Σ

⟨10038161003816100381610038161003816119879119895

10038161003816100381610038161003816 119889119898⟩) (24)


1003816100381610038161003816100381610038161003816intΣ⟨119879 119889119898⟩

1003816100381610038161003816100381610038161003816le |119879| + int

Σ⟨|119879| 119889119898⟩ (25)



2([0 1]) rarr 119871

2([0 1]) such




(119879119891) (119909) = int

119909

0

119891 (119910) 119889119910 119909 isin [0 1] for 119891 isin 1198712([0 1])

(26)






2(120583) is


C such that 119879 = 119879119896 where

119896 =

infin

sum

119895=1

120601119895 otimes 120595119895 (120583 otimes 120583) -ae (27)

with suminfin119895=1 1206011198952


tr (119879) = intΣ⟨119879 119889119898⟩ = int

Σ

(119909 119909) 119889120583 (119909) (28)

hold




119876(119861)





Σ⟨119879 119889119898⟩ is finite

and in this case

tr (119879) = intΣ⟨119879 119889119898⟩ (29)


2(120583) rarr 119871

2(120583)


2(120583) and int

Σ⟨119879 119889119898⟩ is finite


sup119895

sup119899ge119898


= sup119899ge119898

E (119896 | E119899 otimesE119899) (119909 119909)

(30)


sup119895

intΣ

⟨119876 (119861119895) 119879119876 (119861119895) 119889119898⟩ le intΣ⟨119879 119889119898⟩ (31)


By choosing 119861119895 = cup119895



= E (119896 | E119899 otimesE119899) (119909 119909) 120594119861119895(119909)

(32)

for all 119899 119895 = 1 2 so

intΣ

⟨119876 (119861119895) 119879119876 (119861119895) 119889119898⟩ = int119861119895

⟨119879 119889119898⟩ 997888rarr intΣ⟨119879 119889119898⟩

(33)


tr (119876 (119861119895) 119879119876 (119861119895)) = int119861119895

⟨119879 119889119898⟩ (34)


(119876 (119861119895) 119879119876 (119861119895) 119906 119906) le 1199062 tr (119876 (119861119895) 119879119876 (119861119895))

le 1199062intΣ⟨119879 119889119898⟩

(35)



10038161003816100381610038161003816tr (119876 (119861119895) 119879119876 (119861119895))

10038161003816100381610038161003816

le 119862intΣ⟨119879 119889119898⟩

(36)



2(120583)) = C1(119871

2(120583)) and

tr (119879) = intΣ⟨119879 119889119898⟩ 119879 isin C1 (119871

2(120583)) (37)





infin

sum

119895=1

10038161003816100381610038161003816119888119895

10038161003816100381610038161003816] (119862119895 times 119863119895) lt infin (38)

and we can write

119896 (119909 119910) =

infin

sum

119895=1

119888119895 120594119862119895times119863119895(119909 119910) (39)




intΣ

⟨119876 (1198812) 119879119876 (1198811) 119889119898⟩ = intΣ

⟨119876 (11988111198812) 119879 119889119898⟩

= intΣ

⟨119879119876 (11988111198812) 119889119898⟩

(40)


1003816100381610038161003816100381610038161003816intΣ

⟨119876 (119881) 119879 119889119898⟩

1003816100381610038161003816100381610038161003816le 119881infin int

Σ

⟨119879 119889119898⟩ isin [0infin] (41)


(119876 (1198821) 119896119876 (1198822))sim(119909 119909) = (119876 (1198821) 119876 (1198822) 119896)

sim(119909 119909)

= (119896119876 (1198821) 119876 (1198822))sim(119909 119909)

(42)

is equal to

infin

sum

119895=1

119888119895 120594119862119895cap119863119895cap1198821cap1198822(119909) (43)



tr (119861119879) = tr (119879119861) (44)


2(120583))


2(120583)) the operator 119861119879may



intΣ

⟨11987911198792 119889119898⟩ = intΣ

⟨11987921198791 119889119898⟩ (45)




intΣ

⟨11987911198792 119889119898⟩ = intΣ

(intΣ

1 (119909 119910) 2 (119910 119909) 119889120583 (119910)) 119889120583 (119909)

(46)



intΣ

(intΣ

2 (119910 119909) 1 (119909 119910) 119889120583 (119909)) 119889120583 (119910) = intΣ

⟨11987921198791 119889119898⟩

(47)



ΞintΣ⟨119891(120585) 119889119898⟩ 119889](120585) lt infin we


intΞ

intΣ

⟨119891 (120585) 119889119898⟩ 119889] (120585) = intΣ⟨119879 119889119898⟩ (48)

intΞ

⟨119891 (120585) 119906 V⟩ 119889] = ⟨119879119906 V⟩ (49)


intΣtimesΞ

⟨119891 119889 (119898 otimes ])⟩ = intΣ⟨119879 119889119898⟩ (50)


119860119891 119889] isin C1(119883) such that

⟨(int119860

119891 119889]) 119906 V⟩ = int119860

⟨119891 (120585) 119906 V⟩ 119889] le ⟨119879119906 V⟩ (51)









intΣtimesΞ

⟨119891 119889 (119898 otimes ])⟩ = intΣ

⟨intΞ

119891119889] 119889119898⟩ (52)

= intΞ

intΣ

⟨119891 (120585) 119889119898⟩ 119889] (120585) (53)


119860intΣ⟨119891(120585) 119889119898⟩ 119889](120585) for

every 119860 isin B



⟨(int119860

119891 119889]) 119906 V⟩

= int119860

intΣ

intΣ

119896120585(119909 119910)119906 (119909)V(119910)119889120583(119909)119889120583(119910)119889](120585)(54)


intΣ

⟨int119860

119891 119889] 119889119898⟩ = int119860

intΣ

119896120585 (119909 119909) 119889120583 (119909) 119889] (120585)

= int119860

intΣ

⟨119891 (120585) 119889119898⟩ 119889] (120585)(55)





intΣ⟨119872 (119860) 119889119898⟩ le int

Σ⟨119872 (Ξ) 119889119898⟩ 119860 isin B

intΣ

⟨119872(119891) 119889119898⟩ = intΞ

119891 119889 ⟨119872119898⟩ le10038171003817100381710038171198911003817100381710038171003817infin int

Σ⟨119872 (Ξ) 119889119898⟩

(56)




References

















Volume 2014




Journal of











Journal of


Function Spaces






Algebra










By choosing 119861119895 = cup119895



= E (119896 | E119899 otimesE119899) (119909 119909) 120594119861119895(119909)

(32)

for all 119899 119895 = 1 2 so

intΣ

⟨119876 (119861119895) 119879119876 (119861119895) 119889119898⟩ = int119861119895

⟨119879 119889119898⟩ 997888rarr intΣ⟨119879 119889119898⟩

(33)


tr (119876 (119861119895) 119879119876 (119861119895)) = int119861119895

⟨119879 119889119898⟩ (34)


(119876 (119861119895) 119879119876 (119861119895) 119906 119906) le 1199062 tr (119876 (119861119895) 119879119876 (119861119895))

le 1199062intΣ⟨119879 119889119898⟩

(35)



10038161003816100381610038161003816tr (119876 (119861119895) 119879119876 (119861119895))

10038161003816100381610038161003816

le 119862intΣ⟨119879 119889119898⟩

(36)



2(120583)) = C1(119871

2(120583)) and

tr (119879) = intΣ⟨119879 119889119898⟩ 119879 isin C1 (119871

2(120583)) (37)





infin

sum

119895=1

10038161003816100381610038161003816119888119895

10038161003816100381610038161003816] (119862119895 times 119863119895) lt infin (38)

and we can write

119896 (119909 119910) =

infin

sum

119895=1

119888119895 120594119862119895times119863119895(119909 119910) (39)




intΣ

⟨119876 (1198812) 119879119876 (1198811) 119889119898⟩ = intΣ

⟨119876 (11988111198812) 119879 119889119898⟩

= intΣ

⟨119879119876 (11988111198812) 119889119898⟩

(40)


1003816100381610038161003816100381610038161003816intΣ

⟨119876 (119881) 119879 119889119898⟩

1003816100381610038161003816100381610038161003816le 119881infin int

Σ

⟨119879 119889119898⟩ isin [0infin] (41)


(119876 (1198821) 119896119876 (1198822))sim(119909 119909) = (119876 (1198821) 119876 (1198822) 119896)

sim(119909 119909)

= (119896119876 (1198821) 119876 (1198822))sim(119909 119909)

(42)

is equal to

infin

sum

119895=1

119888119895 120594119862119895cap119863119895cap1198821cap1198822(119909) (43)



tr (119861119879) = tr (119879119861) (44)


2(120583))


2(120583)) the operator 119861119879may



intΣ

⟨11987911198792 119889119898⟩ = intΣ

⟨11987921198791 119889119898⟩ (45)




intΣ

⟨11987911198792 119889119898⟩ = intΣ

(intΣ

1 (119909 119910) 2 (119910 119909) 119889120583 (119910)) 119889120583 (119909)

(46)



intΣ

(intΣ

2 (119910 119909) 1 (119909 119910) 119889120583 (119909)) 119889120583 (119910) = intΣ

⟨11987921198791 119889119898⟩

(47)



ΞintΣ⟨119891(120585) 119889119898⟩ 119889](120585) lt infin we


intΞ

intΣ

⟨119891 (120585) 119889119898⟩ 119889] (120585) = intΣ⟨119879 119889119898⟩ (48)

intΞ

⟨119891 (120585) 119906 V⟩ 119889] = ⟨119879119906 V⟩ (49)


intΣtimesΞ

⟨119891 119889 (119898 otimes ])⟩ = intΣ⟨119879 119889119898⟩ (50)


119860119891 119889] isin C1(119883) such that

⟨(int119860

119891 119889]) 119906 V⟩ = int119860

⟨119891 (120585) 119906 V⟩ 119889] le ⟨119879119906 V⟩ (51)









intΣtimesΞ

⟨119891 119889 (119898 otimes ])⟩ = intΣ

⟨intΞ

119891119889] 119889119898⟩ (52)

= intΞ

intΣ

⟨119891 (120585) 119889119898⟩ 119889] (120585) (53)


119860intΣ⟨119891(120585) 119889119898⟩ 119889](120585) for

every 119860 isin B



⟨(int119860

119891 119889]) 119906 V⟩

= int119860

intΣ

intΣ

119896120585(119909 119910)119906 (119909)V(119910)119889120583(119909)119889120583(119910)119889](120585)(54)


intΣ

⟨int119860

119891 119889] 119889119898⟩ = int119860

intΣ

119896120585 (119909 119909) 119889120583 (119909) 119889] (120585)

= int119860

intΣ

⟨119891 (120585) 119889119898⟩ 119889] (120585)(55)





intΣ⟨119872 (119860) 119889119898⟩ le int

Σ⟨119872 (Ξ) 119889119898⟩ 119860 isin B

intΣ

⟨119872(119891) 119889119898⟩ = intΞ

119891 119889 ⟨119872119898⟩ le10038171003817100381710038171198911003817100381710038171003817infin int

Σ⟨119872 (Ξ) 119889119898⟩

(56)




References

















Volume 2014




Journal of











Journal of


Function Spaces






Algebra











intΣ

(intΣ

2 (119910 119909) 1 (119909 119910) 119889120583 (119909)) 119889120583 (119910) = intΣ

⟨11987921198791 119889119898⟩

(47)



ΞintΣ⟨119891(120585) 119889119898⟩ 119889](120585) lt infin we


intΞ

intΣ

⟨119891 (120585) 119889119898⟩ 119889] (120585) = intΣ⟨119879 119889119898⟩ (48)

intΞ

⟨119891 (120585) 119906 V⟩ 119889] = ⟨119879119906 V⟩ (49)


intΣtimesΞ

⟨119891 119889 (119898 otimes ])⟩ = intΣ⟨119879 119889119898⟩ (50)


119860119891 119889] isin C1(119883) such that

⟨(int119860

119891 119889]) 119906 V⟩ = int119860

⟨119891 (120585) 119906 V⟩ 119889] le ⟨119879119906 V⟩ (51)









intΣtimesΞ

⟨119891 119889 (119898 otimes ])⟩ = intΣ

⟨intΞ

119891119889] 119889119898⟩ (52)

= intΞ

intΣ

⟨119891 (120585) 119889119898⟩ 119889] (120585) (53)


119860intΣ⟨119891(120585) 119889119898⟩ 119889](120585) for

every 119860 isin B



⟨(int119860

119891 119889]) 119906 V⟩

= int119860

intΣ

intΣ

119896120585(119909 119910)119906 (119909)V(119910)119889120583(119909)119889120583(119910)119889](120585)(54)


intΣ

⟨int119860

119891 119889] 119889119898⟩ = int119860

intΣ

119896120585 (119909 119909) 119889120583 (119909) 119889] (120585)

= int119860

intΣ

⟨119891 (120585) 119889119898⟩ 119889] (120585)(55)





intΣ⟨119872 (119860) 119889119898⟩ le int

Σ⟨119872 (Ξ) 119889119898⟩ 119860 isin B

intΣ

⟨119872(119891) 119889119898⟩ = intΞ

119891 119889 ⟨119872119898⟩ le10038171003817100381710038171198911003817100381710038171003817infin int

Σ⟨119872 (Ξ) 119889119898⟩

(56)




References

















Volume 2014




Journal of











Journal of


Function Spaces






Algebra
















Volume 2014




Journal of











Journal of


Function Spaces






Algebra









Documents

Research Article Lattice Trace Operatorsdownloads.hindawi.com/journals/joper/2014/629502.pdf · Research Article Lattice Trace Operators BrianJefferies School of Mathematics,e University