The Dixmier trace and the Wodzicki residue for global pseudo-differential operators on compact manifolds

H Revista Integración, temas de matemáticas

EsueladeMatemátias

UniversidadIndustrialdeSantander

Vol. 38,N

◦

1,2020,pág. 6779

DOI:http://dx.doi.org/10.18273/revint.v38n1-2020006

The Dixmier trae and the Wodziki residue

for global pseudo-dierential operators on

ompat manifolds

Duván Cardona

a ∗

, César Del Corral

b

a

GhentUniversity,DepartmentofMathematis:Analysis,LogiandDisrete

Mathematis,Ghent,Belgium.

b

UniversidaddelosAndes,DepartmentofMathematis,Bogotá,Colombia.

Abstrat. In this note, we announe the results of our investigation on

the Dixmier trae and the Wodziki residue for pseudo-dierential opera-

tors on ompat manifolds. We give formulae for the Dixmier trae and

the non-ommutative residue (also alled Wodziki's residue) of invariant

pseudo-dierentialoperators on ompatmanifolds with orwithoutbound-

ary. For every losed manifold, the notion of global symbol for invariant

pseudo-dierentialoperatorswillbebasedontheFourieranalysisassoiated

to everyelliptiand positiveoperator (developedby M.Ruzhansky,V. Tu-

runenandJ.Delgado). Inpartiular,foreahompatLiegroupwewilluse

itsrepresentationtheory. Fortheanalysisofoperatorsonompatmanifolds

withboundary,wewillusethenon-harmonianalysisassoiatedwithbound-

aryvalued problems(developedbyM.Ruzhansky,N. Tokmagambetov,and

J.Delgado).

Keywords:Dixmiertrae,nonommutativeresidue,globaloperators,repre-

sentationtheory.

MSC2010:81Q10,47B10.

La traza de Dixmier y el residuo de Wodziki para

operadores pseudodifereniales globales sobre

variedades ompatas

Resumen. En estanota seanunianlosresultadosdenuestra investigaión

sobre la traza de Dixmier y el residuo de Wodziki para operadores pseu-

dodifereniales sobre variedades ompatas. Se alula la traza de Dixmier

y el residuo no onmutativo (residuo de Wodziki) de operadores pseudo-

difereniales invariantes sobre variedades ompatas on o sin borde. Para

0

∗

E-mail: duvan306gmail.om

Reeived: 17July2019, Aepted:16January2020.

Toitethisartile:D.CardonaandC.delCorral,TheDixmiertraeandtheWodzikiresidueforglobal

pseudo-dierentialoperatorsonompatmanifolds,Rev. Integr.temasmat. 38(2020),No. 1,6779.

adavariedaderrada(suave,ompataysin borde),seemplealanoiónde

símbologlobalquevienedadaporelanálisisdeFourierasoiadoaadaopera-

dorelíptioypositivo(desarrolladoporM.RuzhanskyandV.Turunenpara

para grupos de Lie y porM. Ruzhansky, N. Tokmagambetovy J. Delgado

paravariedadeserradas). En partiular,para adagrupodeLie ompato,

seusa su teoríade representaión.Respeto alanálisisde operadoressobre

variedades on borde,se usa el análisis no armónio asoiado a problemas

onvaloresdefrontera(introduidoporM.Ruzhansky,N.Tokmagambetov,

yJ.Delgado).

Palabras lave:TrazadeDixmier,residuonoonmutativo,operadorglobal,

teoríaderepresentaiones.

1. Introdution

Inthiswork,welassifytheDixmiertraeabilityofsomeinvariantoperatorsonompat

manifolds andweapply theseresultsin orderto study theWodzikiresidueof pseudo-

dierentialoperators.ThisworkisasubsequentanalysistotheprogramofM.Ruzhansky

andJ.Delgadoonthelassiationofglobaloperatorsinidealsontainedinthealgebra

of bounded operators on Lebesgue spaes (see [16, 17, 18, 19, 20, 21℄). Thetopi has

beenof intense researhin thelast yearsnot onlyforoperators onompat manifolds,

but also onnon-ompat manifoldsand other spaeswithout adierentiablestruture

(seee.g. [1,3,4,5,6,8,9,10,11,14,15,27,24℄). Thenoveltyoftheannounedresults

isthat weusethematrix-valuedglobalquantisationdevelopedbyRuzhansky,Turunen,

Tokmagambetov and Delgado, insteadof the lassial results on the subjet based on

the lassialnotion ofthe loal symbol vialoalizations (see, e.g. Hörmander[26℄), as

in [13, 23, 25, 33, 34℄ and referenes therein. We summarise our investigation in the

followingresults:

Weprovideneessaryandsuientonditionsinorderthatglobalinvariantpseudo-

dierentialoperatorsonamanifoldwithorwithoutboundarybelongtotheDixmier

lass

L ^(1,∞) (L ² ).

WeprovidesuientandneessaryonditionsinordertoobtainDixmiertraeabil-

ityforatypeofpseudo-dierentialoperatorswithglobalsymbolsintheHörmander

lasseson ompatLie groups,and weexpress our resultsbyusing therepresen-

tationtheoryof thesegroups. Also,weuseConnes'traetheoremin orderto get

formulaeforthenonommutativeresidueof lassialpseudo-dierentialoperators

onompatLie groups.

For a smooth ompat manifold with or without boundary, we nd riteria in

terms of the global symbols in order that the orresponding operators belong to

the Marinkiewiz ideal

L ^(p,∞) (L ² ), 1 < p < ∞ ,

^{aswell asto the Dixmier ideal}

L ^(1,∞) (H )

^where

H = L ² (M )

^.

Our mainresultwill bepresentedinSetion 3. InSetion 2weprovidesomebasison

thetheoryofpseudo-dierentialoperators,theWodzikiresidueandtheDixmiertrae.

Finally,in Setion4weprovidesomeexamples.

2. Preliminaries

Inorderto presentourmain resultswereallsomedenitions(seeConnes[13℄). Let

H

be aHilbert spae,

L (H )

^{bethe algebraof bounded linear operators on}

H

^and

K (H )

betheidealof ompatoperatorson

H.

^{Aompat operator}

A ∈ K (H )

^{belongsto the}

Dixmierlass

L ^(1,∞) (H)

^if

X

1≤n≤N

s n (A) = O(log(N )),

^as

N → ∞ ,

where

{ s n (A) }

^{denotesthesequeneofsingularvaluesof}

A

^{,whihonsistsofthepoints}

in thespetrumof

√ A ^∗ A

^{. The sequene}

s n (A)

^{anbe arrangedin inreasingorder so}

that

L ^(1,∞) (H )

^{maybeendowedwiththenorm}

k A k L ^{(1, ∞ )} (H) = 1

log N sup

N >1

X

n≤N

s n (A).

If

A ∈ L ^(1,∞) (H )

^{, thefuntional}

k A k L ^(1,∞) (H )

^{denesanorm. Some ideals,loselyre-}

latedtotheDixmierideal,areMarinkiewizideals

L ^(p,∞) (H ),

^{denedbythosebounded}

linearoperators

A

^on

L ²

^{satisfyingtheeigenvalueondition:}

X

1≤n≤N

s n (A) = O(N ^(1−1/p) ), N → ∞ ,

where

1 < p < ∞ .

^Onthelass

L ^(p,∞) (L ² )

^{theusualnormisgivenby}

k A k L ^(p,∞) (H) := sup

N ≥1

N ^{( p 1 )−1} X

1≤n≤N

s n (A).

The main objetshereare pseudo-dierentialoperators, whih weintrodueasfollows

([16, 26, 31℄). A pseudo-dierentialoperator

A

^{is an operator dened by the integral}

representation

Af (x) = Z

R ⁿ

e ^ix·ξ σ ^A (x, ξ) ˆ f (ξ) dξ,

⁽¹⁾

where thefuntion

σ ^A (x, ξ)

^{alledthesymbolof}

A

^{satisesestimatesofthetype(see}

Hörmander[26℄)

| ∂ _x ^α ∂ _ξ ^β σ ^A (x, ξ) | ≤ C α,β (1 + | ξ | ) ^m−|β| .

⁽²⁾

Here

f ˆ

^{denotes theeulideanFouriertransformofthefuntion}

f

^and

m

^istheorderof

thepseudo-dierentialoperator

A

^{. Ifweonsideraompatmanifoldwithoutboundary}

M

^{of dimension}

κ ,

^apseudo-dierentialoperator

A

^on

M

^{anbedened byusing the}

notion of loal symbol; this means that forany loal hart

U

^{, the operator}

A

^{has the}

form

Au(x) = Z

T _x ^∗ U

e ^ix·ξ σ ^A (x, ξ) u(ξ) b dξ.

Apseudo-dierentialoperator

A

^{isalledlassial,if}

σ ^A

^{admitsanasymptotiexpansion}

σ ^A (x, ξ) ∼ P ∞

j=0 σ _m−j ^A (x, ξ)

^{insuhawaythateahfuntion}

σ m−j (x, ξ)

^ishomogeneous

in

ξ

^oforder

m − j

^for

ξ 6 = 0

^{. Theset oflassial}pseudo-dierentialoperatorsoforder

m

^isdenotedby

Ψ ^m _cl (M )

^{. For}

A ∈ Ψ ^m _cl (M )

^,andfor

x ∈ M

^,

res

x (A) = Z

|ξ|=1

σ − κ (x, ξ) dξ

denes a loal densitywhih anbeglued over

M

^{. In this ase, the}non-ommutative residueof

A

^{isdened bytheexpression}

res

(A) = 1 κ (2π) ^κ

Z

M

Z

|ξ|=1

σ ₋ ^A κ (x, ξ) dξ dx.

⁽³⁾

The followingfat is due to Connes,[13℄: if

A ∈ Ψ ⁻ _cl ^κ (M )

^{is apositiveoperator, then}

Tr

ω (A) =

^res

(A).

In the ase of a manifold

M

^{with boundary, the typial algebra of} pseudo-dierential operatorsis knownastheBoutetde Monvelalgebra(BdM). This algebraisformed by

matrixvaluedpseudo-dierentialoperatorsassoiatedwithboundaryvalueproblems(see

[2℄):

A

=

P + + G K

T S

: C ^∞ (M ) ⊕ C ^∞ (∂M ) → C ^∞ (M ) ⊕ C ^∞ (∂M ).

ThenonommutativeresidueandtheDixmiertraeforanoperator

P

^{inBdM'salgebra}

havebeenstudied e.g. inFedosov[23℄andNestandShrohe[28℄.

In order to study the Dixmiertraeand thenon-ommutative residue,wewill use the

notion of global pseudo-dierentialoperator. If weonsider alosed manifold

M

^(i.e.,

a ompat manifold without boundary) and

H = L ² (M ),

^{there exists a global Fourier}

analysis assoiatedto everypositiveellipti pseudo-dierentialoperator

E

^on

M

^whih

givesforertainpseudo-dierentialoperators

A,

^alled

E

-invariants,adisreteFourier representationoftheform

Af (x) = X ∞

l=0

h σ A,E (l) f b (l), e l (x) i ^{C dl} ,

⁽⁴⁾

where

e l (x) := (e ^m _l ) 1≤m≤d l

^and

{ e ^m _l : l ∈ N , 1 ≤ m ≤ d l }

^{is abasisof}

L ² (M )

^onsisting

of eigenfuntions

λ l , l ∈ N ₀ ,

^of

E.

^Thefuntion

σ A,E

^{isalled thematrixvaluedsymbol}

of

A

^{with respet to}

E

^{. The} quantisation proedure assoiating to everyoperator

A

ating in

C ^∞ (M )

^{the matrix valued symbol}

σ A,E

^{was introdued by J. Delgado and}

M. Ruzhansky, [20℄. Also, If

M = G

^{is a ompat Lie group and}

E = −L G

^{is the}

positiveLaplae-Beltramioperatoron

G,

^{RuzhanskyandTurunenin[31℄giveadisrete}

representationto everypseudo-dierentialoperator

A

^atingon

C ^∞ (G)

^{in termsofthe}

representationtheoryofthegroup

G

^{,inthefollowingway:}

Af (x) = X

[ξ]∈ G b

d ξ

^Tr

[ξ(x)σ A (x, ξ) f b (ξ)].

⁽⁵⁾

Here

G b

^{denotestheunitarydual of}

G.

Ruzhansky-Turunen'salulusthat givesahar- aterisationfortheHörmanderlasses

Ψ ^m _ρ,δ (G)

^{onaompatLiegroup}

G

^{byusingglobal}

symbols. Infat,

A ∈ Ψ ^m _ρ,δ (G)

^{,ifandonlyif,itsglobalsymbol}

σ A (x, ξ)

^{asin(5)satises}

k ∆ ^α _ξ ∂ _x ^β σ A (x, ξ) k ^op ≤ C h ξ i m−ρ|α|+δ|β| , x ∈ G, [ξ] ∈ G, b

^where

h ξ i := (1 + λ [ξ] ) ^{1 2}

^and

{ λ [ξ] }

is thespetrumof

−L G ,

^{whihanbeenumeratedby}

[ξ] ∈ G b

^.

Let us observethat if

M = G

^{is a ompat Lie group and the operator}

E ∈ Ψ ^ν _+e (G)

is notinvariant(this meansthat theglobal symbol

σ E (x, ξ)

^{depends of bothvariables,}

thespatial variable

x ∈ G,

^{andtheFouriervariables}

[ξ] ∈ G b

^{),thenin loal}oordinates,

E

-invariantsoperators

A

^are

E

-multipliers,buttheyarenotFouriermultipliersbeause in loaloordinatestheirsymbolsaredependingonthespatialvariables.

Intheformulationofourresultforompatmanifoldswithboundary,weusetheglobal

quantization ofpseudo-dierentialoperators onompat manifoldswith boundarydue

to J. Delgado, M. Ruzhansky and N. Tokmagambetov [30℄ whih we briey desribe

as follows. Let

M

^{bea ompat manifold with boundary}

∂M

^{and let}

L

^{be a pseudo-}

dierentialoperatoron

M

^{satisfyingsomeboundaryonditionson}

∂M.

^Weassumethat

L

^{hasdisrete spetrum}

{ λ ξ | ξ ∈ I}

^{. If}

σ A,L : M × I → C

is a suitablefuntion, the operator

A

^{assoiatedwith}

σ A

^{an bewrittenas}

Af (x) = X

ξ∈I

u ξ (x)σ A,L (x, ξ) f b (ξ),

^for

f ∈ C _L ^∞ (M ),

⁽⁶⁾

where

f b := F ^L (f )

^{denotes the}

L

^{-Fouriertransform of}

f

^{introdued in [30℄.}

L

^-Fourier

multipliersareboundedlinearoperators

A : C _L ^∞ (M ) → C _L ^∞ (M )

^satisfying

F ^L (Af )(ξ) = σ A,L (ξ) F ^L (f )(ξ),

^forall

f ∈ C _L ^∞ (M ),

^{and for somefuntion}

σ A,L : I → C

depending onlyontheFouriervariables

ξ ∈ I

^{. Inthisase,}

σ A,L (ξ)

^{isalled the}

L

^-symbolof

A

^.

3. The Dixmier Trae and the non-ommutative residue for global

pseudo-dierential operators

In thissetion, wepresentourmain resultsforglobal operatorsonompat manifolds.

Westart with the ase of ompat manifolds without boundary. Here,

κ

denotes the dimensionofaxedlosedmanifold.

Theorem 3.1. Let

M

^{be a}

κ

-dimensional ompat manifold without boundary and let

E ∈ Ψ ^ν _+e (M )

^{be a positive ellipti} pseudo-dierential operator on

M.

^If

A : L ² (M ) → L ² (M )

^isan

E

^{-invariantboundedoperator with}matrix-valuedsymbol

(σ A,E (l)) l

^,thenwe

have that

A

^{is Dixmiertraeable,i.e.,}

A ∈ L ^(1,∞) (L ² (M ))

^{if,and onlyif,}

τ(A) := 1

dim(M ) lim

N →∞

1 log N

X

l:(1+λ l ) ν ¹ ≤N

Tr

( | σ A,E (l) | ) < ∞ ,

⁽⁷⁾

where

σ A,E

^{isasin (4). Moreover, if}

A

^ispositive,

τ(A) =

^Tr

w (A).

If

A

^isan

E

^{-invariantoperator, then}

A ∈ L ^(p,∞) (L ² (M ))

^{if,andonly if,}

γ p (A) := sup

N ≥1

N ^{dim M( 1 p −1)} · X

l:(1+λ l ) ^{ν 1} ≤N

Tr

( | σ A,E (l) | ) < ∞ ,

⁽⁸⁾

where

1 < p < ∞ .

^{In thisase,}

k A k L ^{(p, ∞ )} (L ² (M)) ≍ γ p (A)

^.

Let

M = G

^{beaompatLiegroupandlet}

G b

^{betheunitarydualof}

G.

^Ifwedenote

by

σ A (x, ξ)

^{the matrixvaluedsymbol assoiated to}

A,

^{thenunder the ondition}

k ∆ ^α _ξ ∂ _x ^β σ A (x, ξ) k ^op ≤ C h ξ i ^−κ−|α| , x ∈ G, [ξ] ∈ G, b ∀ α ∈ R ⁿ ,

⁽⁹⁾

the operator

A

^isDixmiermeasurable.

If

A ∈ Ψ ⁻ _cl ^κ (G)

^{isalassialandpositive}pseudo-dierentialoperatoranditssymbol admitsan asymptotiexpansioninhomogeneousomponentsof the form:

σ ^A (x, ξ) ∼ X ∞

j=0

a m−j (x)σ ^{A m − j} (ξ),

⁽¹⁰⁾

then

res

(A) = 1 dim(G)

Z

G

a −κ (x)dx × lim

N→∞

1 log N

X

ξ:hξi≤N

d ξ

^Tr

( | σ A − κ (ξ) | ).

⁽¹¹⁾

The expression (11)in thepreeding theorem givesaformula for thenonommutative

residue of lassialoperators in terms of irreduible representationson

G

^{. Weobserve}

that ourapproahprovideformulaeforthenonommutativeresidueofoperatorsonthe

torus,ofadierentwayinrelationwiththereentworkofPietsh[29℄.

Nowwepresentourresultonerningtoglobaloperatorsonamanifoldwithboundary.

WiththenotationsaboveourresultontheDixmiertraeabilityofoperatorsonmanifolds

with boundaryanbeenuniatedofthefollowingway. Wewrite

I = { ξ l : l ∈ N ₀ } .

Theorem3.2. Let

M

^{beaompat manifoldwithboundary}

∂M.

^If

A : L ² (M ) → L ² (M )

isaboundedFouriermultiplier,and

L

^isaself-adjointoperator on

L ² (M ),

^thenwehave

the following assertions:

A

^{is Dixmiertraeableif, andonlyif,}

τ ^′ (A) := lim

N →∞

1 log N

X

l≤N

| σ A,L (ξ l ) | < ∞ .

⁽¹²⁾

Inthis ase, for

A

^{apositive operator,}

τ ^′ (A) =

^Tr

ω (A).

Moreover, if

L

^{isan operator of order}

m

^{satisfying the Weyl eigenvalue ounting}

formula,that is,

N L (λ) := # { l : | λ ξ l | ^{m 1} ≤ λ } = C 0 λ ^{dim M} + O(λ ^{dim M −1} ), λ → ∞ ,

⁽¹³⁾

then

A

^{isDixmiertraeable if,onlyif (f. (12)),}

τ ^′ (A) = 1

dim M lim

N→∞

1 log N

X

l:|λ _ξl | ^{m 1} ≤N

| σ A,L (ξ l ) | < ∞ .

⁽¹⁴⁾

Inthis ase, for

A

^{apositive operator,}

τ ^′ (A) =

^Tr

ω (A).

A ∈ L ^(p,∞) (L ² (M ))

^{if,andonly if,}

γ _p ^′ (A) := sup

N≥1

N ^{( 1 p −1)} X

l≤N

| σ A,L (ξ l ) | < ∞ ,

⁽¹⁵⁾

where

1 < p < ∞ .

^{In this ase,}

k A k L ^{(p, ∞)} (L ² (M )) ∼ γ _p ^′ (A)

^{. Moreover,if}

L

^satises

(13),we have

γ _p ^′ (A) ≍ sup

N≥1

N ^{dimM ( p 1 −1).} X

l: |λ _ξl | ^{m 1} ≤N

| σ A,L (ξ l ) | .

⁽¹⁶⁾

On the other hand, let A

= [P + , T ] ^t

^{be a olumn ellipti operator in the Boutet de}

Monvel algebra of order

n

^{. We assume the existene of}

L

^{satisfying as in Ruzhansky-}

Tokmagambetov's alulus. Anyparametrix

R

^of

P T

^{isDixmier traeable andTr}

ω (R) =

res

(R) = lim N →∞ 1 ln N

P

l≤N | (σ P,L (ξ l )) ⁻¹ | ,

^here

(σ P,L (ξ l )) l∈ N ₀

^{denotestheglobalsymbol}

oftheoperator

P

^withrespetto

L.

^{Thisresidue'sformulaisthesameforallparametrixof}

A, R

^andindependentoftheboundaryonditionandindependentoftheaveragefuntion

ω

^{. If inaddition}

L

^{satisesthe WeylEigenvalueCountingFormula,wealso have}

Tr

w (A) = 1

dim(M ) lim

N →∞

1

log N . X

l : |λ _ξl | ^{m 1} ≤N

| σ A,L (ξ l ) | .

⁽¹⁷⁾

The proofs of these theorems and the orresponding analysis developed in this global

settinganbefoundinCardonaandDelCorral[12℄. WereferthereadertoCardona,Del

Corral andKumar[7℄ wheretheDixmier traeof disretepseudo-dierentialoperators

hasbeeninvestigated.

4. Examples

In this setion we provide some examplesin relation with ourmain results. First, we

onsider aDixmiertraeable Besselpotentialassoiatedto

L.

^{Also,weonsider anex-}

ampleofaDixmiertraeableoperatoronthemanifoldwithboundary

M = [0, 1].

^Later,

weonsider theaseofBesselpotentialonSU

(2) ∼ = S ³

andSU

(3)

respetively.

Example4.1. Let

M

^{beaompatmanifoldwithboundary}

∂M

^and

L

^{asinthepreeding}

setions. Let us onsider the model operator

L M

^{self adjoint on}

L ² (M ).

^{If we assume}

that

− L

^{isapositive operator oforder}

ν,

^{andsomepositive realnumber}

s 1

^satises

X

l∈ N ₀

h ξ l i ^−s = ∞ , X

l∈ N ₀

h ξ l i ^{−s ′} < ∞

⁽¹⁸⁾

for all

0 < s ≤ s 1 < s ^′ < ∞

^{,thenwiththe notationsabove theoperator}

A := (I − L) ^{− s ν 1}

is Dixmiertraeableon

L ² (M ).

^{Forthe proof,weombine that theglobalsymbol of}

A

^is

given by

σ A,L (ξ l ) = (1 − λ ξ l ) ^{− s ν 1} ≍ h ξ l i ^{−s 1} .

As onsequeneof (18)wehavethat

h ξ l i ^{−s 1} = O(l ⁻¹ ), l → ∞ ,

^and

X

l≤N

h ξ l i ^{−s 1} = O(log N), N → ∞ .

⁽¹⁹⁾

Hene, byTheorem3.2 weobtain

Tr

w ((I − L) ^{− s ν 1} ) ≍ lim

N →∞

1 log N

X

l≤N

h ξ l i ^{−s 1} < ∞ .

⁽²⁰⁾

Example 4.2. Consider

M = [0, 1]

^{andthe operator}

L = − i _dx ^d

^on

M ^◦ = (0, 1)

^{with the}

domain

D(L) = { f ∈ W ₂ ¹ [0, 1] | af (0) + bf(1) + Z 1

0

f (x)q(x) dx = 0 } ,

where

a 6 = 0

^,

b 6 = 0

^{, and}

q ∈ C ¹ [0, 1]

^{, here}

W ₂ ¹ [0, 1] := { f ∈ L ² [0, 1] | f ^′ ∈ L ² [0, 1] }

^.

Under the assumption that

a + b + R 1

0 q(x) dx = 1,

^{we have the inverse}

L ⁻¹

^{exists and}

is boundedfrom

L ² [0, 1]

^to

D(L)

^{. The operator}

L

^{has adisretespetrum whih an be}

enumerated by

λ j = 2πj − i ln( − a/b) + α j ,

^for

j ∈ Z ,

^{where the sequene}

α j

^satises

that for any

ǫ > 0

^,

P

j∈Z | α j | ^1+ǫ < ∞

^{. If}

m j

^{denotes the} multipliity of the eigenvalue

λ j

^,then

m j = 1

^{for suientlylarge}

| j |

^{,andthe systemof} eigenfuntionsare given by

u jk = (ix) ^k

k! e ^{iλ j x} ,

^for

0 ≤ k ≤ m j − 1

^and

j ∈ Z .

Let

j 0 ∈ N

large enough so that

m j = 1

^for

| j | ≥ j 0

^{. The global symbol}

σ L,L (x, j )

^of

L

with respetto

L

^{isgiven by}

σ L,L (x, j) =

_x ^2k

k! ² λ j , x ∈ [0, 1]; 0 ≤ k ≤ m j − 1, | j | ≤ j o , λ j , | j | ≥ j 0 ,

for some

j 0 ∈ N .

^{Itfollows fromthefuntionalalulusthat}

L ⁻¹

^{hasadisretespetrum}

givenby

λ ⁻¹ _j

^with

j ∈ Z

. Sine

| λ j | =O(j)

^for

j ∈ Z

,wehave

P

|j|≤N | λ ⁻¹ _j | =O ln(2N +1)

.

Therefore,

L ⁻¹

^isapseudo-dierentialoperatorwhihliesin

L ^(1,∞) (L ² [0, 1])

^withsymbol

σ L ^{− 1} ,L (ξ) = (σ L,L (ξ)) ⁻¹ , | ξ | ≥ j 0 ,

^and

Tr

ω (L ⁻¹ ) = lim

N →∞

1 ln (2N + 1)

X

|j|≤N

| λ ⁻¹ _j |

= lim

N →∞

1 ln (2N + 1)

X

j 0 ≤|j|≤N

| (σ L,L (x, j)) ⁻¹ |

= lim

N →∞

1 ln (2N + 1)

X

|j|≤N

2πj + i ln( − a/b) + α j

(2π j +

^Re

(α j )) ² + (

^Im

(α j ) + ln(b/a)) ² .

Example4.3. Letusassumethat

A

^isaleft-invariantoperatoronSU

(2) ∼ = S ³ .

^Wereall

that the unitarydual ofSU

(2)

^(see

[

³¹

]

^{) anbe identiedas}

c

SU

(2) ≡ { [t l ] : 2l ∈ N , d l := dim t l = (2l + 1) } .

⁽²¹⁾

Thereareexpliitformulaefor

t l

^{asfuntion ofthe Euleranglesintermsofthe so-alled}

Legendre-Jaobipolynomials, see

[

³¹

]

^{. In thisase,if}

A

^{isDixmiertraeablewithsymbol}

σ([t l ]) ≡ σ(l)

^,then

Tr

w (A) = lim

N →∞

1 log P

l≤ ^N ₂ ,l∈ ¹ ₂ N ₀ d ² _l X

l≤ ^N ₂ ,l∈ ¹ ₂ N ₀

d [t l ]

^Tr

[ | σ(l) | ]

= lim

N →∞

1 log P

l≤ ^N ₂ ,l∈ ¹ ₂ N ₀ (2l + 1) ² X

l≤ ^N ₂ ,l∈ ¹ ₂ N ₀

(2l + 1)

^Tr

[ | σ(l) | ]

= lim

N →∞

1

log[ ¹ ₆ (N + 1)(2N ² + 7N + 1)]

X

l≤ ^N ₂ ,l∈ ¹ ₂ N ₀

(2l + 1)

^Tr

[ | σ(l) | ].

Forexample, if

A = (I −L

^SU

(2) ) ^{− κ 2} ,

^{istheBesselpotentialoforder}

− κ = − dim

^SU

(2) =

− 3,

^then

A

^{isDixmiertraeable,}

σ(l) = (1 + l(l + 1)) ^{− 3 2}

^and

Tr

w ((I − L

^SU

(2) ) ^{− 3 2} )

= lim

N →∞

1

log[ ¹ ₆ (N + 1)(2N ² + 7N + 1)]

X

l≤ ^N ₂ ,l∈ ¹ ₂ N ₀

(2l + 1) ² [1 + l(l + 1)] ^{− 3 2}

= lim

N →∞

1

log[ ¹ ₆ (N + 1)(2N ² + 7N + 1)]

X N

n=0

(n + 1) ² h 1 + n

2 ( n

2 + 1) i − ³ ₂

= lim

N →∞

1

log[ ¹ ₆ (N + 1)(2N ² + 7N + 1)]

N+1 X

n=1

n ²

8 [n ² + 3] ^{− 3 2}

∼ 0.03935...

^{(numerialevidene).}

A similarresultanbeobtainedifweonsiderthe operator

(I − L ^sub ) ^{− 3 2}

^,where

L ^sub :=

D ² ₁ + D ² ₂

^{is the} sub-Laplaian on SO

(3);

^here

D 1

^and

D 2

^{are the} derivatives in the orresponding variables tothe Euleranglesfor SO

(3)

^(see

[

³¹

,

³²

]

^).

Example4.4. Let

A

^bealeft-invariantpseudo-dierentialoperatorDixmiertraeableon SU

(3).

^{TheLiegroupSU}

(3)

^{hasdimension8and}

3

^{positivesquareroots}

α, β

^and

ρ

^with

the property

ρ = ¹ ₂ (α + β + ρ).

^{Wedene theweights}

σ = 2 2 α + 1

3 β, τ = 1 3 α + 2

3 β.

⁽²²⁾

With the notationsabove,the unitary dualof SU

(3)

^{anbe identiedwith}

c

SU

(3) ≡ { λ := λ(a, b) = aσ + bτ : a, b ∈ N ₀ , d λ = 1

2 (a + 1)(b + 1)(a + b + 2) } .

⁽²³⁾

The Dixmiertraeof

A,

^{onedenoteditsfullsymbol by}

σ(λ) ≡ σ(λ(a, b))

^{,isgivenbythe}

expression

Tr

w (A)

= lim

N →∞

1 log[ P

0≤a+b≤N d λ(a,b) ] × X

0≤a+b≤N

1

2 (a + 1)(b + 1)(a + b + 2)

^Tr

[ | σ(λ(a, b)) | ]

= lim

N →∞

1

log[ ₁₂₀ ¹ (N + 1)(N + 2)(N + 3)(N + 4)(2N + 5)] × X

(a,b):0≤a+b≤N

1

2 (a + 1)(b + 1)(a + b + 2)

^Tr

[ | σ(λ(a, b)) | ].

Sine, the eigenvaluesof theLaplaian

L

^SU

(3)

^on

SU (3)

^{are ofthe form}

− c(λ) = − 1

9 (a ² + b ² + ab + 3a + 3b),

⁽²⁴⁾

the symbol ofthe operator

A = (I − L

^SU

(3) ) ^{− 8 2}

^{(whih isDixmiertraeable) isgiven by}

σ(λ) = (1 + c(λ)) ⁻⁴ .

⁽²⁵⁾

Hene,

Tr

w ((I − L

^SU

(3) ) ^{− 8 2} )

= lim

N→∞

1

log[ ₁₂₀ ¹ (N + 1)(N + 2)(N + 3)(N + 4)(2N + 5)] × X

(a,b):0≤a+b≤N

1

36 (a + 1) ² (b + 1) ² (a + b + 2) ² [1 + (a ² + b ² + ab + 3a + 3b)] ⁻⁴ .

Remark 4.5. Similarexamplesto theonsidered aboveanbeobtainedif wetake, for

example,every

n

^-torus

T ⁿ ∼ = R ⁿ / Z ⁿ

ortheompatLiegroupSO

(3) ∼ = RP ³ ,

^spheres

S ⁿ

and arbitraryrealandomplexprojetivespaes.

Example 4.6. Letus assume that

G

^{isa ompat Liegroup ofdimension}

κ ,

^andletus

dene

A := a(x)( −L G ) ^{− κ 2}

^{. We onsider}

a(x)

^{positive and smooth in order that}

A

^will

beapositive operator. Theglobalsymbol of

A

^{isgiven by}

σ A (x, ξ) := a(x)λ ⁻ _[ξ] ^{κ 2} I d ξ , λ [ξ] 6 = 0,

⁽²⁶⁾

andby using (11)wehavethat

res

(A) = 1 dim(G)

Z

G

a(x)dx × lim

N →∞

1 log N

X

ξ:hξi≤N

d ² _ξ λ ⁻ _[ξ] ^{κ 2} .

⁽²⁷⁾

In partiular, for

G = T ^κ ,

^where

T ≡ [0, 1), d ξ = 1,

^and

λ [ξ] = 1/4π ² | ξ | ² , ξ ∈ Z .

^Inthis

ase,

N lim →∞

1 log N

X

ξ:hξi≤N

d ² _ξ λ ⁻ _[ξ] ^{κ 2}

≍ lim

N→∞

1 log N

X

ξ∈Z ^κ ,1≤|ξ|≤N

| ξ | ^{− κ} ≍ lim

N →∞

1 log N

Z

1≤|ξ|≤N

| ξ | ^{− κ} dξ

≍ lim

N→∞

1

log N log(N ) = 1.

Consequently,

res

(A) ≍ 1 κ

Z

T κ

a(x 1 , · · · , x κ )dx 1 · · · dx κ ,

⁽²⁸⁾

whih isawellknown fat(seee.g. Connes [13 ℄).

Aknowledgements. TheauthorsthankVishveshKumarforusefuldisussionsonthe

subjet.

Referenes

[1℄ Barraza E.S. and Cardona D., On nulear Lp multipliers assoiated to the harmoni

osillator, in Analysisand Partial Dierential Equations: Perspetives from Developing

Countries. SpringerProeedingsinMathematis&Statistisvol275(edsRuzhanskyM.

andDelgadoJ.)Springer,Cham(2019),3141.

[2℄ Boutet deMonvelL., Boundary problemsfor pseudo-dierentialoperators, AtaMath.

126(1971),No.12,1151.

[3℄ ChatzakouM., Delgado J.mand RuzhanskyM., On a lass of anharmoniosillators,

arXiv:1811.12566.

[4℄ CardonaD.,Nulearpseudo-dierentialoperatorsinBesovspaesonompatLiegroups,

J.FourierAnal.Appl.23(2017),No.5,12381262.

[5℄ CardonaD.,Abriefdesriptionofoperatorsassoiatedtothequantumharmoniosillator

onShatten-vonNeumannlasses,Rev.Integr. temasmat.36(2018),No.1,4957.

[6℄ CardonaD.,OnthenuleartraeofFourierIntegral Operators,Rev.Integr. temasmat.

37(2019),No.2,219249.

[7℄ CardonaD.,DelCorral C.andKumarV.,Dixmiertraesfordisretepseudo-dierential

operators,arXiv:1911.03924.

[8℄ Cardona D. and Kumar V., Multilinear analysis for disrete and periodi pseudo-

dierentialoperators inLpspaes,Rev.Integr.temasmat.36(2018),No.2,151164.

[9℄ Cardona D. and Kumar V.,

L ^p −

boundednessand

L ^p

-nulearity of multilinear pseudo- dierentialoperators on

Z ⁿ

andthe torus

T ⁿ

, J.Fourier Anal. Appl.25(2019), No. 6, 29733017.

[10℄ CardonaD.andKumarV.,Thenuleartraeofvetor-valuedpseudo-dierentialoperators

withappliationstoIndextheory,arXiv:1901.10010.

[11℄ Cardona D. andBarraza E.S., Charaterization ofnulear pseudo-multipliersassoiated

totheharmoniosillator,Politehn.Univ.Buharest Si.Bull.Ser.AAppl.Math. Phys

80(2018),No.4,163172.

[12℄ Cardona D. and DelCorral C., The Dixmiertraeand the nonommutativeresidue for

multipliersonompatmanifolds,arXiv:1703.07453.

[13℄ ConnesA.,Nonommutativegeometry,AademiPressIn.,SanDiego,1994.

[14℄ Delgado J. and Ruzhansky M., Shatten-von Neumann lasses of integral operators,

arXiv:1709.06446.

[15℄ DelgadoJ.andRuzhanskyM.,FouriermultipliersinHilbertspaes,arXiv:1707.03062.

[16℄ Delgado J. and RuzhanskyM.,

L ^p

-nulearity,traes, and Grothendiek-Lidskiiformula onompatLiegroups,J.Math. Pures Appl.102(2014),No.1,153172.

[17℄ DelgadoJ.andRuzhanskyM.,Shattenlassesonompatmanifolds:Kernelonditions,

J.Funt.Anal.267(2014),No.3,772798.

[18℄ Delgado J. and Ruzhansky M., Kernel and symbol riteria for Shatten lasses and r-

nulearityonompatmanifolds,C.R.Aad. Si.Paris.352(2014),No.10,779784.

[19℄ Delgado J.and RuzhanskyM.,

L ^p

-bounds for pseudo-dierential operators onompat Liegroups, arXiv:1605.07027.

[20℄ Delgado J. andRuzhansky M., Fourier multipliers, symbolsand nulearity onompat

manifolds,arXiv:1404.6479.

[21℄ DelgadoJ.,RuzhanskyM.andTokmagambetovN.,Shattenlasses,nulearityandnon-

harmonianalysisonompatmanifoldswithboundary,arXiv:1505.02261.

[22℄ DixmierJ.,Existenedetraesnonnormales,C.R.Aad.Si.Paris.SérA-B262(1966),

11071108.

[23℄ Fedosov B., Golse F., LeihtnamE. and ShroheE., The NonommutativeResidue for

ManifoldswithBoundary,J.Funt. Anal.142(1996),No.1,131.

[24℄ GhaemiM. B., JamalpourbirganiM. and Wong M. W.,Charaterizations, adjointsand

produtsof nulearpseudo-dierentialoperators onompatand Hausdorgroups, Po-

litehn.Univ.Buharest Si.Bull.Ser.AAppl.Math.Phys.79(2017),No.4,207220.

[25℄ GrubbG.andShroheE.,Traeexpansionsandthenonommutativeresidueformanifolds

withboundary,J.ReineAngew.Math.536(2001),167207.

[26℄ HörmanderL.,TheAnalysisofthelinearpartialdierentialoperatorsIII,Springer-Verlag,

Berlin,1985.

[27℄ Kumar V.n and Wong M. W.,

C ^∗

-algebras,

H ^∗

-algebras and trae ideals of pseudo- dierentialoperatorsonloallyompat,Hausdorandabeliangroups, J.Pseudo-Dier.

Oper.Appl.10(2019),No.2,269283.

[28℄ NestR.andShroheE.,Dixmier'straeforboundaryvalueproblems,Manusriptamath.

Springer-Verlag96(1998),No.2,203218.

[29℄ PietshA.Traesandresiduesofpseudo-dierentialoperatorsonthetorus,Integr.Equ.

Oper.Theory.83(2015),No.1,123.

[30℄ RuzhanskyM.andTokmagambetovN.,NonharmoniAnalysisofBoundaryValueProb-

lems,InternationalMathematis ResearhNoties,(2016),No.12,35483615.

[31℄ RuzhanskyM.,TurunenV.,Pseudo-DierentialOperatorsandSymmetries.,Bakground

AnalysisandAdvaned Topis,Birkhäuser-Verlag,Basel,2010.

[32℄ RuzhanskyM., Wirth,J.,

L ^p

FouriermultipliersonompatLiegroups.,Math.Z.,280 (2015),No.3-4,621642.

[33℄ Shrohe,E.,NonommutativeResidues,Dixmier'sTrae, andHeatTrae Expansionson

Manifolds withBoundary., Amer.Math. So.Providene, Providene, RI, No.161186,

1999.

[34℄ Wodziki M., Nonommutative residue. I. Fundamentals., Yu. I. Manin (ed.), Leture

NotesinMath.,1289,Springer,Berlin,1987.