Recent results on truncated Toeplitz operators

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by  Isabelle Chalendar

RECENT RESULTS ON TRUNCATED TOEPLITZ OPERATORS
ISABELLECHALENDAR,EMMANUELFRICAIN,ANDDANTIMOTIN
6
1
0 ABSTRACT. TruncatedToeplitzoperatorsarecompressionsofToeplitzoperators
2 onmodelspaces;theyhavereceivedmuchattentioninthelastyears.Thissurvey

n article presents several recent results, which relate boundedness, compactness,
a and spectra of these operators to propertiesof theirsymbols. We also connect
J
thesefactswithpropertiesofthenaturalembeddingmeasuresassociatedtothese

7 operators.

]
A
F
. 1. INTRODUCTION
h
t
a Truncated Toeplitz operators on model spaces have been formally introduced
m
by Sarason in [34], although some special cases have long ago appearedin liter-
[ ature, most notably as model operators for completely nonunitary contractions

1 with defect numbers one and for their commutant. This new area of study has
v beenrecentlyveryactiveandmanyopenquestionsposedbySarasonin[34]have
0
nowbeensolved. See[5,8,12,21,9,20,36,19,35,6,13]. Nevertheless,thereare
1
5 stillbasicandinterestingquestionswhichremainmysterious.
1
The truncated Toeplitz operators live on the model spaces KΘ, which are the
0
. closedinvariantsubspacesforthebackwardshiftoperatorS∗ actingontheHardy
1
0 space H2 (see Section 2 for precise definitions). Given a model space KΘ and a
6 function φ ∈ L2 = L2(T), the truncated Toeplitz operator AΘ (or simply A if
φ φ
1
thereisnoambiguityregardingthemodelspace)isdefinedonadensesubspace
:
v of KΘ as the compression to KΘ of multiplication by φ. The function φ is then
i
X calledasymboloftheoperator. AnalternatewayofdefiningatruncatedToeplitz
r operatorisbymeansofameasure;incaseφisbounded,thenapossiblechoiceof
a
Θ
thedefiningmeasurefor A isφdm(withmLebesguemeasure).
φ
Notethatthesymbolortheassociatedmeasureareneveruniquelydefinedby
theoperator. FromthisandotherpointsofviewthetruncatedToeplitzoperators
havemuchmoreincommonwithHankelOperatorsthanwithToeplitzoperators.
Thispointofviewwillbeoccasionallypursuedthroughoutthepaper.
We intend to survey several recent results that are mostly scattered in the lit-
erature. They focus on the relation between the operator and the symbol or the
measure. Obviously the nonuniqueness is a main issue, and in some situations
it may be avoided by considering the so-called standard symbol of the operator.
2010MathematicsSubjectClassification. 30J05,30H10,46E22.
Keywordsandphrases. TruncatedToeplitzoperators,modelspaces,compactness.
1
2 CHALENDAR,FRICAIN,ANDTIMOTIN
The properties under consideration are boundedness, compactness, and spectra.
Mostoftheresultspresentedareknown,andourintentionisonlytoputthemin
contextandemphasizetheirconnections, indicatingthe relevantreferences. Part
oftheembeddingpropertiesofmeasureshavenotappearedexplicitelyinthelit-
erature,sosomeproofsareprovidedonlywherereferencesseemedtobelacking.
The structure of the paper is the following. After a preliminary section with
generalities about Hardy spaces and model spaces, we discuss in section 3 Car-
leson measures, first for the whole H2 and then for model spaces. Truncated
Toeplitz operators are introduced in Section 4, where one also discusses some
boundednessproperties.Section5isdedicatedtocompactnessoftruncatedToeplitz
operators,andSection6toitsrelationtoembeddingmeasures. Thelasttwosec-
tionsdiscussSchatten–vonNeumannandspectralproperties,respectively.
2. PRELIMINARIES
Forthecontentofthissection,[17]isaclassicalreferenceforgeneralfactsabout
Hardyspaces,while[26]canbeusedforToeplitzandHankeloperatorsaswellas
formodelspaces.
2.1. Functionspaces. Recallthatthe Hardyspace Hp of theunit diskD = {z ∈
C : |z| <1}isthespaceofanalyticfunctions f onDsatisfyingkfk <+∞,where
p
2π dt 1/p
kfk = sup |f(reit)|p , (1≤ p< +∞).
p
0≤r<1(cid:18)Z0 2π(cid:19)
ThealgebraofboundedanalyticfunctionsonDisdenotedbyH∞. Wedenotealso
Hp = zHpandHp =zHp. Alternatively,Hpcanbeidentified(viaradiallimits)to
0 −
thesubspaceoffunctions f ∈ Lp = Lp(T)forwhich fˆ(n) = 0foralln < 0. Here
TdenotestheunitcirclewithnormalizedLebesguemeasurem.
Inthecasep=2,H2becomesaHilbertspacewithrespecttothescalarproduct
inheritedfromL2 andgivenby
hf,gi = f(ζ)g(ζ)dm(ζ), f,g∈ L2.
2
T
Z
The orthogonal projection from L2 to H2 will be denoted by P+. The space H−2
ispreciselytheorthogonalof H2,andthecorrespondingorthogonalprojectionis
P− = I−P+.
ThePoissontransformofafunction f ∈ L1 is
1−|z|2
(2.1) fˆ(z)= f(ξ) dξ, z ∈D.
T |1−ξz¯|2
Z
SupposenowΘisaninnerfunction,thatisafunctioninH∞whoseradiallimits
areofmodulusonealmosteverywhereonT. Itsspectrumisdefinedby
(2.2) s(Θ):={ζ ∈D : liminf |Θ(λ)| =0}.
λ∈D,λ→ζ
RECENTRESULTSONTRUNCATEDTOEPLITZOPERATORS 3
Equivalently,ifΘ = BSisthedecompositionofΘintoaBlaschkeproductand
asingular innerfunction, thenρ(Θ)istheunionbetweentheclosureofthe limit
points of the zerosof B and the support of the singular measureassociated to S.
Wewillalsodefine
ρ(Θ)=s(Θ)∩T.
Wedefinethecorrespondingshift-coinvariantsubspacegeneratedbyΘ(alsocalled
modelspace)bytheformulaKp = Hp∩ΘHp,where1 ≤ p < +∞. Wewillbees-
Θ 0
pecially interested in the Hilbert case, that is when p = 2. In this case, we also
denotebyKΘ =KΘ2 anditiseasytoseethatKΘ isalsogivenbythefollowing
KΘ = H2⊖ΘH2 = f ∈ H2 : hf,gi=0,∀g∈ H2 .
n o
The orthogonal projection of L2 onto KΘ is denoted by PΘ. Itis wellknown (see
[26,page34])thatPΘ = P+−ΘP+Θ¯. SinceP+ actsboundedlyonLp,1 < p < ∞,
this formula shows that PΘ canalso be regardedasa bounded operator from Lp
intoKp,1< p< ∞.
Θ
ThespacesH2 andKΘ arereproducingkernelspacesovertheunitdiscD. The
respectivereproducingkernelsare,forλ ∈D,
1
k (z) = ,
λ 1−λ¯z
1−Θ(λ)Θ(z)
kΘ(z) = .
λ 1−λ¯z
Evaluationsatcertainpointsζ ∈ Tmayalsobeboundedsometimes;thishap-
penspreciselywhenΘhasanangularderivativeinthesenseofCaratheodoryat
ζ[1].InthiscasethefunctionkΘζ (z) = 1−Θ1(−ζζ)¯zΘ(z) isinKΘ,anditisthereproducing
kernelforthepointζ.
Iteasytocheckthat,if f,g ∈ KΘ,then fg ∈ H1∩z¯Θ2H−1 ⊂ KΘ12. Inparticular,
if f,garealsobounded,then fg ∈K . So(kΘ)2 ∈K forallλ∈D.
Θ2 λ Θ2
ThemapCΘ definedonL2 by
(2.3) CΘf =Θz¯f¯;
is a conjugation (i.e. CΘ is anti-linear, isometric and involutive), which has the
convenientsupplementarypropertyofmappingKΘ preciselyontoKΘ.
2.2. One-componentinnerfunctions. In view of their main role in the study of
operatorsonmodelspaces,wedevotethissubsectiontoaparticularclassofinner
functions. Fixanumber0<ǫ <1,anddefine
(2.4) Ω(Θ,ǫ)={z ∈D : |Θ(z)| <ǫ}.
ThefunctionΘiscalledone-componentifthereexistsavalueofǫforwhichΩ(Θ,ǫ)
isconnected.(Ifthishappens,thenΩ(Θ,δ)isconnectedforeveryǫ< δ<1.)One-
component functionshave beenintroducedbyCohn [15]. Anextensivestudyof
thesefunctionsappearsin[4,3];allresultsquotedbelowappearin[3].
4 CHALENDAR,FRICAIN,ANDTIMOTIN
Theabovedefinitionisnotverytransparent. Infact,one-componentfunctions
areratherspecial: afirstimmediatereasonisthattheymustsatisfym(ρ(Θ)) = 0.
Thiscondition,ofcourse,isnotsufficient,butitsuggestsexaminingsomesimple
cases.
The setρ(Θ) isemptyforfinite Blaschkeproducts, whichareone-component.
The next simplest case is when ρ(Θ) consists of just one point. One can prove
z+ζ
easily that the elementary singular inner functions Θ(z) = ez−ζ (for ζ ∈ T) are
indeedone-component.
SupposethenthatΘisaBlaschkeproductwhosezerosa tendnontangentially
n
toasinglepointζ ∈T. If
|ζ−a |
(2.5) inf n+1 >0,
n≥1 |ζ−an|
then Θ is one-component. So, in particular, if 0 < r < 1 and Θ is the Blaschke
productwith zeros 1−rn, n ≥ 1, then Θ is one-component. If condition (2.5) is
notsatisfied,thenusuallyΘisnotone-component. Adetaileddiscussionofsuch
Blaschkeproductsisgivenin[3],includingthedeterminationoftheclassesC (Θ)
p
(seeSubsection3.2).
∞
One-componentinnerfunctionscanbecharacterizedbyanestimateonthe H
norm of the reproducing kernels kΘ. While for a general inner function Θ we
λ
have kkΘk∞ = O(1−|λ|−1), this estimate can be improved for one-component
λ
functions: Θ is one-component if and only if there exists a constant C > 0 such
thatforeveryλ∈D,wehave
1−|Θ(λ)|
kkΘλk∞ ≤ C 1−|λ| .
2.3. Multiplication operators and their cognates. For φ ∈ L∞, we denote by
Mφf =φf themultiplicationoperatoronL2;wehavekMφk =kφk∞.TheToeplitz
operatorT : H2 −→ H2 andtheHankeloperatorH : H2 −→ H2 = L2⊖H2 are
φ φ −
givenbytheformulae
Tφ = P+Mφ, Hφ = P−Mφ.
In the case where φ is analytic, T is just the restriction of M to H2. We have
φ φ
Tφ∗ = TφandHφ∗ = P+MφP−.
Itshouldbenotedthat,whilethesymbolsofM andT areuniquelydefinedby
φ φ
theoperators,thisisnotthecasewithH . Indeed,itiseasytocheckthatH = H
φ φ ψ
ifandonlyifφ−ψ ∈ H∞. SostatementsaboutHankeloperatorsoftenimplyonly
theexistenceofasymbolwithcorrespondingproperties.
TheHankeloperatorshavetherangeanddomainspacesdifferent. Itissome-
times preferableto work with an operator acting on a single space. For this, we
introduceinL2 theunitarysymmetryJ definedby
J(f)(z)= z¯f(z¯).
RECENTRESULTSONTRUNCATEDTOEPLITZOPERATORS 5
WehavethenJ(H2) = H2 andJ(H2) = H2. DefineΓ : H2 → H2 by
− − φ
(2.6) Γ =JH .
φ φ
ObviouslypropertiesofboundednessorcompactnessarethesameforH andΓ .
φ φ
Thedefinitionof M , T and H canbeextendedtothecasewhenthesymbol
φ φ φ
φ is only in L2 instead of L∞, obtaining (possibly unbounded) densily defined
operators. Then M and T are bounded if and only if φ ∈ L∞ (and kM k =
φ φ φ
kTφk =kφk∞). Thesituationismorecomplicatedfor Hφ. Namely,Hφisbounded
ifandonlyifthereexistsψ ∈ L∞ withH = H ,and
φ ψ
kHφk =inf{kψk∞ : Hφ = Hψ}
ThisisknownasNehari’sTheorem;see,forinstance,[24,p. 182]. Moreover(but
wewillnotpursuethisinthesequel)anequivalentconditionisP−φ ∈ BMO(and
kHφkisthenanormequivalenttokP−φkBMO).
Relatedresultsareknownforcompactness. TheoperatorsM andT arenever
φ φ
compact except in the trivial case φ ≡ 0. Hartman’s Theorem states that H is
φ
compact if and only if there exists ψ ∈ C(T) with H = H ; or, equivalently,
φ ψ
P−φ ∈ VMO. If we know that φ is bounded, then Hφ is compact if and only if
φ ∈C(T)+H∞.
3. CARLESONMEASURES
3.1. EmbeddingofHardyspaces. Letusdiscussfirstsomeobjectsrelatedtothe
Hardyspace;wewillafterwardsseewhatanalogousfactsaretrueforthecaseof
modelspaces.
A positive measure µ on D is called a Carleson measure if H2 ⊂ L2(µ) (such
an inclusion is automatically continuous). It is known that this is equivalent to
Hp ⊂ Lp(µ)forall1 ≤ p < ∞. Carlesonmeasurescanalsobecharacterizedbya
geometricalcondition,asfollows. ForanarcI ⊂Tsuchthat|I| <1wedefine
S(I) ={z ∈D :1−|I| <|z| <1andz/|z| ∈ I}.
ThenµisaCarlesonmeasureifandonlyif
µ(S(I))
(3.1) sup < ∞.
|I|
I
Condition(3.1)iscalledtheCarlesoncondition.
Theresultcanactuallybeextended(see[10])tomeasuresdefinedonD. Again
the characterizationdoesnot dependon p, and itamounts to the factthat µ|T is
absolutelycontinuouswithrespecttoLebesguemeasurewithessentiallybounded
density,whileµ|D satisfies(3.1).
ThereisalinkbetweenHankeloperatorsandCarlesonmeasuresthathasfirst
appearedin[29,39];acomprehensivepresentationcanbefindin[28,1.7].Letµbe
6 CHALENDAR,FRICAIN,ANDTIMOTIN
afinitecomplexmeasureonD. DefinetheoperatorΓ[µ]onanalyticpolynomials
bytheformula
hΓ[µ]f,gi = zf(z)g(z¯)dµ(z).
D
Z
Note thatif µ is supported on T, then the matrix of Γ[µ] in the standardbasisof
H2is(µˆ(i+j)) ,whereµˆ(i)aretheFouriercoefficientsofµ.
i,j≥0
Then the operator Γ[µ] is bounded whenever µ is a Carleson measure. Con-
versely,ifΓ[µ]isbounded,thenthereexistsaCarlesonmeasureνonD suchthat
Γ[µ] =Γ[ν].
It iseasy tosee thatif dµ = φdm for some φ ∈ L∞, then Γ[µ] = Γ , where Γ
φ φ
hasbeendefinedby(2.6)andistheversionofaHankeloperatoractingonasingle
space.
Analogousresultsmaybeprovedconcerningcompactness. Inthiscasetherel-
evantnotionisthatofvanishingCarlesonmeasure,whichisdefinedbytheproperty
µ(S(I))
(3.2) lim =0.
|I|→0 |I|
NotethatvanishingCarlesonmeasurescannothavemassontheunitcircle(inter-
vals containing a Lebesgue point of the corresponding density would contradict
thevanishingcondition). ThentheembeddingHp ⊂ Lp(µ)iscompactifandonly
ifµisavanishingCarlesonmeasure.
AsimilarconnectionexiststocompactnessofHankeloperators.Ifµisavanish-
ingCarlesonmeasureonD,thenΓ[µ] iscompact. Conversely,ifΓ[µ] iscompact,
thenthereexistsavanishingCarlesonmeasureνonDsuchthatΓ[µ] = Γ[ν].
3.2. Embeddingofmodelspaces. Similarquestionsformodelspaceshavebeen
developed starting with the papers[15, 16] and [38]; however, the results in this
case are less complete. Let us introduce first some notations. For 1 ≤ p < ∞,
define
C (Θ) ={µfinitemeasureonT : Kp ֒→ Lp(|µ|)isbounded},
p Θ
C+(Θ) ={µpositivemeasureonT : Kp ֒→ Lp(µ)isbounded},
p Θ
V (Θ) ={µfinitemeasureonT : Kp ֒→ Lp(|µ|)iscompact},
p Θ
V+(Θ) ={µpositivemeasureonT : Kp ֒→ Lp(µ)iscompact}.
p Θ
ItisclearthatC (Θ)andV (Θ)arecomplexvectorialsubspacesofthecomplex
p p
measuresontheunitcircle. UsingtherelationsKΘ2 = KΘ⊕ΘKΘ andKΘ·KΘ ⊂
K1 ,itiseasytoseethatC (Θ2) =C (Θ),C (Θ2)⊂ C (Θ),andV (Θ2)⊂ V (Θ).
Θ2 2 2 1 2 1 2
Itisnaturaltolookforgeometricconditionstocharacterizetheseclasses.Things
are,however,morecomplicated,andtheresultsareonlypartial.Westartbyfixing
anumber0< ǫ <1;thenthe(Θ,ǫ)-Carlesonconditionassertsthat
µ(S(I))
(3.3) sup < ∞,
|I|
I
RECENTRESULTSONTRUNCATEDTOEPLITZOPERATORS 7
wherethesupremumistakenonlyovertheintervals|I|suchthatS(I)∩Ω(Θ,ǫ)6=
∅. (RememberthatΩ(Θ,ǫ)isgivenby(2.4).)
Itisthenprovedin[38]thatifµsatisfiesthe(Θ,ǫ)-Carlesoncondition,thenthe
embeddingKp ⊂ Lp(µ)iscontinuous. TheconverseistrueifΘisone-component;
Θ
inwhichcasethe embeddingcondition doesnotdependon p, while fulfillingof
the (Θ,ǫ)-Carleson condition does not depend on 0 < ǫ < 1 (see Theorem 3.1
below).
Asconcernsthegeneralcase,itisshownbyAleksandrov[3]thatiftheconverse
istrueforsome1 ≤ p < ∞,thenΘisone-component. Also,Θisone-component
if and only if the embedding condition does not depend on p. More precisely,
thenexttheoremisprovedin[3](notethataversionofthisresultfor p ∈ (1,∞)
alreadyappearsin[38]).
Theorem3.1. ThefollowingareequivalentforaninnerfunctionΘ:
(1) Θisone-component.
(2) Forsome0 < p < ∞and0 < ǫ < 1,C (Θ)concideswiththeclassofmeasures
p
thatsatisfythe(Θ,ǫ)-Carlesoncondition.
(3) Forall0 < p < ∞ and 0 < ǫ < 1, C (Θ) concideswiththeclassofmeasures
p
thatsatisfythe(Θ,ǫ)-Carlesoncondition.
(4) TheclassC (Θ)doesnotdependon p∈ (0,∞).
p
Inparticular,ifΘisonecomponent,thensoisΘ2,whenceC (Θ2) = C (Θ2) =
1 2
C (Θ).
2
NotethatageneralcharacterizationofC (Θ)hasrecentlybeenobtainedin[22];
2
however,thegeometriccontentofthisresultisnoteasytosee.
ThequestionofcompactnessoftheembeddingKp ⊂ Lp(µ)inthiscaseshould
Θ
berelatedtoavanishingCarlesoncondition. Infact,therearetwovanishingcon-
ditions, introducedin[14]. Whatiscalledthereinthesecondvanishingconditionis
easiertostate. Wesaythatµsatisfiesthesecond(Θ,ǫ)-vanishingcondition[7,14]if
foreachη > 0thereexistsδ > 0suchthatµ(S(I))/|I| < η whenever|I| < δand
S(I)∩Ω(Θ,ǫ)6= ∅. Thefollowingresultisthenprovedin[7].
Theorem3.2. Ifthepositivemeasure µ satisfiesthesecond (Θ,ǫ)-vanishingcondition,
thentheembeddingKp ⊂ Lp(µ)iscompactfor1< p< ∞.
Θ
TheconverseistrueincaseΘisone-component.
In other words, the theoremthus statesthatpositive measuresthatsatisfythe
second vanishingcondition arein V+(Θ) forall1 < p < ∞, and theconverse is
p
trueforΘone-component.
To discuss the case p = 1, we have to introduce what is called in [14] the first
vanishingcondition. Letuscallthesupremumin(3.3)the(Θ,ǫ)-Carlesonconstant
ofµ. Define
(3.4) H ={z ∈D :dist(z,ρ(Θ))<δ},
δ
8 CHALENDAR,FRICAIN,ANDTIMOTIN
and µ (A) = µ(A∩ H ). Then µ are also Θ-Carleson measures, with (Θ,ǫ)-
δ δ δ
Carleson constants decreasing when δ decreases. We say that µ satisfies the first
(Θ,ǫ)-vanishingconditioniftheseCarlesonconstantstendto0whenδ→0.
Itisshownin[7]thatthefirstvanishingconditionimpliesthesecond,andthat
theconverseisnottrue: thereexistmeasureswhichsatisfythesecondvanishing
conditionbutnotthefirst.
Thenexttheoremisprovedin[14].
Theorem3.3. Ifapositivemeasureµsatisfiesthefirst(Θ,ǫ)-vanishingcondition,then
µ∈ V+(Θ)for1≤ p< ∞.
p
In case µ ∈ C (Θ), we will denote by ι : Kp → Lp(|µ|) the embedding
p µ,p Θ
(whichisthenknowntobeaboundedoperator). Thenµ ∈ V (Θ)meansthatι
p µ,p
iscompact. Wewillalsowriteι insteadofι .
µ µ,2
4. TRUNCATEDTOEPLITZOPERATORS
LetΘbeaninner functionand φ ∈ L2. ThetruncatedToeplitzoperator A =
φ
Θ
A ,introducedbySarasonin[34],willbeadenselydefined,possiblyunbounded
φ
operatoronKΘ. ItsdomainisKΘ∩H∞,onwhichitactsbytheformula
∞
Aφf = PΘ(φf), f ∈ KΘ∩H .
If A thus defined extends to a bounded operator, that operator is called a TTO.
φ
TheclassofallTTOsonKΘ isdenotedbyT(Θ),andtheclassofallnonnegative
TTO’sonKΘ isdenotedbyT(Θ)+.
AlthoughtheseoperatorsarecalledtruncatedToeplitz,theyhavemoreincom-
mon with Hankel operators H , or rather with their cognates Γ , which act on a
φ φ
single space. As a first example of this behavior, we note that the symbol of a
truncatedToeplitzoperatorsisnotunique. Itisprovedin[34]that
(4.1) A = A ⇐⇒φ −φ ∈ ΘH2+ΘH2.
φ1 φ2 1 2
LetusdenoteSΘ = L2⊖(ΘH2+ΘH2);itiscalledthespaceofstandardsymbols.
It follows from (4.1) that every TTO has a unique standard symbol. One proves
in[34,Section3]thatSiscontainedinKΘ+KΘ asasubspaceofcodimensionat
mostone;thislastspaceissometimeseasiertoworkwith.
ItisoftenthecasethattheassumptionΘ(0)=0simplifiescertaincalculations.
For instance, in that case we have precisely S = KΘ+KΘ; we will see another
example in Section 7. Fortunately, there is a procedure to pass from a general
inner Θtoonethathasthisproperty: itiscalledtheCrofoottransform. For a ∈ D
letΘ begivenbytheformula
a
Θ(z)−a
Θ (z) = .
a 1−a¯Θ(z)
RECENTRESULTSONTRUNCATEDTOEPLITZOPERATORS 9
IfwedefinetheCrofoottransformby
1−|a|2
J(f):= f,
1−a¯Θ
p
then JisaunitaryoperatorfromKΘ toKΘ ,and
a
(4.2) JT(Θ)J∗ =T(Θ ).
a
Inparticular,ifa = Θ(0),thenΘ (0) = 0,and(4.2)allowsthetransferofproper-
a
tiesfromTTOsonKΘ toTTOsonKΘ.
a
Especiallynice propertiesareexhibited by TTOswhich have ananalytic sym-
bol φ ∈ H2 (of course, this is never a standard symbol). It is a consequence of
interpolationtheory[33]that
{AΘ ∈T(Θ) : φ ∈ H2}={AΘ}′
φ z
Θ
(A iscalledacompressedshift,oramodeloperator).
z
One should also mentioned that other two classes of TTOs have alreadybeen
studiedin differentcontexts. First, the classicalfinite Toeplitz matricesareTTOs
with Θ(z) = zn writtenin the basisof monomials. Secondly, TTOswith Θ(z) =
z+1
ez−1 correspond,aftersomestandardtransformations,toaclassofoperatorsalter-
natelycalledToeplitzoperatorsonPaley–Wienerspaces[31],ortruncatedWiener–
Hopfoperators[11].
There is an alternate manner to introduce TTOs, related to the Carleson mea-
suresintheprevioussection. Foreveryµ∈ C (Θ)thesesquilinearform
2
(f,g)7→ fg¯dµ
Z
Θ
isbounded,andthereforethereexistsaboundedoperator Aµ onKΘ suchthat
Θ
(4.3) hA f,gi= fg¯dµ.
µ
Z
Θ
Itisshownin[34,Theorem9.1]that A thusdefinedisactuallyaTTO.Infact,
µ
the converse is also true, as stated in Theorem 4.2 below. An interesting open
questionisthecharacterizationofthemeasuresµforwhich A =0.
µ
ThedefinitionofTTOsdoesnotmakeprecisetheclassofsymbols φ ∈ L2 that
produceboundedTTOs. Afirstremarkisthatthestandardsymbolofabounded
truncatedToeplitzoperatorisnotnecessarilybounded. Togiveanexample,con-
sider an inner function Θ with Θ(0) = 0, for which there exists a singular point
ζ ∈TwhereΘhasanangularderivativeinthesenseofCaratheodory.Itisshown
then in [34, Section 5] that kΘ ⊗kΘ is a bounded rank one TTO with standard
ζ ζ
symbolkΘ+kΘ−1,andthatthislastfunctionisunbounded.
ζ ζ
10 CHALENDAR,FRICAIN,ANDTIMOTIN
A natural question is therefore whether every bounded TTO has a bounded
symbol(suchasisthecasewithHankeloperators). Inthecaseof T with φana-
φ
lytic, the answer is readilyseento be positive, being provedagainin [33]; more-
over,
inf{kψk∞ : ψ ∈ H∞, AΘψ = AΘφ}= kAΘφk.
Thefirstnegativeanswerforthegeneralsituationhasbeenprovidedin[6],and
the counterexample is again given by the rank one TTO kΘ⊗kΘ. The following
ζ ζ
resultisprovedin[6].
Theorem4.1. SupposeΘhasanangularderivativeinthesenseofCaratheodoryinζ ∈T
(equivalently,kΘ ∈ L2),butkΘ 6∈ Lpforsomep∈ (2,∞).ThenkΘ⊗kΘhasnobounded
ζ ζ ζ ζ
symbol.
Amoregeneralresulthasbeenobtainin[5],whereonealsomakesclearthere-
lationbetweenmeasuresandTTO.Inparticular,onecharacterizestheinnerfunc-
tions Θ which have the propertythateverybounded TTOon KΘ hasa bounded
symbol.
Theorem4.2. SupposeΘisaninnerfunction.
(1) ForeveryboundedTTO A ≥ 0thereexistsapositivemeasureµ ∈ C+(Θ)such
2
thatA = AΘ.
µ
(2) Forevery bounded A ∈ T(Θ) thereexistsa complexmeasure µ ∈ C (Θ) such
2
thatA = AΘ.
µ
(3) AboundedTTO A ∈ T(Θ)admitsaboundedsymbolifandonlyif A = AΘ for
µ
someµ∈ C (Θ2).
1
(4) Every bounded TTO on KΘ admits a bounded symbol if and only if C1(Θ2) =
C (Θ2).
2
Inparticular,asshownbyTheorem3.1,thesecondconditionissatisfiedifΘis
one-component(sincethenallclassesC (Θ)coincide). Itisstillanopenquestion
p
whether Θ one-component is actuallyequivalent to C (Θ2) = C (Θ2). (Asmen-
1 2
tionedpreviously,Θisone-componentifandonlyifΘ2 isone-component.) Such
aresultwouldbeasignificantstrengtheningofTheorem3.1.
Asageneralobservation,onemaysaythat,ifΘisone-component, thenTTOs
on KΘ havemanypropertiesanalogoustothose of Hankeloperators. Thisisthe
classofinnerfunctionsforwhichthecurrenttheoryismoredeveloped.
5. COMPACT OPERATORS
Surprisingly enough, the first result about compactness of TTOs dates from
1970. In [1, Section 5] one introduces what are, in our terminology, TTOs with
continuoussymbol,andoneprovesthefollowingtheorem.
Theorem5.1. IfΘisinnerandφiscontinuousonT,then AΘ iscompactifandonlyif
φ
φ|ρ(Θ)=0.