Truncated Toeplitz operators and boundary values in nearly invariant subspaces

0.13 MB·English

by  Andreas Hartmann

TRUNCATED TOEPLITZ OPERATORS AND BOUNDARY VALUES IN NEARLY
INVARIANT SUBSPACES
ANDREASHARTMANN&WILLIAMT.ROSS
1
1
0
ABSTRACT. WeconsidertruncatedToeplitzoperatoronnearlyinvariantsubspacesoftheHardy
2 spaceH2. Ofsomeimportanceinthiscontextistheboundarybehaviorofthefunctionsinthese

n spaceswhichwewilldiscussinsomedetail.
a
J

9
1

1. INTRODUCTION
]
V
C If H2 is the classical Hardy space [Dur70], we say a (closed) subspace M ⊂ H2 is nearly
. invariant when
h
f
at f ∈M,f(0) =0⇒ ∈M.
z
m
Thesesubspaceshavebeencompletelycharacterizedin[Hit88,Sar88]andcontinuedtobestud-
[
ied in [AR96, HSS04, MP05, KN06, AK08, CCP10]. In this paper, we will examine certain

1
propertiesofnearly invariantsubspaces. Weaimto accomplishthreethings.
v
1
First,inTheorem3.1weobservethatmuchofwhatwealreadyknowabouttruncatedToeplitz
7
7 operators on model spaces [Sar07] can be transferred, via a unitary operator to be introduced
3 below,mutatismutandistotruncated operators onnearly invariantsubspaces.
.
1
Secondly, in Theorem 4.5 we will show that every function in a nearly invariant subspace M
0
1 has a finite non-tangential limit at ∣ζ∣ = 1 if and only if (i) g has a finite non-tangential limit at
1
ζ and (ii) the reproducing kernel functions for M are uniformly norm bounded in Stolz regions
:
v withvertexat ζ. Thisparallelsaresult byAhern and Clark inmodelspaces.
i
X
Third, in order to better understand the self-adjoint rank-one truncated Toeplitz operators on
r
a nearlyinvariantsubspaces,wewilldiscussthenon-tangentiallimitsoffunctionsinthesespaces.
It turns out (Theorem 5.2) there is a kind of dichotomy: If every function in a nearly invariant
subspace M has a boundary limit at a fixed point ∣ζ∣ = 1 then either ζ is a point where every
1
function in M (where g is the extremal function for M - see definition below) has a finite
g
non-tangentiallimitoreveryfunctioninM hasnon-tangentiallimit0 at ζ.
Date:January20,2011.
1991MathematicsSubjectClassification. 30B30,47B32,47B35.
Key wordsandphrases. truncatedToeplitzoperator,kernelfunction,nearlyinvariantsubspace, non-tangential
limits.
ThisworkhasbeendonewhilethefirstnamedauthorwasstayingatUniversityofRichmondastheGaineschair
inmathematics.Hewouldliketothankthatinstitutionforthehospitalityandsupportduringhisstay.Thisauthoris
alsopartiallysupportedbyANRFRAB.
1
2 ANDREASHARTMANN&WILLIAMT.ROSS
A word concerning numbering in this paper: in each section, we have numbered theorems,
propositions,lemmas,corollaries andequationsconsecutively.
2. PRELIMINARIES
If H2 is the Hardy space of the open unit disk D (with the usual norm ∥ ∥) and I is inner,
let K = H2 IH2 be the well-studied model space [Nik86]. Note that H2,⋅as well as K , are
I I
regarded, via⊖non-tangential boundary values on T = ∂D, as subspaces of L2 = L2(T,dθ/2π)
[Dur70]. Itis wellknownthatH2 isa reproducingk∶ernel Hilbertspacewithker∶nel
1
k (z) = ,
λ
∶ 1 λz
−
as isK withkernel
I
1 I(λ)I(z)
kI(z) = − .
λ
∶ 1 λz
Note that kI is bounded and that finite linear comb−inations of them form a dense subset of K .
λ I
AlsonotethatifP is theorthogonalprojectionofL2 onto K , then
I I
(P f)(λ)= f,kI , λ∈D.
I λ
⟨ ⟩
Note that for every λ ∈ D this formula extends to f ∈ L1 = L1 T,dθ 2π so that we can define
P forL1-functions(which ingeneral doesnotgivean H1-func(tion)./ )
I
By a theorem of Ahern and Clark [AC70], every function f ∈ K has a finite non-tangential
I
limitat aboundarypointζ ∈T, i.e.,
f ζ = limf λ
( )∶ ∠λ→ζ ( )
exists for every f ∈ K , if and only if ζ ∈ ADC I . Here ζ ∈ ADC I means that I has a finite
I
( ) ( )
angularderivativeinthesenseofCarathe´odory,meaning
limI λ =η ∈T
∠λ→ζ ( )
and
limI′ λ exists.
∠λ→ζ ( )
Moreover,wheneverζ ∈ADC I , thelinearfunctionalf ↦f ζ is continuouson K givingus
I
( ) ( )
akernel functionkI forK at theboundarypointζ. That istosay
ζ I
f ζ = f,kI f ∈K .
ζ I
( ) ⟨ ⟩ ∀
Sarason [Sar07] begana study,taken up by others, ofthe truncatedToeplitz operatorson K .
I
These operators are defined as follows: For ϕ ∈ L2, define the truncated Toeplitz operator A
ϕ
denselyon theboundedf ∈K by
I
A f =P ϕf .
ϕ I
( )
Let denotetheA whichextendtobeboundedoperatorsonK . Certainlywhenϕisbounded,
I ϕ I
T
then A ∈ , but there are unbounded ϕ which yield A ∈ . Moreover, there are bounded
ϕ I ϕ I
T T
truncatedToeplitzoperatorswhichcannotberepresentedbyaboundedsymbol[BCF+10]. Much
TRUNCATEDTOEPLITZOPERATORSANDBOUNDARYVALUESINNEARLYINVARIANTSUBSPACES 3
is known about these operators (see the Sarason paper [Sar07] for a detailed discussion)but we
listafew interestingfacts below:
(1) is aweakly closedlinearsubspaceofoperators onK .
I I
T
(2) A ≡0ifand onlyifϕ∈IH2 IH2.
ϕ
+
(3) The operator Cf = zfI (considered as boundary functions on T) defines an isometric,
∶
anti-linear, involution on K for which CA C = A∗ = A . The operator C is called a
I ϕ ϕ ϕ
conjugation. Thismakes acollectionofcomplexsymmetricoperators[GP06, GP07].
I
T
(4) A boundedoperatorAon K belongsto ifand onlyiftherearefunctionsϕ ,ϕ ∈K
I I 1 2 I
T
so that
A A AA∗ = ϕ kI kI ϕ .
z z 1 0 0 2
− ( ⊗ )+( ⊗ )
In theabove,weusethenotationf g fortherank-oneoperator
⊗
f g h = h,g f.
( ⊗ )( )∶ ⟨ ⟩
(5) An operatorAon K isarank-onetruncatedToeplitzoperatoron ifand onlyifitcan be
I
writtenas constantmultipleofoneofthefollowingthreetypes
kI CkI, CkI kI, kI kI
λ λ λ λ ζ ζ
⊗ ⊗ ⊗
where λ ∈ D, ζ ∈ ADC I . The two first ones are the non-selfadjoint and the last ones
( )
are theselfadjointtruncatedToeplitzoperators.
(6) Sedlock[Sed](seealso[GRW])showedthateverymaximalalgebrain canbewritten
I
T
as thecommutantofageneralizationoftheClark unitaryoperator.
(7) In [CGRW10] they show, for two inner functions I and I , that is spatially isomor-
1 2 I1
T
phic to if and only if either I = ψ I φ or I = ψ I z φ for some disk
I2 1 2 1 2
T ○ ○ ○ ( ) ○
automorphismsφ,ψ.
As we have already mentioned, a nearly invariant subspace M is a closed subspace of H2
such that
f
f ∈M,f 0 =0⇒ ∈M.
( ) z
A result ofHitt [Hit88] says that if g (called the extremalfunction for M) is theuniquesolution
totheextremalproblem
(2.1) sup Reg 0 g ∈M, g =1 ,
{ ( )∶ ∥ ∥ }
thenthere isan innerfunctionI so that
M =gK
I
and moreover,themap
U K M, U f =gf,
g I g
∶ →
isisometric.
Note that necessarily we have the condition I 0 = 0 since g ∈ M = gK and so 1 ∈ K . If
I I
one were to choose any g and any inner I, then g(K) is, in general, not even a subspace of H2.
I
4 ANDREASHARTMANN&WILLIAMT.ROSS
Sarason showsin[Sar88]thatwhen I 0 =0,everyisometricmultiplieron K takes theform
I
( )
a
(2.2) g = ,
1 Ib
−
where a and b are in the unit ball of H∞ with a2 b2 = 1 a.e. on T. As a consequence, gK is
I
a(closed)nearly invariantsubspaceofH2 with∣ e∣x+tre∣m∣ alfunctiong as in(2.1).
Remark 2.3. From now on, whenever we speak of a nearly invariant subspace M = gK , we
I
will always assume that I 0 = 0 and that g is extremal for M (as in (2.1)) and of the form in
( )
(2.2). We will say that gK is a nearly invariant subspace with extremal function g and suitable
I
innerfunctionI withI 0 =0.
( )
Our first step towards defining a truncated Toeplitz operator on the nearly invariant subspace
M =gK , as Sarason did forK ,is tounderstandP , theorthogonalprojectionofL2 ontoM.
I I M
Lemma 2.4. Let M = gK be a nearly invariant subspace with extremal function g and associ-
I
atedinner functionI, I 0 =0. Then
( )
P f =gP gf , f ∈M.
M I
( )
Proof. To show that the map f gP gf is indeed P , we need to show that this map is the
I M
identityon M andvanishesonL↦2 M(. )
⊖
Letf =gh∈gK withh∈K . Thenforevery λ∈D wehave
I I
P ggh λ = ggh,kI = gh,gkI .
I λ λ
( )( ) ⟨ ⟩ ⟨ ⟩
But,fromourpreviousdiscussion,multiplicationbytheextremalfunctiong isanisometryfrom
K ontoM andso
I
gh,gkI = h,kI =h λ .
λ λ
⟨ ⟩ ⟨ ⟩ ( )
Thus
g λ P gf λ =f λ ,
I
( )( )( ) ( )
inotherwords,f gP gf is theidentityonM.
I
↦ ( )
Tofinishtheproof,weneed toshowthat themapf gP gf vanisheson L2 M. Clearly,
I
when f ∈H2 = h h∈H2,h 0 =0 then P gf =0.↦When(f ∈)H2 M we hav⊖e
0 I
{ ∶ ( ) } ( ) ⊖
P gf λ = gf,kI = f,gkI =0
I λ λ
( )( ) ⟨ ⟩ ⟨ ⟩
sincegkI ∈M and f M.
λ ⊥ ∎
Corollary2.5. Thereproducingkernel forM =gK isgiven by
I
1 I λ I z
kM z =g λ g z − ( ) ( ).
λ
( ) ( ) ( ) 1 λz
−
Proof. ObservethatkM z = P k z =g z P gk z and
λ ( ) ( M λ)( ) ( )( I λ)( )
P gk z = gk ,kI = k ,gkI = gkI λ =g λ kI z .
I λ λ z λ z z λ
( )( ) ⟨ ⟩ ⟨ ⟩ ( )( ) ( ) ( )
∎
TRUNCATEDTOEPLITZOPERATORSANDBOUNDARYVALUESINNEARLYINVARIANTSUBSPACES 5
3. TRUNCATED TOEPLITZ OPERATORS ON NEARLY INVARIANT SUBSPACES
Weare nowabletointroducetruncatedToeplitzoperatorsonnearly invariantsubspaces. Cer-
tainlywheneverϕ isaboundedfunctionwecan useLemma2.4tosee thattheoperator
AMf =P ϕf =gP gϕf , f ∈M,
ϕ M I
∶ ( )
iswell-defined and bounded.
The most general situation is when the symbol ϕ is a Lebesgue measurable function on T
and g 2ϕ ∈ L2. Then, for every bounded h ∈ K , the function g 2ϕh belongs to L2 and so
I
P g∣ 2∣ϕh ∈K , and,by theisometricmultiplierproperty ofg inK∣ ∣, weget
I I I
(∣ ∣ )
2
P ϕh =gP g ϕh ∈gK =M.
M I I
( ) (∣ ∣ )
Notethatbytheisometricpropertyofg,thesetgK∞,whereK∞ isthesetofboundedfunctions
I I
inK , is densein gK . Hence, inthissituation,AM is denselydefined.
I I ϕ
Let M denote the densely defined AM, g 2ϕ ∈ L2, which extend to be bounded on M and
ϕ
T ∣ ∣
recall that U K M,U f =gf,isunitary.
g I g
∶ →
Theorem3.1. LetM =gK beanearlyinvariantsubspacewithextremalfunctiongandsuitable
I
inner function I, I 0 = 0. Then for any Lebesgue measurable function ϕ on T with g 2ϕ ∈ L2
( ) ∣ ∣
we have
Ug∗AMϕ Ug =A∣g∣2ϕ.
Proof. Let h∈K ,then
I
AMϕ Ugh=AMϕ gh=gPI g 2ϕh=UgA∣g∣2ϕh.
∣ ∣
∎
Theaboveshowsthatthemap
AM A
ϕ ∣g∣2ϕ
↦
establishesaspatialisomorphismbetween M (theboundedtruncatedToeplitzoperatorsonM)
T
and (theboundedtruncatedToeplitzoperatorsonK )induced,viaconjugation,bytheunitary
I I
T
operatorU K M =gK ,i.e., M =U U∗.
g I I g I g
∶ → T T
InviewoftheresultsinSarason’spaper[Sar07]onecanusetheabovespatialisomorphismto
provethefollowingfacts:
(1) isaweakly closedlinearsubspaceofoperators onM.
M
T
(2) AM ≡0ifand onlyif g 2ϕ∈IH2 IH2.
ϕ
∣ ∣ +
(3) Recalling the conjugation C mentioned earlier, define C = U CU∗. This defines a
g g g
∶
conjugation on M and C AC = A∗ for every A ∈ M, and since AM ∗ = AM (as
g g T ( ϕ ) ϕ
can be verified by direct inspection), we see that M is also a collection of complex
T
symmetricoperators(withrespect toC ).
g
6 ANDREASHARTMANN&WILLIAMT.ROSS
(4) IfS =U A U∗,thenaboundedoperatorAonM belongsto M ifandonlyifthereare
g g z g
∶ T
functionsϕ ,ϕ ∈M so that
1 2
A S AS∗ = ϕ kM kM ϕ .
g g 1 0 0 2
− ( ⊗ )+( ⊗ )
(5) Recall from our earlier discussion of [Sar07] that A is a rank-one truncated Toeplitz
operatoronK ifandonlyifitcanbewrittenasconstantmultipleofoneofthefollowing
I
threetypes(two ofwhichare non-selfadjoint,and oneofwhich isselfadjoint)
kI CkI, CkI kI, kI kI
λ λ λ λ ζ ζ
⊗ ⊗ ⊗
whereλ∈D,ζ ∈ADC I . It followsimmediatelythat forλ∈D,
( )
U kI CkI U∗ =U kI U CkI =gkI gCkI,
g λ λ g g λ g λ λ λ
( ⊗ ) ⊗ ⊗
U CkI kI U∗ =U CkI U kI =gCkI gkI,
g λ λ g g λ g λ λ λ
( ⊗ ) ⊗ ⊗
are rank-one truncated Toeplitz operators on M = gK . What are the self-adjoint trun-
I
cated Toeplitz operators on M? The result here is the following: Suppose that ζ ∈
ADC I , then
( )
gkI gkI
ζ ζ
⊗
is a self-adjoint rank-one truncated Toeplitz operator on M. Conversely, if A is a non-
trivial rank-one self adjoint truncated Toeplitz operator on M, then there exists a ζ ∈
ADC I sothatA=gkI gkI.
( ) ζ ⊗ ζ
(6) Recall from ourearlierdiscussionthateverymaximalalgebrain can bewrittenasthe
I
T
commutant of a generalization of the Clark unitary operator. Conjugating by U we get
g
an analogousresultforthemaximalalgebrasof M.
T
(7) Again from our earlier discussion, for two inner functions I and I , that is spatially
1 2 I1
T
isomorphicto if and only if either I = ψ I φ or I = ψ I z φ for some disk
I2 1 2 1 2
T ○ ○ ○ ( )○
automorphisms φ,ψ. Using Theorem 3.1 we get the exact same result for M1, M2.
T T
Noticehowthisresult isindependentofthecorrespondingextremalfunctionsg and g .
1 2
4. EXISTENCE OF NON-TANGENTIAL LIMITS
With trivial examples one can show that there is no point ζ ∈ T such that every f ∈ H2 has a
non-tangentiallimitatζ. However,therearemodelspacesK andζ ∈Tsothateveryf ∈K has
I I
a non-tangentiallimitat ζ. Thissituationwas thoroughlydiscussedby Ahern and Clark [AC70]
withthefollowingtheorem. Forζ ∈Tand α >1 let
z ζ
Γ ζ = z ∈D ∣ − ∣ <α
α
( )∶ { ∶ 1 z }
−∣ ∣
bethestandardStolzdomainswithvertexat ζ.
Theorem 4.1 (Ahern-Clark). Let I be an inner function. Every function f ∈ K has a non-
I
tangential boundary limit at ζ ∈ T if and only if for every fixed Stolz domain Γ ζ at ζ, the
α
( )
family
F = kI λ∈Γ ζ
α,ζ λ α
∶ { ∶ ( )}
TRUNCATEDTOEPLITZOPERATORSANDBOUNDARYVALUESINNEARLYINVARIANTSUBSPACES 7
isuniformlynormbounded,i.e.,
1 I λ 2
sup{∥kλI∥2 = −1 ∣ (λ2)∣ ∶λ∈Γα(ζ)}<∞.
−∣ ∣
Theoriginal proof ofthis theorem involveda technical lemmausing operatortheory. We will
givea new proofwhich avoids thistechnical lemma. This willallowus to considerthesituation
ofnearly invariantsubspaceswheretheoperatortheorylemmaofAhern-Clark nolongerworks.
Also note that the uniform boundedness of the families F is equivalent to the condition ζ ∈
α,ζ
ADC I .
( )
Proofof Theorem 4.1. From the uniform boundedness principle it is clear that if every function
in K has a non-tangential limit at ζ ∈ T, then the kernels kI are uniformly bounded in Stolz
I λ
domainsΓ ζ .
α
( )
Letusconsiderthesufficiencypart. If,forfixedαandζ,thefamilyF isuniformlybounded,
α,ζ
then there exists a sequence Λ = λ ⊂ Γ ζ such that kI converges weakly to some
kI,Λ ∈K , i.e.,foreveryfunction∶f {∈Kn},n≥1 α( ) λn
ζ I I
(4.2) f λ = f,kI f,kI,Λ = fΛ as n .
( n) ⟨ λn⟩→⟨ ζ ⟩ ∶ ζ →∞
Observethat
f λ f 0
Azf λ = ( )− ( )
λ
( )( )
and
f λ N−1f j λj
ANz f λ = ( )−∑λjN=0 ̂( ) ,
( )( )
wheref j isthej-thFouriercoefficient off. As aresultwehave
̂
( )
∞
ANf 2 = f j 2 0 asN .
∥ z ∥ j∑=N∣̂( )∣ → →∞
Apply(4.2)to thefunctionsANf to get
z
f λ N−1f j λj
AzNf,kλI = AzNf λ = ( )−∑λjN=0 ̂( ) .
⟨ ⟩ ( )( )
Hence
f λ N−1f j λj
⟨AzNf,kζI,Λ⟩ = nl→im∞⟨AzNf,kλIn⟩=nli→m∞ ( n)−∑λjNn=0 ̂( ) n
fΛ N−1f j ζj
= ζ −∑j=0 ̂( ) .
ζN
8 ANDREASHARTMANN&WILLIAMT.ROSS
Nowusingthefact that ANf 0 when N , weobtainfrom ANf,kI,Λ 0 that
∥ z ∥→ →∞ ⟨ z ζ ⟩→
fΛ N−1ζjf j
lim ζ −∑j=0 ̂( ) =0,
N→∞ ζN
and, sincethedenominatorin theabovelimitisharmless,weget
∞
f j ζj =fΛ.
̂ ζ
j∑=0 ( )
Thismeansthat theFourierseries forf convergesat ζ.
NowtakeanysequenceΛ = λ convergingnon-tangentiallytoζ. Inviewoftheuniform
n n≥1
boundedness of the family F∶ {, th}ere exists a weakly convergent subsequence Λ′ = kI .
α,ζ ∶ { λnl}l≥1
Repeating theaboveproof, weobtain
∞
ζjf j =fΛ′.
̂ ζ
j∑=0 ( )
In other words, for every sequence λ converging non-tangentially to ζ there exists a sub-
n n≥1
sequence λ such that f λ { } convergesto thesamelimit
nj j≥1 nj j≥1
{ } { ( )}
∞
l = f j ζj.
ζ
∶ j∑=0 ̂( )
It followsthatf has afinitenon-tangentiallimitat ζ.
∎
We will now prove an analog of Theorem 4.1 for nearly invariant subspaces where the sit-
uation is a bit more complicated. Certainly a necessary condition for the existence of finite
non-tangential limits for every function in gK at ζ is that g has a finite non-tangential limit at
I
ζ (since indeed g is extremal and so g ∈ gK ). As we will see with the next example, it is not
I
possible to deduce the existence of the non-tangential limits for every function in gK merely
I
from theuniformnormboundednessofthereproducing kernelsin aStolzangles.
Example 4.3. Givenany innerfunction I, with I 0 =0, wehavealready cited Sarason’s result
[Sar88]statingthat everyisometricmultiplieron K( )isoftheform
I
a
(4.4) g = ,
1 Ib
−
whereaandbarefunctionsintheballofH∞ with a2 b2 =1a.e.onT. LetΛ = 1 4−n
1 n≥1
and Λ = 1 2−n , which are both interpolatin∣g∣se+qu∣e∣nces. Observe that Λ co{nsi−sts of}the
2 n≥1 2
sequence{Λ −to wh}ich we have added in a certain way the (pseudohyperbolic)midpoints of two
1
consecutive points of Λ . Let B be the Blaschke product whose zeros are Λ . In particular, B
1 i i 1
vanishesonΛ andisbigonΛ Λ ,say B λ ≥δ >0forλ∈Λ Λ (thiscomesfromthefact
1 2 1 1 2 1
that B is an interpolating Blas∖chke prod∣uct(, a)n∣d that on Λ Λ ∖we are pseudohyperbolically
1 2 1
∖
farfrom thezeros ofB ).
1
Next,define
1 1
a= B
1
2 + 4
TRUNCATEDTOEPLITZOPERATORSANDBOUNDARYVALUESINNEARLYINVARIANTSUBSPACES 9
which is a function oscillating on Λ between the values 1 2 (on Λ ) and a λ 1 2 ≥ δ 4,
2 1
λ∈Λ Λ . / ∣ ( ) − / ∣ /
2 1
∖
By construction we have 1 4 ≤ a ≤ 3 4. In particular there is an outer function b in the ball
0
of H∞ such that a2 b0 2 =/1 a.e∣.∣on T/. Moreover 0 < √7 4 ≤ b0 = √1 a2 ≤ √15 4 < 1
a.e. on T and this∣ex∣ te+n∣ds∣by themaximum/minimumprincip/lesto∣th∣edisk (−b ∣b∣eing oute/r). As
0
a consequence, for every inner function J, the function g = a 1 Jb will be a bounded outer
0
function(actuallyinvertible). Wewillusethisfact inparticul/a(rfo−rJ =)IB :
2
a a
g = = .
1 Jb 1 I B b
0 2 0
− −
=±bin(4.4)
Then
1 1 =0 whenλ∈Λ1
g λ = a λ δ
∣ ( )− 2∣ ∣ ( )− 2∣{ ≥ 4 whenλ∈Λ2 Λ1.
∖
Hence g has no limitat ζ =1. Choosenowan innerfunction I such thatζ =1∈ADC I . Then,
the kernels kM = g λ gkI are uniformly bounded in a fixed Stolz region Γ 1 at ζ =(1.)Indeed
λ λ α
g isbounded,theke(rn)elskI areuniformlyboundedinΓ 1 by1∈ADC I ((s)eeTheorem4.1),
λ α
and kM = g λ kI . ( ) ( )
λ λ
∥ ∥ ∣ ( )∣∥ ∥
Theorem 4.5. Let M =gK bea non-trivialnearlyinvariantsubspacewithextremal functiong
I
and corresponding inner function I. Then every function in M has a finite non-tangential limit
atζ ∈Tif andonlyif thefollowingtwo conditionsaresatisfied:
(1) g has afinitenon-tangentiallimitatζ.
(2) ForeveryStolzregionΓ ζ atζ,thefamilyFM = kM λ∈Γ ζ isuniformlynorm
bounded,i.e., α( ) α,ζ ∶ { λ ∶ α( )}
1 I λ 2
sup{∥kλM∥2 =∣g(λ)∣2 1− (λ)2∣ ∶λ∈Γα(ζ)}<∞.
−∣ ∣
Proof. If every function in M has a finite non-tangential limit at ζ, then (1) holds since g ∈ M.
Condition(2)holdsby theuniformboundednessprinciple.
Nowsupposethatconditions(1)and (2)hold.
Case1: ζ ∈ADC I . In this case, everyfunction in M has a non-tangentialboundarylimitat
ζ sinceeveryfunctio(n)inK has as wellas g does.
I
Case2: ζ ∈ ADC I . From the Ahern-Clark theorem (Theorem 4.1), we know that ζ ∈
ADC I ifan/d onlyi(f )
( ) 1 I λ 2
kλI 2 = −1 ∣ (λ2)∣
∥ ∥
−∣ ∣
is uniformly bounded in Stolz regions Γ ζ . Hence, if ζ ∈ ADC I , then there is a sequence
α
Λ = λ in Γ ζ such that kI 2( ) . Recall t/hat kM(=)g λ gkI so that kM =
∶ { n}n≥1 α( ) ∥ λn∥ → +∞ λ ( ) λ ∥ λn∥
10 ANDREASHARTMANN&WILLIAMT.ROSS
g λ kI . Inorderthatthislastexpressionisuniformlyboundeditisnecessarythatg λ
∣0(whne)n∣∥nλn∥ . ( n)→
→∞
Now, since FM is a uniformly bounded family, there exists a subsequence Λ′ = λ so
thatkM convergα,eζs weaklyto afunctionkM,Λ′. ∶ { nj}j≥1
λnj ζ
Letus introducethefollowingoperator
Q =U ANU∗
N g z g
from M =gK toitself. Let f =gh∈gK =M. Recall thatU∗f =h. Then
I I g
h z N−1h m zm
QNf z = UgAzNUg∗ f z = UgANz h z =g z ( )−∑mzN=0̂( )
( )( ) (( ) )( ) (( )( ) ( )
f z g z N−1
= h m zm.
( ) ( )
zN − zN m∑=0̂( )
NowsinceQ f ∈M, weget, fromtheweak convergenceofkM kM,Λ′ as j ,
N λnj → ζ →∞
f λ g λ N−1
(4.6) ⟨QNf,kζM,Λ′⟩=jl→im∞(QNf)(λnj)=jl→im∞( (λNnnjj) − (λNnnjj) m∑=0̂h(m)λmnj).
Thelastsumappearingaboveisaharmlesspolynomialconvergingtosomenumberwhenj .
Alsoλ ζ and g λ 0when j . Hence thelimitin (4.6)isequal to →∞
nj → ( nj)→ →∞
fΛ′
Q f,kM,Λ′ = ζ ,
N ζ ζN
⟨ ⟩
where
fΛ′ = f,kM,Λ′ ,
ζ ζ
⟨ ⟩
whichexistsbyconstruction.
Since
Q f = U ANh = gANh = ANh 0 as N ,
N g z z z
∥ ∥ ∥ ∥ ∥ ∥ ∥ ∥→ →∞
wehave
fΛ′
lim ζ = lim Q f,kM,Λ′ =0
N→∞ ζN N→∞⟨ N ζ ⟩
whichis possibleprecisely whenfΛ′ =0.
ζ
Theaboveconstructionisindependentofthechoiceofthesequence λ sothatforevery
n n≥1
such sequence there is a subsequence λ such that f λ 0 {whe}n j . As in the
proofofTheorem4.1 weconcludethat{ nljim}j≥1 f λ exist(sannjd)i→s, infact, equ→al t∞o0.
λ→ζ
∠ ( )
∎