Uniform minimality, unconditionality and interpolation in backward shift invariant spaces

0.23 MB·English

by  Eric Amar

UNIFORM MINIMALITY, UNCONDITIONALITY AND INTERPOLATION IN
BACKWARD SHIFT INVARIANT SPACES
ERICAMAR&ANDREASHARTMANN
9
0
0
ABSTRACT. We discuss relations between uniform minimality, unconditionality and interpola-
2
tionforfamiliesofreproducingkernelsinbackwardshiftinvariantsubspaces.Thisclassofspaces

n containsasprominentexamplesthePaley-Wienerspacesforwhichitisknownthatuniformmin-
a
imalitydoesingeneralneitherimplyinterpolationnorunconditionality. Hence,contrarilytothe
J
situation of standard Hardy spaces (and other scales of spaces), changing the size of the space
2
seems in this context necessary to deduce unconditionalityor interpolation from uniform mini-
2
mality.Suchachangecantaketwodirections:loweringthepowerofintegration,or“increasing”

] thedefininginnerfunction(e.g.increasingthetypeinthecaseofPaley-Wienerspace).
V
C
.
h
t
a 1. INTRODUCTION
m
[ A famous result by Carleson states that a sequence of points S = {a } in the unit disk D =
k
1 {z ∈ C : |z| < 1} is an interpolating sequence for the space H∞ of bounded analytic functions
v on D, meaning that every boundedsequence on S can be interpolatedby a function f in H∞ on
1
S, i.e.H∞|S ⊃ l∞, ifand onlyifthesequenceS satisfies theCarleson condition:
0
5
(1.1) inf |B (a)| = δ > 0,
3 a
a∈S
.
1
0 where B = b is the Blaschke product vanishingexactly on S \{a}, and b (z) = |a| a−z
a u6=a u a a 1−az
9
(see [Ca58]). We willwriteS ∈ (C) forshort when S satisfies(1.1). Obviouslyin thissituation
0 Q
: we also have the embedding H∞|Λ ⊂ l∞, so that S ∈ (C) is equivalent to H∞|Λ = l∞.
v
Subsequently it was shown by Shapiro and Shields [SS61] that for p ∈ (1,∞) a similar result
i
X
holds:
r
a Hp|S ⊃ lp(1−|a|2) = {(v ) : (1−|a|2)|v |p < ∞}
a a∈S a
a∈S
X
if and only if S ∈ (C). Again, it turns out that we also have Hp|S ⊂ lp(1−|a|2) (the measure
(1 − |a|2)δ is a so-called Carleson measure), so that S ∈ (C) is equivalent to Hp|S =
a∈S a
lp(1 − |a|2). Considering reproducing kernels k (z) = (1 − az)−1 the interpolation condition
P a
and the Carleson condition can be restated in terms of geometric properties of the sequence
(ka)a∈S. More precisely the Carleson condition is equivalent to (ka/kkakp′)a being uniformly
minimal in Hp′, and the interpolating condition Hp|S = lp(1 − |a|) to (ka/kkakp′)a∈S being
an unconditional sequence in Hp′ (precise definitions will be given below). Hence, another
way of stating the interpolation result in Hardy spaces is to say that a sequence of normalized
Date:January22,2009.
1991MathematicsSubjectClassification. 30E05,31A05.
Keywordsandphrases. Uniformminimality,unconditionalbases,modelspaces,Paley-Wienerspaces,interpo-
lation,one-componentinnerfunctions.
1
2 ERICAMAR&ANDREASHARTMANN
reproducing kernels in Hp′ is uniformly minimal if and only if it is an unconditional basis in
its span (since interpolation in the scale of Hardy spaces does not depend on p, the distinction
between p and p′ may appear artificial here). This special situation is not isolated. It turns out
tobetrueintheBergman space(see[SchS98]), andinFockspacesandPaley-Wienerspacesfor
certain indicesofp (see[SchS00]).
Morerecently,in[Am08]thefirstnamed authorhasgivenamethodallowingtodeduceinter-
polationfromuniformminimalitywhenthesizeofthespaceisincreasedbyloweringthepower
of integration. This result requires that the underlying space is the closure of a uniform algebra,
and appliesinparticulartoHardy spaces on theball.
We would like to use some of the methods discussed in [Am08] to show that uniform mini-
mality implies unconditionality in a bigger space for certain backward shift invariant subspaces
Kp forwhichthePaley-Wienerspacesareaparticularinstance. Recall thatforaninnerfunction
I
I,Kp = Hp∩IHp (whenconsideredasaspaceoffunctionsonT),whichisequaltotheorthog-
I 0
onalcomplementofIH2 whenp = 2. NotealsothatthesespacesareprojectedsubspacesofHp
(1 < p < ∞), and the projection — orthogonal when p = 2 — is given by P = IP I, where
I −
P = Id−P andP istheRieszprojectionsoff(eit) = a eint ∈ Lp(T)ontotheanalytic
− + + n∈Z n
part a eint. Wewouldliketodraw theattentionofthereader tothespecialsituationwhen
n≥0 n P
I(z) = I (z) := exp(2τ(z+1)/(z−1)). Then,thespaceKp isisomorphictothePaley-Wiener
P τ I
space PWp of entire functions of exponential type τ and p-th power integrable on the real line
τ
(see Section 3). By the Paley-Wiener theorem, PW2 is isometrically isomorphic to L2(−τ,τ).
τ
Already in this “simple” case the description of interpolating sequences is not known (see more
comments below). There exist sufficient density conditions for interpolation (or unconditional-
ity) when p = 2. They allow to check that a certain uniform minimal sequence, which is not
unconditional, becomes unconditional when we “increase” the inner function meaning that we
replaceI byI1+ε,ε > 0. (ItiswellknownthatK2 ⊂ K2 andevenK2 = K2 +IK2 .) The
I I1+ε I1+ε I Iε
density conditions for p = 2 do not seem to generalize to p 6= 2 (see Proposition 3.2 and com-
ments at the end of Section 3), so that there is no easy argument that could show that lowering
the integration power without changing I is sufficient to deduce unconditionality from uniform
minimality. Thismakestheproblemextremelydelicate. So,inthegeneralsituationthatwecon-
sider and where density or other usable conditions are not known, it seems extremly difficult to
deduce interpolationfrom uniform minimalityonly be increasing the space in one direction (ei-
theraddingfactors toI orloweringtheintegrationpowerp). Let usmentionhoweverthatunder
theassumptionI(λ ) → 0 theequivalencebetween uniform minimalityand unconditionalityin
n
K2 has been establishedin[HNP81](see also[Fr99] foravectorvaluedversionofthisresult).
I
Our results will require some conditions on the inner function such as being one-component.
ThismeansthatthelevelsetL(I,ε) = {z ∈ D : |I(z)| < ε}oftheinnerfunctionI isconnected
for some ε ∈ (0,1) (which is for instance the case for I ). One-component inner functions
τ
appear in work by Aleksandrov, Treil-Volberg etc. in the connection with embedding theorems
and Carleson measures.
Asaconsequenceofourdiscussionswestatehere asampleresult:
Theorem 1.1. Let I be a one-component singular inner function, S ⊂ D, 1 < p ≤ 2. Suppose
that supa∈S |I(a)| < 1. If (kaI/kkaIkp′)a∈S is uniformly minimal in KIp′, where 1/p + 1/p′ =
UNIFORMMINIMALITYANDUNCONDITIONALITY 3
1, then for every ε > 0 and for every s < p, S is an interpolating sequence for Ks and
I1+ε
(kaI/kkaIks′)a∈S isan unconditionalsequenceinKIs1′+ε, 1/s+1/s′ = 1.
Asalreadypointedout,acharacterizationofinterpolatingsequencesalreadyforPaley-Wiener
spaces is unknown for general p (when p = ∞ Beurling gives a characterization, and for
0 < p ≤ 1, see [Fl95]; a crucial difference between these cases and 1 < p < ∞ is the bounded-
ness of the Hilbert transform on Lp). For the case of completeinterpolating sequences in PWp,
τ
i.e. interpolating sequences for which the interpolating functions are unique, these are charac-
terized in [LS97] appealing to the Carleson condition and the Muckenhoupt Ap-condition for
somefunctionassociatedwiththegeneratingfunctionofS. Sufficientconditionsarepointedout
in [SchS00] using a kind of uniform zero-set condition in the spirit of Beurling. Such a condi-
tion cannot be necessary since there are complete interpolating sequences in the Paley-Wiener
spaces. Anotherapproach isbasedoninvertibilitypropertiesofP |Kp,whereB = b and
I B a∈S a
discussed in the seminal paper [HNP81] (see also [Ni02]). Once having observed that the Car-
Q
lesonconditionforS isnecessary(undertheconditionsup |I(a)| < 1),andso(k /kk k )
a∈S a a p a∈S
is an unconditional basis for Kp, the left invertibility of P |Kp guarantees that (kI/kkIk )
B I B a a p a∈S
is still an unconditional sequence. The invertibility properties of P |Kp can be reduced to the
I B
invertibility properties of a certain Toeplitz operator (T ). Again, and also in this approach,
IB
onecanfeelanessentialdifferencebetweencompleteinterpolatingsequencesandnotnecessary
complete interpolating sequences. The case of complete interpolating sequences corresponds to
invertibility of T , and a criterion of invertibility of Toeplitz operators is known. This is the
IB
theoremofDevinatzandWidom(seee.g.[Ni02,TheoremB4.3.1])forp = 2andRochberg (see
[Ro77])for1 < p < ∞,andagainitisbasedontheMuckenhoupt(A )condition(ortheHelson-
p
Szego˝ condition in case p = 2), this time for some function h ∈ Hp such that IB = h/h. A
useful description of left-invertibility of Toeplitz operators, the situation corresponding to gen-
eral not necessarily complete interpolating sequences, is not available. For the case p = 2 an
implicitconditionisgivenin[HNP81],andaconditionbasedonextremalfunctionsinthekernel
oftheadjointT can befoundin [HSS04].
IB
The paper is organized as follows. In the next section we introduce the necessary material on
uniformminimality,dualboundednessandunconditionality. Acharacterizationofunconditional
bases of point evaluations (or reproducing kernels) will be given in terms of interpolation and
embedding. We will also discuss some Carleson-type conditions which are naturally connected
with embedding problems. Section 3 is devoted to a longer discussion of the situation in the
Paley-Wienerspaces. Weessentiallyputtheknownmaterialintheperspectiveofourwork. This
should convince the reader that it is difficult to get better result. In the last section we give our
mainresultTheorem 4.5which as aspecialcase containsTheorem1.1.
2. PRELIMINARIES
2.1. Geometric properties of families of vectors of Banach spaces. We begin with some ob-
servationsintheclassical Hp concerning therelationbetween uniformminimalityand uncondi-
tionality. Recall that the reproducing kernel of Hp in a ∈ D is given by k (z) = (1 − az)−1.
a
The Carleson condition infa∈S |Ba(a)| ≥ δ > 0 can then be restated as (ka/kkakp′)a∈S being a
uniformlyminimalsequenceinHp′ (whichisequivalenthereto(k /kk k ) beinguniformly
a a p a∈S
minimal in Hp). Let us explain this a little bit more. By definition a sequence of normalized
4 ERICAMAR&ANDREASHARTMANN
vectors(x ) in aBanach spaceX is uniformlyminimalif
n n
(2.1) infdist(x , x ) = δ > 0.
n k
n
k6=n
_
(Here x denotes the closed linear span of the vectors x .) By the Hahn-Banach theorem this
i i i
is equivalent to the existence of a sequence of functionals (ϕ ) in X∗ such that ϕ (x ) = δ ,
W n n n k nk
whereδnk istheusualKroneckersymbol,andsupnkϕkX∗ < ∞. In oursituation,setting
B k
a a
ϕ = kk k ,
a a p
B (a)k (a)
a a
weget
k
b
hϕ , i = δ .
a ab
kk k
b p
Since kk k ≃ (1−|a|2)1−1/s we moreover have sup kϕ k < ∞. Another way of viewing
a s a∈S a q
the uniform minimality condition when p = 2 is given in terms of angles: a sequence (x )
n n
of vectors in a Hilbert space is uniformly minimal if the angles between x and x are
n k6=n k
uniformlyboundedaway from zero.
W
A notion closely related with uniform minimality is that of dual boundedness (see [Am08]).
Let usgiveageneral definition
Definition 2.1. Let X ⊂ Hol(Ω) be a reflexive Banach space of holomorphic functions on a
domain Ω. Suppose that the point evaluations E are continuous for every z ∈ Ω. A sequence
z
S ⊂ Ω is called dual-bounded if the sequence (Ea/kEakX∗)a∈S of reproducing kernels is uni-
formlyminimal.
Again, by the Hahn-Banach theorem this means that there exists a sequence (ρ ) of ele-
a a∈S
mentsinX (= X∗∗)withuniformlyboundednormsupa∈S kρakX < ∞andhρa,Eb/kEbkX∗i =
δab, i.e.ρa(b) = δabkEbkX∗.
Thisconditionistermed weak interpolationin [SchS00].
Let us discuss the unconditionality. Recall that a basis (x ) of vectors in a Banach space X
n n
is an unconditional basis if for every x ∈ X, there exists a numerical sequence (α ) such that
n
thesum α x convergesto x, and forevery sequenceofsignsε = (ε ), thesum ε α x
n n n n n n n n
converges in X to a vector x with norm comparable to kxk. We will discuss the interpolation
P ε P
conditionHp|Λ ⊃ lp(1−|a|2)inthelightofthisdefinitionusingreproducingkernels. Firstrecall
from [SS61] that we haveHp|Λ = lp(1−|a|2). Let B = B bethe Blaschke product vanishing
S
on S. Set Kp = Hp ∩ BHp, where Hp = zHp. The space Kp is a backward shift invariant
B 0 0 B
subspace. Also, Kp = k , and Hp = Kp + BHp (Kp = P Hp is a projected space).
B a∈S a B B B
So the interpolation condition is equivalent to Kp|Λ = lp(1−|a|2), and since the interpolation
W B
problem has unique solution in Kp, we have for every f ∈ Kp, kfkp ≃ (1 − |a|2)|f(a)|p.
B B p n
Clearly under this condition the functions ϕ introduced above exist and are in Kp. Then for
a P B
everyfinitesequence(v ) andeverysequence ofsigns(ε ) wehave
a n
k ε v ϕ kp ≃ (1−|a|2)|ε |p|v |p ≃ (1−|a|2)|v |p
a a a p a n n
a∈S a∈S a∈S
X X X
which shows that (ϕ ) is an unconditional basis in Kp. Then (k /kk k ) is also an uncondi-
a a B a a p
tionalbasisinKp.
B
UNIFORMMINIMALITYANDUNCONDITIONALITY 5
Again,theunconditionalitycanbeexpressedintermsofangleswhenp = 2: asequence(x )
n n
of vectors in a Hilbert space is unconditional if the angles between x and x is
k∈σ k k∈N\σ k
uniformlyboundedaway from zero foreveryσ ∈ N.
W W
So the interpolation results tell us that in Hp a sequence of reproducing kernels is uniformly
minimalifandonlyifitisanunconditionalsequence. Suchresultsalsoholdinotherspaceslike
e.g.Bergmanspaces(see[SchS98])andinFockandPaley-Wienerspacesforcertainvaluesofp
(see[SchS00]).
Wewillbeinterestedinthesituationin backward shiftinvariantsubspaces Kp.
I
2.2. Unconditional bases and interpolation. In this section we will establish a general link
between unconditional basis on the one hand and interpolation with an additional embedding
property on the other hand. It turns out that this link can be reformulated, in the spirit of [Ni78,
Theorem1.2], in abstract terms without appealing to the notion of interpolation. We will start
withthisgeneral resultbefore comingback to thespecial contextofinterpolation.
Suppose that X is a reflexive Banach space, and let (y ) be a sequence of normalized ele-
n n
ments in X∗ that we suppose at least minimal: dist(y , y ) > 0 for every n ∈ N. We set
n k6=n k
Y = y and N := Y⊥ ⊂ (X∗)∗ = X. By the minimality condition there exists a sequence
n W
(xn)nW∈ X∗∗ = X such thathxn,ykiX=X∗∗,X∗ = δn,k, n,k ∈ N.
Forasequencespacel,weconsiderthecanonicalsystem{e } wheree = (δ ) . Thespace
n n n nk k
l will be called ideal if whenever (a ) ∈ l and |b | ≤ |a |, n ∈ N, then (b ) ∈ l. Recall also
n n n n n
that a familyof vectors in a Banach space is called fundamental ifit generates a denseset in the
Banach space. Observethat thecanonical systemis an unconditionalbasisin l ifand only ifl is
ideal andthecanonical systemis fundamentalinl.
Weobtainthefollowingresult.
Proposition 2.2. Let X be a reflexive Banach space. With the above notation, the following
assertionsareequivalent.
(1) Thesequence(y ) isan unconditionalbasisinY = y .
n n n n
(2) Thesequence(x +N) is anunconditionalbasisin X/N (in generalnotnormalized).
n n W
(3) There exists two reflexive Banach sequence spaces l , l , in which the respective canoni-
1 2
cal systemsareunconditionalbases andsuchthat
(i) The setof generalizedFouriercoefficients ofX containsl :
1
{hx,yniX,X∗)n : x ∈ X} ⊃ l1,
(ii) for everyµ = (µ ) ∈ l ,
n n 2
k µnynkX∗ . kµkl2;
n
X
moreover l ≃ l∗ and the duality of l and l∗ ≃ l2 is given by h(α ) ,(µ ) i =
2 1 1 1 n n n n l1,l2
α µ .
n n n
P
This theorem is in the spirit of [Ni78, Theorem 1.2]. However, in Nikolski’s theorem there
does not really appear the condition (i) together with an embedding of type (ii). The condition
(i)willlateronplay theroˆleoftheinterpolationpart.
6 ERICAMAR&ANDREASHARTMANN
Proof. . Observe first that Y∗ = (X∗)∗/Y⊥ = X/N. Moreover, for every u ∈ N = Y⊥,
hx +u,y i = hx ,y i = δ ,andhence((y ) ,(x +N) )isabiorthogonalsystemin(Y,Y∗).
n k n n nk n n n n
By the general theory (see for instance [Sing70, Corollary I.12.2, Theorem II.17.7]) we obtain
theequivalenceof(1)and (2).
Letus nowprovethat(1) and(2) imply(3). By [Ni78, Theorem 1.1]thesequence(y ) isan
n n
unconditional sequence in Y if and only if the multiplier space mult(y ) := {µ = (µ ) : T :
n n n µ
Lin(y ) −→ Lin(y ), α y 7−→ µ α y extendsto a boundedoperator on Y} is
n n finite n n finite n n n
equal to l∞. And this, by [Ni78, Lemma 1.2] is equivalent to the existence of a sequence space
P P
l in which the canonical system is an unconditional basis such that (y ) is a l -basis, which
2 n n 2
meansthat
T : l −→ Y
2
(µ ) 7−→ µ y
n n n n
n
X
is an isomorphism. Note that Y is reflexive as a closed subspace of the reflexive Banach space
X∗, and sois l .
2
Forexactlythesamereason,by(2)thereexistsasequencespacel withtherequiredproperties
1
such that
S : l −→ X/N
1
(α ) 7−→ α x =: x +N
n n n n α
n
X
is an isomorphism. Notethat X/N isreflexiveas a quotientspace ofthereflexiveBanach space
X, and so is l . Take (α ) ∈ l , then S((α ) ) = x + N ∈ X/N for a suitable x ∈ X.
1 n n 1 n n α α
Now (hx ,y i) = (h α x ,y i) = (α ) (note that α x +N converges in X/N). So
α n n k k k n n n n k k k
(α ) ∈ {(hx,y i) : x ∈ X}.
n n n P P
Finally, since l ≃ X/N, l ≃ Y and Y∗ = X/N we have l∗ ≃ l and by reflexivity l ≃ l∗.
1 2 2 1 2 1
Moreover,bytheidenficationmapswecan writefor(α ) ∈ l and (µ ) ∈ l ≃ l∗:
n n 1 n n 2 1
h(α ) ,(µ ) i = h α x +N,µ y i = α µ hx ,y i = α µ .
n n n n l1,l2 n n k n X/N,Y n k n n X,Y n n
n n,k n
X X X
Wefinishbyshowingthat (3)implies(1). By (ii),theoperatorT isboundedand byconstruc-
tion onto, so that we are done if we can show that T is left invertible: kµk . kTµk . Now by
l2 Y
(i)for(α ) ∈ l , thereexistsx ∈ X suchthat α = hx ,y i. Let usintroducetheoperator
n n 1 α n α n
A : l −→ X/N
1
(α ) 7−→ x +N.
n n α
This operator is well defined (if we choose x′ with hx′ ,y i = α , then hx′ − x ,y i = 0 for
α α n n α α n
everynandx′ −x ∈ N). Itisalsolinear. Letuscheckthatitsgraphisclosed. Forthisconsider
α α
a sequence (αN) converging to (α ) in l . Since the canonical basis is an unconditional basis
n n n n 1
in l , we obtain coordinate-wise convergence: αN → α when N → ∞. We assume that
1 n n
A((αN) ) = x + N → x + N. Note that A((α ) ) = x + N. Then for every n we
n n αN n n α
have hx,y i = lim hx ,y i = lim αN = α = hx ,y i. So x − x ∈ N and
n N→∞ αN n N→∞ n n α n α
x+N = A((α ) ). By theclosedgraph theoremAis bounded.
n n
UNIFORMMINIMALITYANDUNCONDITIONALITY 7
Let us show that A∗ : (X/N)∗ = Y → l∗ is the left inverse to T (modulo the isomorphism
1
from l∗ to l ). Equivalently it is sufficient to show that T∗A : l → l∗ is an isomorphism. Note
1 2 1 2
thatfor(α ) ∈ l and (µ ) ∈ l , wehave
n n 1 n n 2
hT∗A(αn)n,(µn)nil2∗,l2 = hA(αn)n,T(µn)niX/N,Y = hxα +N, µnyniX/N,Y
= µnhxα,yniX,X∗ X
n
X
= µ α
n n
n
X
Byassumptionthisisequaltoh(α ) ,(µ ) i sothatforevery(α ) ∈ l and(µ ) ∈ l ,we
n n n n l1,l2 n n 1 n n 2
have
hT∗A(αn)n,(µn)nil2∗,l2 = h(αn)n,(µn)nil1,l2.
HenceT∗Aistheidentity(modulotheidentificationbetween l and l∗). (cid:4)
1 2
ItisinterestingtonotethatwhenX isaHilbertspacemorecanbesaidaboutthestructureofl:
itisclear thatthenl = l2. However,byaresult ofLindenstraussand Zippin(see[LZ69]), ifina
BanachspaceX everytwonormalizedunconditionalbasesareisomorphictoeachother,thenX
is isomorphicto one of the following spaces c , l1 or l2. In other words the general theory does
0
not yield l = lp when (x ) is an unconditional basis in (a subspace of) X = Lp (Pelczynski
n n
constructed actually unconditional bases in lp which are not equivalent to the canonical basis,
[Pe60]).
Let X be Banach space of holomorphicfunctions on D, such that the point evaluations E in
a
a ∈ D are continuous in X. A sequence S ⊂ D is called l-interpolating for a sequence space l
(defined on S) if for every sequence v = (va)a∈S with (va/kEakX∗)a∈S ∈ l there is a function
f ∈ X withf(a) = v , i.e.
a
X|S ⊃ l(1/kEakX∗) := {v = (va)a∈S : (va/kEakX∗)a∈S ∈ l}.
Since, kEak(Hp)∗ ≃ kkakp′ ≃ (1−|a|2)−1/p (1 < p < ∞), this definition is consistent with the
definitionswegavebeforeforHp, in whichcasewehad chosen l = lp.
The reader should also note that in the previous subsection we have repeatedly used the fact
that interpolation in Hp, i.e. Hp|S ⊃ lp(1 − |a|2) (we will not consider the case p = ∞ here)
impliesin fact theequalityHp|S = lp(1−|a|2) (thisisShapiro and Shields’result,[SS61]).
In the general case, withoutany further information,we haveto imposean additionalembed-
ding. FortheconvenienceoftheproofinthefollowingresultwewillsupposethatX isreflexive
(and so l will be). We will also need the notion of ideal space. A sequence space l is called
ideal if wheneverv = (v ) ∈ l and w = (w ) is any numerical sequence with |w | ≤ |v | for
n n n n n n
every n then also w ∈ l. This notion appears naturally in the context of free interpolation and
unconditionalbases.
Proposition 2.3. Suppose X ⊂ Hol(D) is reflexive and S is a sequence in D. The following
assertionsareequivalent.
(1) There existsa reflexiveand idealsequencespacel suchthat
(i) S is l-interpolating
8 ERICAMAR&ANDREASHARTMANN
(ii) There is a constant C such that for every finitely supported sequence µ = (µ) ,
a∈S
E
we havek a∈S µakEakaX∗kX∗ ≤ Ckµkl∗,
(2) (E ) isan uPnconditionalsequencein X∗.
a a∈S
A sequence satisfying condition (ii) will be called l∗-Carleson or q-Carleson when l∗ = lq (a
Carleson embeddingforX∗ withrespect to thesequencespace l∗). See Subsection2.3 formore
on Carleson conditions.
Notethat anotherway ofwriting(ii)is
f(a)
∀f ∈ X,∀µ ∈ l∗, | µa | ≤ CkfkXkµkl∗,
a∈S kEakX∗
X
which means that for every f ∈ X, the sequence (f(a)/kEakX∗)a∈S is in (l∗)∗ = l, and hence
(ii)isequivalentto
(2.2) k(f(a)/kEakX∗)a∈Skl ≤ CkfkX,
which means X|Λ ⊂ l(1/kEakX∗) (there will be more discussions on Carleson measures in
Subsection2.3). Wethushave
Corollary 2.4. Suppose X ⊂ Hol(D) is reflexive and S is a sequence in D. The following
assertionsareequivalent.
(1) There existsa reflexiveand idealsequencespacel suchthatX|Λ = l(1/kEakX∗)
(2) (E ) isan unconditionalsequencein X∗ (an l∗-basisin itsspan).
a a∈S
Proofof Proposition2.3. By [Ni78, Theorem 1.1] the sequence (E ) is an unconditional se-
a a∈S
quence in its span if and only if the multiplier space mult(E ) := {µ = (µ ) : T :
a a a∈S µ
Lin(E ) −→ Lin(E ), α E 7−→ µ α E extends to a bounded operator on
a a finite a a finite a a a
X∗ := E } is equal to l∞. And this, by [Ni78, Lemma 1.2] is equivalent to the existence
0 a∈S a P P
ofanidealspacel suchthat(E ) isal -basis,whichmeansthatX ≃ l (E ) := {(α ) :
W 0 a a∈S 0 0 0 a a a∈S
(αakEakX∗) ∈ l0}, in other words the mapping (αa)a∈S 7−→ a∈S αaEa is an isomorphism
from l (E ) ontoX , orequivalently
0 a 0 P
T : l −→ X∗
0
E
a
(β ) 7−→ β
a a∈S a
a∈S kEakX∗
X
is an isomorphism. Note that X is reflexive as a closed subspace of a reflexive Banach space,
0
andsoisl . Setl := l∗ (sothatl∗ = l ). Bytheprecedingargument,(E ) isanunconditional
0 0 0 a a∈S
sequenceinitsspan ifand onlyif
E
(2.3) ckµkl∗ ≤ k µa a kX∗ ≤ Ckµkl∗,
a∈S kEakX∗
X
forsomefixed constantsc,C. Thisyieldsin particular(ii).
WewillcomputetheadjointoperatorT∗ : X −→ l. Let µ ∈ l∗,
E f(a)
hT∗f,µi = hf,Tµi = hf, µ a i = µ .
a a
a∈S kEakX∗ a∈S kEakX∗
X X
UNIFORMMINIMALITYANDUNCONDITIONALITY 9
Hence, the functional T∗f on l∗ is represented by a sequence the entries of which are given by
f(a)/kEakX∗, a ∈ S. In otherwords T∗f = (f(a)/kEakX∗)a∈S ∈ (l∗)∗ = l.
Nowtheleft hand inequalityin (2.3) is equivalentto theleft invertibilityofT which is equiv-
alent to the surjectivity of T∗ i.e. to the fact S is l-interpolating. This show that (2) implies
(1).
Forthe converseimplication,notethat (ii) impliestheright inequalityin (2.3). Moreoverthis
inequalityshowsalsothatT iswelldefinedandbounded. Bytheaboveargumentsthesurjectivity
of T∗ is equivalent to the fact that S is interpolating. On the other hand the surjectivityof T∗ is
equivalenttotheleft invertibilityofT and soto theleftinequalityin(2.3). (cid:4)
Stillthefollowingis true
Corollary 2.5. If S is interpolating for Kp and if there is a constant C such that for every
I
finitely supported sequence µ = (µ)a∈S, we have k a∈S µakaI/kkaIkp′kp′ ≤ Ckµklp′, then
(kaI/kkaIkp′)a∈S isanunconditionalsequenceinKIp′. P
Morepreciselytheconclusionwouldbethat(kI) isanlp′-basisinitsspan. Thisconclusion
a a∈S
can in general not be deduced only from the condition of unconditionality as explained above.
However,in thespecialsituationsup |I(a)| < 1,[HNP81, Theorem6.3,PartieII]showsthat
a∈S
if the reproducing kernels form an unconditional sequence in Kp′ then automatically they form
I
an lp′-basis intheirspan.
2.3. Carleson measures. Let us fix the framework of this subsection. S is a sequence in D, I
an inner function and 1 ≤ q < ∞. For a ∈ S we denote by kI = kI/kkIk the normalized
q,a a a q
reproducingkernel.
Let1 ≤ q < ∞. Recall thatasequenceS iscalled q-Carleson if
∃D > 0,∀µ ∈ lq, µ kI ≤ D kµk .
q a q,a q q
(cid:13) (cid:13)
(cid:13)aX∈S (cid:13)q
(cid:13) (cid:13)
(cid:13) (cid:13)
Wewillalsousethenotionofweak q-Ca(cid:13)rleson sequ(cid:13)ences:
Definition 2.6. Let2 ≤ q < ∞. ThesequenceS iscalled weakly q-Carleson if
∃D > 0,∀µ ∈ lq, |µ |2|kI |2 ≤ D kµk2.
q a q,a q q
(cid:13) (cid:13)
(cid:13)aX∈S (cid:13)q/2
(cid:13) (cid:13)
(cid:13) (cid:13)
Notethatby[Am08, Lemma3.2],the(cid:13)q-Carleson pro(cid:13)pertyimpliestheweakq-Carleson prop-
erty.
Observe also that (lq)∗ = lp, that the dual of Kq can be identified with Kp, and that the
I I
functional of point evaluation E can then be identified with kI. Now, using the notation from
a a
theprecedingsubsection,by(2.2), S isq-Carleson ifand onlyifforeveryf ∈ Kp,
I
|f(a)|p
≤ ckfk ,
kkIkp p
a∈S a p
X
whichmeans thatν := δ /kkIk is aKp-Carleson measure: Kp ⊂ Lp(ν).
a∈S a a p I I
P
10 ERICAMAR&ANDREASHARTMANN
InthespecialsituationwhenI isone-component,thenby aresult byAleksandrov(see(4.1)),
wehave
1−|I(a)|2 1/p′
kkIk ≃ ,
a p 1−|a|2 !
p′ being the conjugated index to p, and so, if S is q-Carleson and I is one-component, then the
measure
1−|a|2
dν = δ
1−|I(a)|2 a
a∈S
X
isKp-Carleson.
I
GeometricCarlesonconditions
In[TV96],thefollowinggeometricnotionofCarlesonmeasureappears. Foraninnerfunction
I and an ε > 0, let L(I,ε) = {z ∈ D : |I(z)| < ε} betheassociated levelset. In thenotationof
[Al02], let C(I) betheset ofmeasures forwhichthereexistsC > 0such that
(2.4) |µ|(S(ζ,r)) ≤ Cr
for every Carleson window S(ζ = eit,h) := {z = reiθ ∈ D : 1 − h < r < 1,|t − θ| < h}
meeting L(I,1/2) (this is of course a weaker notion than the usual one requiring (2.4) on all
Carleson windows; the value ε = 1/2 is of no particular relevance). Let also C (I) be the set
p
of measures for which Kp ⊂ Lp(µ). Strengthening the results of [TV96], Aleksandrov proved
I
in [Al02, Theorem 1.4] that for one component inner functions C(I) = C (I). In other words,
p
the geometric Carleson condition (2.4) on Carleson windows meeting the level set L(I,1/2)
characterizes theKp-Carleson measuresforonecomponentinnerfunctions.
I
Combiningtheseobservations,wegetthefollowingcharacterization.
Fact 2.7. LetI beaone-componentinnerfunction. Thenthefollowingassertionsareequivalent.
(i) S isp′-Carleson
1−|a|2
(ii) ν = δ is Kp-Carleson
a∈S 1−|I(a)|2 a I
(iii) ν (asPdefined in point (ii)) satisfies the geometric Carleson condition (2.4) on Carleson
windowsmeetingthelevelset L(I,1/2).
Question. Do there exist in backward shift invariant subspaces interpolating sequences S that
are notp′-Carleson?
3. PALEY-WIENER SPACES
We will discuss a special class of backward shift invariant subspaces. Let I(z) = ei2πz be the
singular inner function in the upper half plane with sole singularity at ∞ (to fix the ideas, we
havechosen themass oftheassociated singularmeasuretobe 2π). Recall (see[Ni02, B.1]) that
thetransformation
U : Hp(D) −→ Hp(C+)
p
1/p
1 x−i
f 7−→ x → (U f)(x) = f
 p π(x+i)2! (cid:18)x+i(cid:19)
 