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Comparison of singular numbers of composition operators on different Hilbert spaces of analytic
functions
Hervé Queffélec, Pascal Lefèvre, Daniel Li, Luis Rodriguez-Piazza
To cite this version:
Hervé Queffélec, Pascal Lefèvre, Daniel Li, Luis Rodriguez-Piazza. Comparison of singular numbers
of composition operators on different Hilbert spaces of analytic functions. 2020. �hal-02437146�
Comparison of singular numbers of composition operators on different Hilbert spaces of analytic functions
Pascal Lefèvre, Daniel Li,
Hervé Queffélec, Luis Rodríguez-Piazza
January 17, 2020
Abstract. We compare the rate of decay of singular numbers of a given com- position operator acting on various Hilbert spaces of analytic functions on the unit diskD. We show that for the Hardy and Bergman spaces, our results are sharp. We also give lower and upper estimates of the singular numbers of the composition operator with symbol the “cusp map” and the lens maps, acting on weighted Dirichlet spaces.
MSC 2010primary: 47B33 ; secondary: 46B28 ; 30H10 ; 30H20
Key-words approximation numbers ; composition operator ; Hardy space ; Hilbert spaces of analytic functions ; Schatten classes ; singular numbers ; weighted Bergman space ; weighted Dirichlet space
1 Introduction
Composition operators are mainly studied on Hilbert spaces of analytic func- tions, and more specifically on the Hardy spaceH2, the Bergman spaceB2, and the Dirichlet spaceD=D2. It is well known, thanks to the Littlewood subor- dination principle, that every analytic self-map ϕ:D →D induces a bounded composition operator Cϕ on H2 and on B2, but not necessarily on D2 ([37, Chapter 1 and Exercises]; see also [11, Section 6.2]). There exist even composi- tion operators which are not bounded onD2but which are in all Schatten classes Sp(H2)andSp(B2)withp >0, of both the Hardy space and the Bergman space ([25, Theorem 2.10]). Nevertheless, for compact composition operators, the fol- lowing results hold: 1) every composition operator which is compact onH2 is compact onB2(see [37, Theorem 3.5] and [33, Theorem 3.5]); 2) every compo- sition operator that is compact onD2 is in all the Schatten classesSp(H2)for p >0([25, Theorem 2.9]); for everyp >0, every composition operator that is in Sp(H2)is inSp(B2). Since the membership in a Schatten classSp of an opera- tor on a Hilbert space means that its approximation numbers areℓp-summable, that suggests that there is a strong link between the approximation numbers
aDn2(Cϕ), aHn2(Cϕ) and aBn2(Cϕ) of the composition operator Cϕ on D2, H2 andB2 respectively.
The aim of this paper is to prove that, indeed, in some sense aDn2(Cϕ) is
“greater” thanaHn2(Cϕ), which is “greater” thanaBn2(Cϕ). We recover then that Cϕ∈Sp(H2)implies thatCϕ∈Sp(B2)(Section 3). In Section 3.6, we also give some results about conditional multipliers.
In Section 4 we give an example with Cϕ compact on H2 but not in any Schatten class Sp(B2) for p < ∞. We prove that Cϕ ∈ Sp(H2) implies that Cϕ ∈ Sp/2(B2)and give an example with Cϕ ∈Sp(H2) but Cϕ ∈/ Sq(B2)for anyq < p/2.
However, our result is not sufficient to explain why the compactness of Cϕ on D2 implies that Cϕ ∈ Sp(H2) for all p > 0. A more subtle relation- ship should exist between aDn2(Cϕ) and aHn2(Cϕ). In fact, for every compo- sition operator Cϕ that is compact on D2, we havelimn→∞
aDn2(Cϕ)1/n
= limn→∞
aHn2(Cϕ)1/n
([29, Theorem 3.1 and Theorem 3.14]); in particular, for symbolsϕsuch thatkϕk∞<1, the numbersaDn2(Cϕ)andaHn2(Cϕ)behave like rn, with r= exp(−1/Cap[ϕ(D)]), and where Cap[ϕ(D)] is the Green capacity ofϕ(D). On the other hand, for the so-called cusp map χ, we have, for some constantsc1> c′1>0([28, Theorem 4.3]):
(1.1) e−c1n/logn.aHn2(Cχ).e−c′1n/logn and, for some constantsc2> c′2>0 ([26, Theorem 3.1]):
(1.2) e−c2√n.aDn2(Cχ).e−c′2√n,
which is much greater. In Section 5.2, we show that the behavior ofan(Cχ)in (1.1) holds in all weighted Dirichlet spacesDα2forα >0(with other constants), and hence (1.2) shows that a jump happens forα= 0. We also look at the lens maps.
2 Notation and background
LetDbe the open unit disk inC. We denotedA=dx dy/π the normalized area measure on D. The normalized Lebesgue measure dt/2π on T = ∂D is denoteddm.
2.1 Hilbert spaces of analytic functions
Recall that the Hardy spaceH2is the space of analytic functionsf:D→C such that:
kfk2H2:= sup
0<r<1
Z
T|f(rξ)|2dm(ξ)<∞. Iff(z) =P∞
k=0ckzk, we havekfk2H2 =P∞ k=0|ck|2.
The Bergman space B2 is the space of analytic functions f: D → Csuch that:
kfk2B2 :=
Z
D|f(z)|2dA(z)<∞. Iff(z) =P∞
k=0ckzk, we havekfk2B2 =P∞ k=0|ck|2
k+1 .
More generally, forγ >−1, theweighted Bergman space B2γ is the space of analytic functionsf: D→Csuch that:
kfk2B2γ= (γ+ 1) Z
D
|f(z)|2(1− |z|2)γdA(z)<∞, and iff(z) =P∞
k=0ckzk, we have:
kfk2B2γ = X∞ k=0
βk|ck|2,
with:
βk = k! Γ(γ+ 2)
Γ(k+γ+ 2) ≈ 1 (k+ 1)γ+1 (the equivalence depends onγ).
HenceB2=B20 andH2 corresponds to the degenerate caseγ=−1.
The Dirichlet space D2 is the space of analytic functions f:D → C such that:
kfk2D2 =|f(0)|2+ Z
D
|f′(z)|2dA(z)<∞. Iff(z) =P∞
k=0ckzk, we havekfk2D2 =|c0|2+P∞
k=0k|ck|2. With the equivalent normk|f|k2D2 =kfk2H2+R
D|f′(z)|2dA(z), we have the more pleasant formk|f|k2D2=P∞
k=0(k+ 1)|ck|2.
More generally, forα >−1, theweighted Dirichlet space Dα2 is the space of analytic functionsf: D→Csuch that:
kfk2Dα2 =|f(0)|2+ (α+ 1) Z
D|f′(z)|2(1− |z|2)αdA(z)<∞, and iff(z) =P∞
k=0ckzk, we have:
kfk2Dα2 = X∞ k=0
βk|ck|2,
withβ0= 1and fork≥1:
βk = k . k! Γ(α+ 2)
Γ(k+α+ 1) ≈ 1 (k+ 1)α−1
(the equivalence depending onα). Another equivalent expression is:
βek= (k+ 1)! Γ(α+ 2) Γ(k+α+ 1) ; we haveβk≤βek ≤2βk.
In particular, forγ >−1:
(2.1) D2=D02, H2=D12 and B2γ =Dγ+22 .
2.2 Composition operators
Any analytic self-mapϕ:D→Ddefines a bounded composition operatorCϕ
on the Hardy spaceH2 (see [33, Section 2.2]) and on every weighted Bergman spaceB2γ forγ >−1([33, Proposition 3.4]), hence on every weighted Dirichlet space Dα2 with α ≥ 1. However, this is not always the case on the weighted Dirichlet spacesDα2 forα <1 ([33, Proposition 3.12]).
For convenience, we assume that ϕis not constant and we say that ϕis a symbol. We denoteϕ∗ the boundary values function ofϕ.
The Carleson window of sizehcentered atξ∈Tis:
(2.2) W(ξ, h) ={z∈D; |z| ≥1−h and −πh≤arg( ¯ξz)< πh}. For every integern≥1 and forj= 0, . . . ,2n−1, we set:
(2.3) Wn,j=W(e2jiπ/2n,2−n). We also use the Carleson boxes
(2.4) S(ξ, h) ={z∈D; |ξ−z|< h}, which satisfyS(ξ, h)⊆W(ξ, h)⊆S(ξ,2πh).
The Hastings-Luecking boxes are defined, for every integer n ≥1 and for 0≤j≤2n−1, as:
(2.5) Rn,j =n
z∈D; 1− 1
2n−1 ≤ |z|<1− 1
2n and 2jπ
2n ≤argz < 2(j+ 1)π 2n
o A measureµonDis aCarleson measure ifsupξ∈Tµ
W(ξ, h)
= O (h). By the Carleson embedding theorem, µ is a Carleson measure if and only if the inclusion mapJµ:H2→L2(µ)is bounded. The automatic boundedness ofCϕ
onH2implies that the pull-back measuremϕ, defined asmϕ(B) =m[ϕ∗−1(B)]
for all Borel sets B ⊆D, is a Carleson measure. This composition operator is compact onH2 if and only ifmϕ is supported byDandsupξ∈Tmϕ[W(ξ, h)] = o (h)([32]).
Similar results hold for composition operators on the weighted Bergman spaces ([33, Theorem 4.3].
2.3 Singular numbers, approximation numbers and Schat- ten classes
LetH be a separable complex Hilbert space andT:H →H be a compact operator. There exist two orthonormal sequences (un) and (vn) and a non- increasing sequence(sn)of non-negative numbers withsn −→
n→∞0 such that, for allx∈H:
(2.6) T(x) =
X∞ n=1
snhx|vniun. This representationT =P∞
n=1snun⊗vn is called the Schmidt decomposition ofT and the numberssn=sn(T)thesingular numbersofT. They are actually the eigenvalues of|T|=√
T∗T rearranged in non-increasing order. In particular s1=kTk.
These numbers have the important “ideal property”:
sn(AT B)≤ kAksn(T)kBk.
It is known (see [5, p. 155]) that, for alln≥1, we havesn(T) =an(T), the nthapproximation number ofT defined as:
(2.7) an(T) = inf
rankR<nkT−Rk.
For p > 0, the Schatten class Sp(H) is the set of all compact operators T:H →H for whichkTkpp:=P∞
n=1[an(T)]p<∞. We haveSq(H)⊆Sp(H)for0< q≤p.
For composition operatorsCϕ, D. Luecking ([30, Corollary 2]) characterized their membership in the Schatten classes.
For γ > −1, let dAγ(z) = (γ+ 1)(1− |z|2)γdA(z). For γ = −1, we set H2 =B2
−1 and dm=dA−1. Then, forγ ≥ −1, the composition operatorCϕ
belongs toSp(B2
γ)if and only if:
(2.8)
X∞ n=1
2Xn−1
j=0
2n(γ+2)Aγ,ϕ(Rn,j)p/2
<∞,
whereAγ,ϕ is the pull-back measure ofAγ byϕ(byϕ∗ forγ=−1).
As usual, the notationA.Bmeans thatA≤C Bfor some positive constant C, which may depend on some parameters, andA≈B means thatA.B and A&B.
3 Comparison of approximation numbers
3.1 Main result
In the introduction, we said that, in some sense, aDn2(Cϕ)is “greater” than aHn2(Cϕ), which is “greater” thanaBn2(Cϕ). This vague statement is made more precise in the following result.
Theorem 3.1. For any symbolϕ, we have, for every n≥1:
(3.1)
Yn
j=1
aBj 2(Cϕ)≤ Yn
j=1
aHj 2(Cϕ)≤ Yn
j=1
aDj 2(Cϕ).
It is understood that ifCϕ is not bounded onD2, thenaDj 2(Cϕ) = +∞. As a consequence, we recover a previous result ([25, Corollary 3.2]; see also [7, Theorem 2.5]).
Corollary 3.2. For any symbolϕ, we have:
1) a) if Cϕ is compact onD2, thenCϕ is compact onH2; b) if Cϕ is compact onH2, thenCϕ is compact onB2. Moreover, for everyp >0, we have:
2) a) if Cϕ∈Sp(D2), then Cϕ∈Sp(H2);
b) if Cϕ∈Sp(H2), thenCϕ∈Sp(B2).
Items 1) a) and 2) a)are not sharp, since we proved in [25, Theorem 2.9]
that if Cϕ is compact onD2, then Cϕ belongs to all Schatten classes Sp(H2), withp >0. However, we will see in Section 4 that the item1) b)is sharp, but 2)b)is not.
We will prove below these results in a more general setting in Theorem 3.12.
3.2 Subordination of sequences
Let S be the set of non-increasing sequencesu= (uj)j≥1 of real numbers.
Ifu, v∈ S, the sequenceuis said to be subordinate to the sequencev, and we writeu≺v, if:
(3.2)
Xn
j=1
uj≤ Xn
j=1
vj for alln≥1.
For example, ifu= (1,1,0,0, . . .)andv= (2,0,0, . . .), we haveu≺v.
We have this basic stability property of this notion (see [39, Theorem 1.16, p. 13]).
Proposition 3.3. Let I be an interval of R and h:I →R be increasing and convex. Then, ifu, v∈ S are sequences of numbers inI, we have:
u≺v =⇒ h(u)≺h(v).
Proof. We recall a two-lines proof. We may assume thathis C2. We fixn≥1 and seta= min{un, vn}. Then, forx∈I andx > a:
h(x) =h(a) + (x−a)h′(a) + Z +∞
a
(x−t)+h′′(t)dt . One easily checks, using (3.2), thatPn
j=1(uj−t)+≤Pn
j=1(vj−t)+for allt≥a.
Hence, thanks to the positivity ofh′(a)andh′′: Xn
j=1
h(uj) =nh(a) +h′(a) Xn
j=1
(uj−a) + Z +∞
a
Xn
j=1
(uj−t)+h′′(t)dt
≤nh(a) +h′(a) Xn
j=1
(vj−a) + Z +∞
a
Xn
j=1
(vj−t)+h′′(t)dt
= Xn
j=1
h(vj).
A stronger notion is that of log-subordination.
Definition 3.4. We say that the sequence u ∈ S of positive numbers is log- subordinateto the sequencev∈ S of positive numbers iflogu≺logv. In other terms, if:
Yn
j=1
uj≤ Yn
j=1
vj for all n≥1. The following result will be useful.
Proposition 3.5. For sequences of positive numbersu, v∈ S, the following two conditions are equivalent:
1) logu≺logv;
2) up≺vp for all p >0.
Proof. Iflogu≺logv, it suffices to apply Proposition 3.3 to the sequenceslogu andlogv and to the functionh(x) = epx to getup≺vp.
Conversely, ifup≺vp for allp >0, we have:
1 n
Xn
j=1
upj 1/p
≤ 1
n Xn
j=1
vpj 1/p
, and lettingpgoing to 0, we get:
Yn j=1
uj
1/n
≤ Yn
j=1
vj
1/n
, i.e. logu≺logv.
Corollary 3.6. Let u, v∈ S be two sequences of positive numbers such that u is log-subordinate tov. Then for N≥n:
(3.3) uN ≤vn/N1 v1n−n/N. In particular, for anyn≥1:
(3.4) u2n≤√
v1vn. Proof. We have:
uNN ≤ YN
j=1
uj ≤ YN
j=1
vj = Yn
j=1
vj
YN
j=n+1
vj ≤vn1vNn−n,
and (3.3) follows. Now, the choiceN= 2ngives (3.4).
Note that the choiceN= [nlogn]can be useful (see [4]).
3.3 Singular numbers
The following Weyl type result is crucial for the proof of our main result. It is certainly known by specialists, but we have not found any reference.
Proposition 3.7. Let T be a compact operator on a separable complex Hilbert spaceH andT =P∞
j=1sjuj⊗vj its Schmidt decomposition. Then, for every integern≥1:
s1· · ·sn= maxdet hT fj |gii
i,j
,
where the supremum is taken over all pairs(fj)1≤j≤nand(gi)1≤i≤nof orthonor- mal systems of lengthnin H.
Proof. First, assume that H is n-dimensional. We may assume that H = ℓn2 and we denote(ei)1≤i≤n its canonical basis.
SinceT(vj) =sjuj, we havedet hT vj|uii
i,j=s1· · ·sn.
Now, if (fj)1≤j≤n and(gi)1≤i≤n are two orthonormal systems, we consider the following diagram:
ℓn2−→U ℓn2−→T ℓn2−→V ℓn2 whereU,V are the unitary operators defined by:
UXn
j=1
tjej
= Xn
j=1
tjfj and V x= Xn
j=1
hx|gjiej.
We observe that
hV T U ej|eii=hT U ej|V∗eii=hT fj|gii, so that:
det hT fj|gii
i,j
=|detV| |detT| |detU|=|detT|=s1· · ·sn.
In the general case, denote by Pn and Qn the orthogonal projections onto Fn:= [f1, . . . , fn]andGn := [g1, . . . , gn]respectively. We can seeFn andGn as isometric copies ofℓn2. Observe thathT fj |gii=hQnT Pnfj |gii. By the above special case, we get, using the ideal property of singular numbers:
det hT fj|gii
i,j
= Yn
j=1
sj(QnT Pn)≤ Yn
j=1
sj(T).
3.4 Comparison principle for operators
For convenience, we say that an operatorU:H →Kbetween Hilbert spaces isunitary if it is a surjective isometry, even ifH 6=K.
V. È Kacnel’son ([15]) proved the following result.
Theorem 3.8 (V. È Kacnel’son). Let H be a separable complex Hilbert space and(ei)i≥0 a fixed orthonormal basis of H. LetA:H →H be a bounded linear operator. We assume that the matrix of A with respect to this basis is lower- triangular: hAej|eii= 0 for i < j.
Let (dj)j≥0 be an increasing sequence of positive real numbers and D the (possibly unbounded) diagonal operator such that D(ej) = djej, j ≥ 0. Then the operatorD−1AD:H →H is bounded and moreover:
(3.5) kD−1ADk ≤ kAk.
In [6], this theorem was extended in the framework of Banach spaces with 1-unconditional basis and used for the study of composition operators, and in [7] to compare the Schatten-class norms of weighted Hilbert spaces of analytic functions.
We have the following generalization (the case n = 1 giving kD−1ADk ≤ kAk).
Theorem 3.9. With the notation of Theorem 3.8, and assuming moreover that Ais compact, we have, for every n≥1:
(3.6)
Yn
j=1
sj(D−1AD)≤ Yn
j=1
sj(A). In other words, the sequence sj(D−1AD)
j is log-subordinate to sj(A)
j. Proof. Let C0 be the right-half plane C0 = {z ∈ C; Rez > 0} and HN = span{ej; j≤N}. We set:
ai,j=hAej |eii and
AN =PNAPN,
where PN is the orthogonal projection from H into H with range HN. We consider, forz∈C0:
AN(z) =D−zANDz:H→H , whereDz(en) =dznen.
If aNi,j(z)
i,j is the matrix of AN(z) on the basis {ej; j ≥ 0} of H, we clearly have:
aNi,j(z) =
( ai,j(dj/di)z ifi, j≤N
0 otherwise.
In particular, we have, by hypothesis:
|aNi,j(z)| ≤sup
k,l |ak,l|:=M , for allz∈C0. SincekAN(z)k2≤ kAN(z)k2HS=P
i,j≤N|aNi,j(z)|2≤(N+ 1)2M2, we get:
kAN(z)k ≤(N+ 1)M for allz∈C0. Let us consider the functionu:C0→C0 defined by:
(3.7) u(z) =
Yn
j=1
sj AN(z) .
This functionuis continuous onC0.
If αdenotes a pair (fj), (gi)of orthonormal systems of length n of H, we set, forz∈C0:
Fα(z) = det hAN(z)fj|gii
i,j,
the functionFαis analytic inC0and continuous onC0. By Proposition 3.7, we haveu= supα|Fα|, so thatuis subharmonic inC0. Moreover:
u(z)≤ kAN(z)kn≤[(N+ 1)M]n forz∈C0, and:
u(z) = Yn
j=1
sj(AN)≤ Yn
j=1
sj(A) forz∈∂C0,
since the operatorDz:H →H is then unitary. Hence we can use the following form of the maximum principle.
Theorem 3.10(maximum principle). LetΩbe an arbitrary domain inC, with Ω 6= C, and u: Ω → R a function subharmonic in Ω, and continuous and bounded above onΩ. Then:
sup
Ω
u= sup
∂Ω
u .
This theorem is proved in [3, Theorem 15.1, p. 190] foru=|f|, withf: Ω→C holomorphic inΩ, and continuous and bounded onΩ, and in [14, Theorem 5.16, p. 232]. It follows that:
sup
Rez≥0
u(z)≤ Yn
j=1
sj(A). In particularu(1)≤Qn
j=1sj(A), or else:
Yn
j=1
sj(D−1AND)≤ Yn
j=1
sj(A).
Now, since the matrix of A−AN is lower-triangular, the inequality (3.5), applied toA−AN, giveskD−1(A−AN)Dk ≤ kA−ANk −→
N→∞0. Moreover, for eachj≥1, the mapT ∈ L(H)7→sj(T)is continuous, since|sj(T1)−sj(T2)| ≤ kT1−T2k. Then, lettingN tend to infinity, we obtain that
sj(D−1AND) −→
N→∞sj(D−1AD), and the result follows.
An alternative proof of Theorem 3.9 can be given using antisymmetric tensor products.
Alternative proof of Theorem 3.9. LetIdenote the set of all increasingn-tuples α= (i1< i2<· · ·< in)of non-negative integers. Let(uα)α∈I be the orthonor- mal basis ofΛn(H), then-th exterior power ofH, defined by:
uα=ei1∧ei2∧ · · · ∧ein, α∈I . We use the general fact that:
Yn
j=1
sj(D−1AD) =kΛn(D−1AD)k, whereΛn denotes then-th skew product.
SinceΛn(U V) = Λn(U)Λn(V)([39, page 10]), we get:
Λn(D−1AD) = Λn(D−1)Λn(A)Λn(D) =
Λn(D)−1
Λn(A)Λn(D)
=: ∆−1Λn(A)∆,
where ∆ is the diagonal operator on the basis (uα) with diagonal elements δα=di1· · ·din ifα= (i1< i2<· · ·< in).
Now, we claim that Λn(A) is lower triangular in the following sense. If α= (i1 < i2 <· · · < in) andβ = (j1 < j2 <· · ·< jn) are two elements ofI, then:
(3.8) δα< δβ=⇒
Λn(A)uβ, uα
= 0.
Indeed, assume that
Λn(A)uβ, uα
6= 0. Since:
Λn(A)uβ, uα
=
Aej1∧Aej1· · · ∧Aejn, ei1∧ei1· · · ∧ein
= det hAejp, eiqi
1≤p,q≤n,
it follows, by definition of determinants, that there exists a permutation σ of {1,2, . . . , n}such that: Y
1≤k≤n
hAejk, eiσ(k)i 6= 0,
implying that iσ(k)≥jk for eachk. But then, sincel7→dl is nondecreasing:
δα= Y
1≤k≤n
diσ(k) ≥ Y
1≤k≤n
djk=δβ. Now, (3.8) allows to apply Theorem 3.8 to get the result.
Remark. We could also remark that the function:
u(z) = Y
1≤j≤n
sj(D−zANDz) =kΛn(D−zANDz)k
is subharmonic since it is a norm, onΛn(H), and hence a supremum of moduli of the holomorphic functions z 7→ l(D−zANDz), for l a linear functional on Λn(H).
Corollary 3.11. With the notation of Theorem 3.8, D−1AD is compact if A is. Moreover, for anyp >0, ifA∈Sp(H), so does D−1AD, and:
kD−1ADkp≤ kAkp. Proof. Since sn(D−1AD)
nis log-subordinate to sn(A)
n, Corollary 3.6 gives the first assertion, and Proposition 3.5 gives the second one.
3.5 Application to composition operators
We consider here general weighted Hilbert spaces of analytic functions onD. Letβ= (βk)k≥0 be a sequence of positive numbers such that:
(3.9) lim inf
k→∞ βk1/k ≥1
(as we will see right after, this condition ensure that the evaluation maps are bounded) and letH2(β) be the Hilbert space of functions f(z) = P∞
k=0ckzk such that:
(3.10) kfk2H2(β):=
X∞ k=0
βk|ck|2<∞.
This is a Hilbert space of analytic functions onDwith a reproducing kernelKa, namely:
(3.11) f(a) =hf, Kai for allf ∈H2(β), because the evaluationsf ∈H2(β)7→f(a)are continuous:
X∞ k=0
ckak ≤
X∞ k=0
βk|ck|2
1/2X∞ k=0
βk−1|a|2k 1/2
<∞, thanks to condition (3.9).
The canonical orthonormal basis ofH2(β)is formed by the normalized mono- mials
(3.12) eβk(z) = zk
√βk
, k= 0,1, . . .; so we have, for alla∈D:
(3.13) kKak2Hω2 = X∞ n=0
|eβn(a)|2= X∞ n=0
1 βn |a|2n.
We refer to [9] or [43] for more on those spaces. See also [16] for an alternative definition.
For example, the weighted Dirichlet spaceDα2corresponds toβk≈(k+1)1−α. In particular, the Hardy spaceH2 corresponds to βk = 1, the Bergman space B2 toβk = 1/(k+ 1), and the Dirichlet spaceD2to βk = (k+ 1).
For the weights
βk= (k+ 1)! Γ(α+ 2) Γ(k+α+ 1) , we get, using the binomial formula P∞
k=0 Γ(k+α)
k! Γ(α) xk = (1−x)−α for |x| < 1, that the reproducing kernels are, fora6= 0:
Kaα(z) = 1 α(α+ 1)
(1−az)¯ −α−1
¯
az , forα >0 ; (3.14)
Ka0(z) = 1
¯
az log 1 (1−az)¯ , (3.15)
(withK0α(z) = 1/(α+ 1)andK00(z) = 1).
Let us point out thatlimα→0+Kaα(z) =Ka0(z).
Let nowϕ:D→Dbe an analytic map. We assume that:
(3.16) ϕ(0) = 0.
This map ϕ induces formally a lower-triangular composition operator Cϕ on H2(β)since:
hCϕ(eβj), eβii= 1
pβiβjhϕj, zii= 0 fori < j .
Remark. We can often omit condition (3.16). In fact, let us consider the automorphisms ϕa: D → D, a ∈ D, given by ϕa(z) = 1a−−¯azz. When Cϕa is bounded on H2(β), then, with a = ϕ(0), the function ψ = ϕa ◦ϕ satisfies ψ(0) = 0andϕ=ϕa◦ψ; henceCϕ=Cψ◦Cϕa andCψ=Cϕ◦Cϕa, so:
kCϕak−1an(Cψ)≤an(Cϕ)≤ kCϕakan(Cψ).
A necessary condition for havingCϕabounded onH2(β)is thatϕa ∈H2(β).
Sinceϕa(z) =a+P∞
k=1¯ak−1(|a|2−1)zk, we haveϕa ∈H2(β)for alla∈D if limk→∞βk1/k = 1.
For weighted Dirichlet spacesDα2, with anyα >−1, the automorphismsϕa
define bounded composition operators onDα2. In fact, we have, forf ∈ Dα2: kf◦ϕak2Dα2=|f(a)|2+ (α+ 1)
Z
D
|f′[ϕa(z)]|2|ϕ′a(z)|2(1− |z|2)αdA(z)
=|f(a)|2+ (α+ 1) Z
D
|f′(w)|2(1− |ϕa(w)|2)αdA(w). Since:
1− |ϕa(w)|2
1− |w|2 = 1− |a|2
|1−¯aw|2, we have1− |ϕa(w)|2≈1− |w|2 and we get:
kf◦ϕak2Dα2.|f(a)|2+ (α+ 1) Z
D
|f′(w)|2(1− |w|2)αdA(w)≈ kfk2Dα2. For α ≥ 0, that follows directly from [44, Theorem 1] (see also [36, Sec- tion 6.12], [16, Theorem 1.3 and Proposition 3.1] or [34, Theorem 3.1]), since ϕa is univalent.
We will write for shortCϕβ to designate the operatorCϕacting onH2(β).
As an application of the general principles of Section 3.4 we have the fol- lowing result, whose first items were previously obtained by I. Chalendar and J. Partington in [6] and [7] (actually (3.b) is also proved in [7], but for values p≥1).
Theorem 3.12. LetH2(β)andH2(γ)be two weighted Hilbert spaces. Assume that γ is dominated by β in the sense that the sequence (βk/γk) is increasing, so that the continuous inclusion H2(β) ⊆H2(γ) holds. Then, for ϕ: D →D withϕ(0) = 0:
1) if Cϕβ is bounded,Cϕγ is bounded as well, andkCϕγk ≤ kCϕβk; 2) if Cϕβ is compact, so isCϕγ;
3) the sequence sγ = sn(Cϕγ)
n≥1 is log-subordinate to the sequence sβ = sn(Cϕβ)
n≥1, so that:
a) s2n(Cϕγ)≤ q
s1(Cϕβ) q
sn(Cϕβ), for alln≥1;
b) Cϕβ∈Sp H2(β)
=⇒Cϕγ ∈Sp H2(γ)
, for any p >0.
Remark. Let us mention that we can apply the previous theorem in the frame- work of weighted Dirichlet spaces. Indeed, let0< β < γ and consider the two weights:
βk =k.k! Γ(β+ 2)
Γ(k+β+ 1) and γk= k.k! Γ(γ+ 2) Γ(k+γ+ 1)
associated with the weighted Dirichlet spaces Dβ2 and Dγ2 respectively, with γ > β, so thatDβ2⊂ Dγ2 . In order to apply our comparison Theorem 3.12, we have to show that the sequence(βk/γk)increases. But
βk
γk
=Γ(β+ 2) Γ(γ+ 2)
Γ(γ+k+ 1)
Γ(β+k+ 1) =:Γ(β+ 2) Γ(γ+ 2)Ak, and, settingh=γ−β >0 andxk =β+k+ 1, we see that:
Ak =Γ(xk+h) Γ(xk) ·
Since the function Γ is log-convex, the map x 7→ Γ(x+h)Γ(x) increases on(0,∞), and we get that the sequence(βk/γk)increases.
Proof of Theorem 3.12. We setdk=p
βk/γk andek(z) =zk.
LetJ:H2(β) →H2(γ) the unitary (onto isometry) and diagonal operator defined byJ(ek) =dkek, for allk≥0.
The operator A = JCϕβJ−1 maps H2(γ) into itself and sn(A) = sn(Cϕβ) for alln ≥ 1 (in particular kAkL(H2(γ)) = kCϕβkL(H2(β))). Moreover, A has a lower-triangular matrix.
Now we consider the diagonal operator D: H2(γ) → H2(γ) defined by D(ek) = dkek. In general, it is an unbounded operator. It is plain that D−1J:H2(β)→ H2(γ)is the canonical inclusion, since (D−1J)(ek) = ek for
allk≥0. Hence(D−1J)Cϕβ =Cϕγ(D−1J), and sinceAJ =JCϕβ, we have the following commutative diagram:
H2(β) C
β ϕ //
J
H2(β)
J
H2(γ)
D
//H2(γ)
A
//H2(γ)
D−1
//H2(γ)
By Theorem 3.9, we get:
logs(D−1AD)≺logs(A)
(so we have, in particular, kD−1ADkL(H2(γ)) ≤ kAkL(H2(γ)) =kCϕβkL(H2(β))).
ButD−1AD =Cϕγ, and this proves Theorem 3.12, using Proposition 3.5 and Corollary 3.6.
Remark. Actually, the same proof gives the following generalization of Theo- rem 3.12.
Theorem 3.13. With the hypothesis of Theorem 3.12, letT:Hol(D)→ Hol(D) be a linear map such that its restriction Tβ to H2(β) is bounded from H2(β) into H2(β) and has a matrix in the canonical basis of H2(β) which is lower- triangular. Then:
1) Tγ is bounded as well, andkTγk ≤ kTβk; 2) if Tβ is compact, so isTγ;
3) the sequence of singular numberss(Tγ) = sn(Tγ)
n≥1 is log-subordinate to the sequences(Tβ) = sn(Tβ)
n≥1, so that:
a) s2n(Tγ)≤p
s1(Tβ)p
sn(Tβ), for alln≥1;
b) Tβ∈Sp H2(β)
=⇒Tγ ∈Sp H2(γ)
, for any p >0.
3.6 Application to conditional multipliers
We first recall the following well-known proposition (and give a short proof, for sake of completeness). Note that this is not shared by the Dirichlet spaces Dα2whenα≤0([41, Theorem 10]; see also [40, Theorem 2.7], [17, Theorem A], and [42, Theorem 4.2]). Recall that it is well-known that the spaceM(H2)of multipliers ofH2is isometric to H∞.
Proposition 3.14. For every γ >−1, the space M(B2
γ) of multipliers ofB2
γ
is isometric to the spaceH∞.
IfH is a Hilbert space of analytic functions onD, containing the constants, and with reproducing kernelsKa,a∈D, then the spaceM(H)of multipliers of H is contained contractively into the spaceH∞.
Proof. Ifhf ∈H for all f ∈ H, then, takingf =1, we haveh ∈H, so his analytic. The same proof as in [1, Proposition 3.1] shows that h ∈H∞. For sake of completeness we give a short different proof.
In fact, we have, for alla∈D:
(3.17) Mh∗(Ka) =h(a)Ka for alla∈D;
hence|h(a)| kKak ≤ kMh∗k kKak, and, sincekKak is not null, that proves that h∈H∞ andkhk∞≤ kMhk.
HenceM(H)jH∞, contractively.
WhenH =B2γ, we have the reverse inclusion. Indeed, for every h∈ H∞, one clearly has hf ∈ B2
γ and khfkB2γ ≤ khk∞kfkB2γ for all f ∈ B2
γ, so the multiplication operatorMh:B2γ →B2γ is bounded with norm≤ khk∞.
Let now ϕ be an analytic self-map of D and H = H2(β) be a weighted Hilbert space of analytic functions onD, with reproducing kernelKa,a∈D, on whichCϕacts boundedly . We denote its multiplier set, respectively multiplier set conditionally toϕ, by:
(3.18) M(H) ={w∈H; wf ∈H for eachf ∈H} and
(3.19) M(H, ϕ) ={w∈H; w(f◦ϕ)∈H for allf ∈H}. We haveM(H)⊆ M(H, ϕ).
The setM(H, ϕ)plays an important role in the study of weighted composi- tion operators.
Definition 3.15. A Hilbert spaceH of analytic functions onD, containing the constants, and with reproducing kernelsKa,a∈D, is said admissible if:
(i) H2 is continuously embedded inH; (ii) M(H) =H∞;
(iii) the automorphisms ofD induce bounded composition operators onH; (iv) kKakH
kKbkH ≤h
1− |b| 1− |a|
for a, b∈Dclose to ∂D, whereh:R+→R+ is an non-decreasing function.
Note that(i)implies thatkfkH≤CkfkH2 for allf ∈H2, for some positive constantC, and so (BH andBH2 being the unit ball ofH andH2respectively):
kKakH= sup
f∈BH
|f(a)| ≥C−1 sup
f∈BH2
|f(a)|=C−1(1− |a|2)−1/2, implying that:
|alim|→1−kKakH=∞.
Examples.
1) The weighted Bergman spaceB2γ, withγ >−1 is admissible.
Indeed, we know that it is continuously embedded inH2=B2
−1; condition (ii) is Proposition 3.14; condition (iii) is satisfied according to the Remark before Theorem 3.12, andkKak2= (1−|a1|2)γ+2, giving(iv).
2) More generally, we have the following result.
Proposition 3.16. For any decreasing sequenceβ such that the automorphisms of D induce bounded composition operators on H2(β), the space H2(β) is ad- missible.
Recall that H2(β) is defined in (3.10). A particular case is obtained as follows. Let ω: (0,1) → R+ be an integrable function such that, for some positive and locally bounded functionρ:R+→R+, we have:
(3.20) ω(y)
ω(x) ≤ρ y
x
for allx, y∈(0,1), and letHω2 be the space of analytic functions f:D→Csuch that:
(3.21) kfk2Hω2 :=
Z
D
|f(z)|2ω(1− |z|2)dA(z)<∞.
Such spaces are used in [16] and in [29]. We haveHω2 =H2(β)with:
(3.22) βn= 2 Z 1
0
r2n+1ω(1−r2)dr= Z 1
0
tnω(1−t)dt . Indeed, since βn = R1
0(1−t)nω(t)dt, the sequence β = (βn)n is decreasing.
Moreover, the fact that the automorphisms ofD induce bounded composition operators onHω2 is proved as in the Remark before Theorem 3.12, namely:
kf◦ϕak2Hω2 = Z
D|f(w)|2|ϕ′a(w)|2ω(1− |ϕa(w)|2)dA(w)
≤
1 +|a| 1− |a|
2Z
D
|f(w)|2ρ
1− |ϕa(w)|2 1− |w|2
ω(1− |w|2)dA(w)
≤
1 +|a| 1− |a|
2
cρ,a
Z
D
|f(w)|2ω(1− |w|2)dA(w)
=:κakfk2Hω2,
where we used that |ϕ′a(w)| ≤ 1+1−||aa||, that 1−|1−|ϕaw(w)|2|2 ≤ 1+1−||aa||, and that ρ is locally bounded.
Note thatB2γ,γ >−1, corresponds to ω(t) = (γ+ 1)tγ.
Proof of Proposition 3.16. Condition (i) is satisfied because β is decreasing.
Moreover, since β is decreasing, Theorem 3.13, applied to T = Mw, with w∈H∞, ensures thatH∞=M(H2)⊆ M H2(β)
, and, for allw∈H∞: kMw: H2(β)→H2(β)k ≤ kMw:H2→H2k=kwk∞. Now, Proposition 3.14 implies thatH∞=M H2(β)
and kMw:H2(β)→H2(β)k=kwk∞
for allw∈H∞.
It remains to show that, for H = H2(β), the condition (iii) implies the condition(iv).
Since H2(β) is isometrically rotation invariant, it is clear that kKak = kK|a|k; hence, by the maximum principle,kKxk ≤ kKyk, for0≤x≤y <1.
Assume now that0< y < x <1. LetT be the disk automorphism:
(3.23) T(z) = 2z+ 1
z+ 2 , z∈D.
The fixed points ofT are 1 and −1, and T(0) = 1/2. We define the sequence (an)n≥0 by induction, with:
a0= 0, an+1=T(an). We see that:
(3.24) 1−an+1= Z 1
an
T′(x)dx= Z 1
an
3
(x+ 2)2dx≤ 3
4(1−an) ; so(an)n is increasing and converges to1. In the same way, we see that:
(3.25) 1−an+1= Z 1
an
T′(x)dx= Z 1
an
3
(x+ 2)2dx≥ 1
3(1−an). Since0< y < x <1, we can find m≤nsuch that:
am−1< y < am, and an−1< x < an.
We have kKxk ≤ kKank and kKyk ≥ kKam−1k. Since CT∗Kz = KT(z) for all z∈D, we have:
kKxk
kKyk ≤ kKank
kKam−1k ≤ kCT∗kn−m+1=αn−m+1, withα=kCTk ≥1. Applying (3.24) and (3.25), we get:
1−y
1−x ≥ 1−am
1−an−1 ≥ 1−am
3(1−an) ≥1 3
4 3
n−m
.
It suffices now to takes≥0 such that(4/3)s=α, and A >0 large enough in order that, with the increasing functionh(t) = max{Ats,1},t >0, we have:
h1−y 1−x
≥A
3sαn−m≥ A 3sα
kKxk
kKyk ≥kKxk kKyk·
Let us come back to the conditional multipliers. In general, we obviously have:
(3.26) H∞⊆ M(H, ϕ)⊆H .
The extreme cases were characterized by Attele ([2]) whenH =H2=B2
−1
(and Contreras and Hernández-Díaz in [8] for the spacesHp) as follows.
Theorem 3.17(Attele). We have:
1) M(H2, ϕ) =H2 if and only ifkϕk∞<1.
2) M(H2, ϕ) =H∞ if and only ifϕ is a finite Blaschke product.
A key tool for the most delicate second necessary condition is the use of inner and outer functions. We no longer have this tool at our disposal for the admissible spacesH =H2(β), but we can nevertheless state the following analog result.
Theorem 3.18. Letϕan analytic self-map ofDandH be an admissible Hilbert space on whichCϕ acts boundedly. We have:
1) M(H2, ϕ)⊆ M(H, ϕ);
2) M(H, ϕ) =H if and only ifkϕk∞<1;
3) M(H, ϕ) =H∞ if and only if ϕis a finite Blaschke product.
Note that the assumption that Cϕ acts boundedly onH is automatically satisfied whenH =H2(β)withβ decreasing, by Theorem 3.12.
Proof. 1) Suppose first that ϕ(0) = 0. Let w ∈ M(H2, ϕ). The weighted composition operatorMwCϕ is bounded onH2, and moreover lower triangular on the canonical basis; applying Theorem 3.13, 1), we get thatMwCϕis bounded onH as well, that isw∈ M(H, ϕ).
In the general case, let ϕ(0) = a, so that (ϕa◦ϕ)(0) = 0. Property (iii) implies that
M(H2, ϕ) =M(H2, ϕa◦ϕ)⊆ M(H, ϕa◦ϕ) =M(H, ϕ),
sincef ∈H2 if and only iff◦ϕa∈H2 andf ∈H if and only if f◦ϕa∈H. 2)The necessary condition is proved as in [2] forH2; we recall some details.
We start from the (obvious, but useful) mapping equation:
(3.27) (MwCϕ)∗(Kz) =w(z)Kϕ(z).
The assumption implies the existence of a constantC such that:
kMwCϕkL(H)≤CkwkH for allw∈H . As a consequence, for givenz∈D:
k(MwCϕ)∗(Kz)kH≤CkwkHkKzkH,
