# Toeplitz Operators on Weighted Bergman Spaces

Authors: Gerardo R. Chacón
Journal: Journal of Function Spaces and Applications
 (2013)

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## Abstract

We characterize the boundedness and compactness of a Toeplitz-type operator on weighted Bergman spaces satisfying the  Bekollé-Bonami condition in terms of the Berezin transform.


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## Body

## 1. Introduction

Let𝔻 denote the unit disc in the complex plane, and let dA denote the normalized Lebesgue measure on 𝔻. For a given nonnegative integrable function u on 𝔻, we define the weighted Bergman spaceA2(u) as the space of all analytic functions on 𝔻 that belong to the weighted space L2(u). That is, f∈A2(u) if f is analytic on 𝔻 and satisfies
(1)∥f∥A2(u)2:=∫𝔻‍|f(z)|2u(z)dA(z).Weighted Bergman spaces have been studied by several authors in different contexts (see, e.g., [1–5]). Most of the researches about Toeplitz-type operators in this spaces have been done by considering radial weights. In this paper we will consider weights satisfying the the socalled Békollé-Bonami condition.Definition 1.
A functionu in L1(𝔻) is said to satisfy the Bekolle-Bonami condition B2 if there exists a constant C>0 such that
(2)1(A(S(I)))2∫S(I)‍u(z)dA(z)∫S(I)‍1u(z)dA(z)≤C
for every interval I⊂∂𝔻. Here, S(I) denotes the Carleson square:
(3)S(I)={reit:eit∈I,1-|I|2π≤r<1}.
ConditionB2 was introduced by Bekolle and Bonami in [6]. They showed that B2 is necessary and sufficient for the Bergman projection to be bounded on L2(udA). Condition B2 was used by Luecking in [7, 8] to study Carleson measures in weighted Bergman spaces. Then in [9], Nakazi and Yamada generalized Luecking's results by introducing a more restrictive condition (A2)∂ and found, under this condition, a characterization for Carleson measures on A2(u) spaces. They also gave several examples of weights satisfying this condition. In [10], Constantin generalized Luecking's result and studied Toeplitz-type operators on spaces A2(u). The characterization of Carleson measures is as follows.Theorem 2 (see [7, 10]).
Suppose thatu>0 a.e. in 𝔻 and satisfies the B2 condition; let dμ=udA and let ν be a positive Borel measure on 𝔻. Then, the following are equivalent.(1)
There exist a constantC>0 such that
(4)∫𝔻‍|f|2dν≤C∫𝔻‍|f|2dμ,
for every polynomial f, that is, ν is a μ-Carleson measure.(2)
There existsr>0 and γ>0 satisfying
(5)ν(Dr(a))≤γμ(Dr(a)),
for all a∈𝔻.Theorem 3 (see [8]).
Suppose thatu∈L1(dA) satisfies condition B2; then, the dual of A2(u) can be identified with A2(u-1). The pairing is given by
(6)〈f,g〉=∫𝔻‍fg¯dA.Remark 4.
Notice that ifu satisfies the B2 condition, then it is not hard to show (see, e.g., [8]) that there is a constant C>0 such that for any function f analytic in 𝔻 and any a∈𝔻; the following holds:
(7)|f(a)|2≤C(∫Dr(a)‍udA)-1∥f∥A2(u)2.
Consequently, evaluation functionals are bounded on A2(u), and so for every complex number a∈𝔻, there exists a function Ku(·,a)∈A2(u) such that
(8)〈f,Ku(·,a)〉A2(u)=f(a),
for every function f∈A2(u); that is, the functions Ku(·,a) are the reproducing kernels for A2(u). Notice also that the set {Ku(·,w):w∈𝔻} is dense in A2(u).
Moreover, using Theorem3, we have that there exists a bounded, bijective, linear operator F1:A2(u-1)→(A2(u))* such that F1(f)(g)=〈g,f¯〉A2 for every f∈A2(u-1) and g∈A2(u). On the other hand, since A2(u) is a Hilbert space, there exists a bounded, bijective, linear operator F2:(A2(u))*→A2(u) such that 〈g,F2(γ)¯〉A2(u)=γ(g) for every γ∈(A2(u))* and g∈A2(u).
Letγa∈(A2(u))* denote the evaluation functional at a; then, F2(γa)=Ku(·,a). On the other hand, if we let Ba(z), with a,z∈𝔻, denote the Bergman kernel(9)Ba(z):=1(1-a¯z)2,
then Ba∈A2(u-1), and the pairing is given by the A2 product; we have that F1(Ba)=γa.
DefineT:A2(u)→A2(u-1) by T=F1-1F2-1. Then T is a bounded operator and T(Ku(·,a))=F1-1(γa)=Ba. Also, the operator T-1:A2(u-1)→A2(u) is well defined, and consequently we have that
(10)∥Ba∥A2(u-1)~∥Ku(·,a)∥A2(u).Definition 5.
Letν be a positive Borel measure on 𝔻. Define for each polynomial p the Toeplitz operator Tν(11)Tνp(z):=∫𝔻‍Ku(z,w)p(w)dν(w).
We will study the conditions under which Toeplitz operators can be extended toA2(u), and we will characterize the boundedness and compactness of Toeplitz operators acting on the A2(u) spaces in terms of the socalled Berezin transform of the measure ν.Definition 6.
Letν be a positive Borel measure on 𝔻; the Berezin transform ν~ of ν is defined as
(12)ν~(a):=1∥Ba∥A2(u)∫𝔻‍|Ba(z)|2dν(z).
From now on, we will assume thatν is a probability measure.
The following result is well known (see [1–5]).Theorem 7.
Suppose thatμ is a finite positive Borel measure on 𝔻. Then, the following are equivalent (here, dAα(z) represents the standard weighted Lebesgue measure (1-|z|2)αdA(z), α>-1). (a)
Tμ is bounded on A2(dAα).(b)
μ~ is a bounded function on 𝔻.(c)
μ is a Carleson measure for A2(dAα),
where(13)μ~(z)=∫𝔻‍|bz(w)|2dμ(w),z∈𝔻
is the Berezin transform of the Toeplitz operator, here
(14)bz(w):=1-|z|2(1-z¯w)2.The proof of this result mainly relies on the behavior of the reproducing kernelsbz(w) when z is close to w and on the characterization of Carleson measures for the Bergman space. In the case of A2(u) spaces, such a characterization is known for Carleson measures (see, e.g., [10]), but we do not have an explicit formula for the reproducing kernels; consequently, we will need to use a different technique. This will be an atomic decomposition developed in terms of the reproducing kernels of A2(u).


## 2. Atomic Decomposition forA2(u)

In this section, we develop a way of expressing functions inA2(u) as a linear combination of reproducing kernels. Similar problems have been studied in [8, 11]. We will use a similar reasoning as in [8], where the problem is studied in terms of the Bergman kernel. We will assume in the rest of this paper that μ:=udA is a probability measure.Definition 8.
Letɛ>0. A sequence {an}⊂𝔻 is called ɛ-separated if inf{ρ(an,am):n≠m}≥ɛ>0. Here, ρ denotes the pseudohyperbolic metric: ρ(z,w):=|z-w|/|1-z¯w|.Theorem 9 (see [8]).
Ifu satisfies condition B2, then there exists an ɛ-separated sequence {an} such that
(15)∥f∥A2(u)2~∑k‍|f(an)|2μ(Dr(an)),
for every function f∈A2(u).Theorem 10.
Suppose thatu satisfies the condition B2. Then, there exists a sequence {an}⊂𝔻 which is ɛ-separated for some ɛ>0, such that any f∈A2(u) has the form
(16)f(z)=∑n‍cnKu(·,an)μ(Dr(an))1/2,
for some sequence {cn}∈l2.Proof.
By Theorem9, we know that there exists an ɛ-separated sequence {an} such that ∥f∥A2(u)2~∑k‍|f(an)|2μ(Dr(an)). Consider the linear operator R:A2(u)→l2 defined by
(17)Rf:={f(an)μ(Dr(an))1/2}.
Then, ∥Rf∥l22=∑n‍|f(an)|2μ(Dr(an))~∥f∥A2(u)2, and so R is a bounded injective linear operator having closed range. Consequently, the dual mapping R*:l2→A2(u) is onto. Now, let {cn}∈l2; then, for f∈A2(u), we have
(18)〈R*{cn},f〉A2(u)=〈{cn},Rf〉l2=∑n‍cnf(an)¯μ(Dr(an))1/2=∑n‍cn∫𝔻‍f(z)¯Ku(z,an)dμ(z)μ(Dr(an))1/2=∫𝔻‍f(z)¯∑n‍cnKu(z,an)μ(Dr(an))1/2dμ(z)=〈∑n‍anKu(·,an)μ(Dr(an))1/2,f〉A2(u).
Here, we justify the interchange of integration and summation as follows. Given a sequence {cn}∈l2 and ɛ>0, there exists N∈ℕ such that if m,l≥N, then
(19)∑n=ml‍|cn|2<ɛ∥R*∥.
Thus,
(20)∑n=ml‍cn∫𝔻‍f(z)¯Ku(z,an)dμ(z)μ(Dr(an))1/2=∫𝔻‍f(z)¯∑n=ml‍cnKu(z,an)μ(Dr(an))1/2dμ(z),∥∑n=ml‍cnKu(z,an)μ(Dr(an))1/2∥A2(u)=sup∥f∥A2(u)=1|〈∑n=ml‍cnKu(z,an)μ(Dr(an))1/2,f〉A2(u)|=sup∥f∥A2(u)=1|∑n=ml‍cn∫𝔻‍Ku(z,an)dμ(z)μ(Dr(an))1/2|=sup∥f∥A2(u)=1|〈R*{c^n},f〉A2(u)|≤∥R*∥∥{c^n}∥l2<ɛ,
where
(21)c^n:={cn,ifn∈{m,…,l}0,otherwise.
Thus,sm:=∑n=1mcnKu(·,an)μ(Dr(an))1/2 forms a Cauchy sequence, and consequently it converges in the A2(u)-norm to
(22)s:=∑n=1∞cnKu(·,an)μ(Dr(an))1/2∈A2(u).
Therefore,
(23)∫𝔻f(z)sm(z)-dμ(z)⟶∫𝔻f(z)s(z)-dμ(z).
Hence,R*{cn}=∑n=1∞cnKu(·,an)μ(Dr(an))1/2, and since R* is surjective, then the result holds.


## 3. Boundedness of Toeplitz Operators

Theorem 11.
Suppose thatu satisfies the B2 condition and that the Toeplitz operator Tν:A2(u)→A2(u) is bounded. Then, the Berezin transform ν~ is bounded.Proof.
First, notice that for every finite sumsm:=∑n=1mλnKu(·,an), an∈𝔻; the following holds:
(24)〈Tν(∑n=1m‍λnKu(·,an)),∑n=1m‍λnKu(·,an)〉A2(u)=∑l=1m‍Tν(∑n=1m‍λnKu(·,an))(al)=∑l=1m‍λl∫𝔻‍Ku(al,z)∑n=1m‍λnKu(z,an)dν(z)=∥∑n=1m‍λnKu(·,an)∥L2(ν)2≲∥∑n=1m‍λnKu(·,an)∥A2(u)2.
So, if {sm} is a Cauchy sequence in A2(u), it is also a Cauchy sequence in L2(ν), and so it converges in L2(ν). Therefore, by Theorem 10, we have that if sm→g in the A2(u)-norm, then sm→g in L2(ν), and consequently for any function f∈A2(u), 〈f,sm〉L2(ν)→〈f,g〉L2(ν). Moreover,
(25)〈Tνf,g〉A2(u)=limm→∞〈Tνf,sm〉A2(u)=〈f,sm〉L2(ν).
Therefore,
(26)〈Tνf,g〉A2(u)=〈f,g〉L2(ν).
Hence, for any a∈𝔻,
(27)〈TνBa,Ba〉A2(u)=〈Ba,Ba〉L2(ν)=ν~(a)∥Ba∥A2(u)2,
and consequently, ν~(a)≤∥Tν∥.We will need the following lemma due to Constantin.Lemma 12 (see [10]).
Suppose thatu satisfies condition B2; then,
(28)∥Ba∥A2(u)2~μ(Dr(a))(1-|a|)4.Proposition 13.
Ifν~ is bounded on 𝔻, then ν is a μ-Carleson measure, where μ=udA.Proof.
Fix0<r<1; if ν~ is bounded, then there exists a constant C>0 such that for every a∈𝔻(29)1∥Ba∥A2(u)2∫Dr(a)‍1|1-a¯z|4dν(z)≤C,
and consequently
(30)ν(Dr(a))≲C(1-|a|2)4∥Ba∥A2(u)2~(1-|a|2)4μ(Dr(a))(1-|a|)4≲μ(Dr(a)),
and by Theorem 2, we obtain the result.Proposition 14.
Suppose thatν is a μ-Carleson measure; then, the Toeplitz operator Tν:A2(u)→A2(u) is bounded.Proof.
For every function,g∈A2(u) define the linear operator g*:A2(u)→ℂ by: g*(f):=〈f,g〉L2(ν). Note that if ν is a μ-Carleson measure, then
(31)|g*(f)|≤∥f∥L2(ν)∥g∥L2(ν)≤∥f∥A2(u)∥g∥L2(ν),
so g*∈(A2(u))*, ∥g*∥≤∥g∥L2(ν), and there exists g~∈A2(u) such that g*(f)=〈f,g~〉A2(u).
We have just proved that for everyg∈A2(u), there exists g~∈A2(u) such that
(32)〈f,g〉L2(ν)=〈f,g~〉A2(u),∀f∈A2(u).
In particular, taking f=Ku(·,a), we have that for every a∈𝔻,
(33)g~(a)=∫‍g(z)Ku(a,z)dν(z)=Tνg(a).
Hence, Tν(A2(u))⊂A2(u). Moreover, since ∥g~∥A2(u)=∥g*∥≤∥g∥L2(ν)≤∥g∥A2(u), then
(34)∥Tνg∥A2(u)≤∥g∥A2(u),
and consequently Tν is bounded.Corollary 15.
ForTν:A2(u)→A2(u) bounded, the following equality holds for every f and g in A2(u):
(35)〈Tνg,f〉A2(u)=〈g,f〉L2(ν).


## 4. Compactness of Toeplitz Operators

In this section, we will characterize compactness of Toeplitz operators in terms of its Berezin transform. We will use the characterization for vanishing Carleson measures given by Constantin.Definition 16.
A positive Borel measureν on 𝔻 is a μ-vanishing Carleson measure if the inclusion operator i:A2(u)→L2(ν) is compact.Theorem 17 (see [10, Theorem 3.3]).
Suppose thatu satisfies the B2 condition. Then ν is a μ-vanishing Carleson measure if and only if for every 0<r<1,
(36)lim|a|→1-ν(Dr(a))μ(Dr(a))=0.Theorem 18.
Suppose thatν is a positive Borel measure on 𝔻. Then the following are equivalent:(a)
Tν is compact.(b)
ν~(a)→0 as |a|→1-.(c)
ν is a μ-vanishing Carleson.Proof.
(a)⇒ (b)
Notice that by Jensen's inequality, we have that(37)μ(Dr(a))≳(1-|a|2)4(∫Dr(a)‍u-1(z)dA(z))-1,
and consequently using Lemma 12, we get
(38)|ba(w)|≲|1-a¯z|-2(∫Dr(a)‍u-1(z)dA(z))⟶0,
as |a|→1-. Thus, the family {ba}a∈𝔻 converges to zero weakly in A2(u) as |a|→1-. We also know from the proof of Theorem 11 that ν~(a)=〈Tνba,ba〉A2(u). Consequently, ν~(a)≤∥Tνba∥A2(u) which, since Tν is compact, converges to zero as |a|→1-.(b)⇒ (c)
We have thatν~(a)=∫𝔻‍|ba(z)|2dν(z), and by Lemma 12, we obtain that
(39)ν~(a)≥∫Dr(a)‍|ba(z)|2dν(z)~ν(Dr(a))(1-|a|2)4∥Ba∥A2(u)2~ν(Dr(a))μ(Dr(a)),
then (c) follows from (b).(c)⇒ (a)
By Corollary15 and Cauchy-Schwarz inequality we have that:
(40)∥Tνf∥A2(u)=sup{|〈Tνf,g〉A2(u)|:∥g∥A2(u)=1}=sup{|〈f,g〉L2(ν)|:∥g∥A2(u)=1}≤∥f∥L2(ν)sup{∥g∥L2(ν):∥g∥A2(u)=1},
and since ν is a μ-Carleson measure, then ∥g∥L2(ν)≲∥g∥A2(u) for every g∈A2(u), so
(41)∥Tνf∥A2(u)≲∥f∥L2(ν),∀f∈A2(u).
Now, iffn is a sequence in A2(u) that converges to zero weakly, then by the compactness of i:A2(u)→L2(ν) we have that ∥fn∥L2(ν) converges to zero. Therefore, ∥Tνfn∥A2(u) converges to zero, and so Tν is compact on A2(u).We have used the fact that the family of normalized Bergman kernels{ba} converges to zero weakly as |a|→1-. This is also true for the family of normalized reproducing kernels of A2(u).Proposition 19.
Ifu satisfies condition B2, then Ku(·,a)/∥Ku(·,a)∥A2(u) converges to zero weakly in A2(u).Proof.
We use the same notation as in the observation after Theorem3. The operator T:A2(u)→A2(u-1) maps Ku(·,a)/∥Ku(·,a)∥A2(u) to Ba/∥Ku(·,a)∥A2(u). Also, by Lemma 12, we have that
(42)|Ba|∥Ku(·,a)∥A2(u)~|Ba|∥Ba∥A2(u-1)≲|Ba|μ(Dr(a)),
which converges to zero uniformly on compact subsets of 𝔻. Consequently, the family {Ba/∥Ku(·,a)∥A2(u)} is uniformly bounded on A2(u-1) and converges to zero uniformly on compact subsets of 𝔻. Since A2(u-1) is a functional space, then Ba/∥Ku(·,a)∥A2(u) converges to zero weakly on A2(u-1).
Now, the operatorT-1:A2(u-1)→A2(u) is bounded, so Ku(·,a)/∥Ku(·,a)∥A2(u) converges to zero weakly in A2(u).



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