# Volterra Type Integral Operators  and Composition Operators on Model Spaces

Authors: Tesfa Mengestie
Journal: Journal of Function Spaces
 (2015)

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## Abstract

We study some mapping properties of Volterra type integral operators and composition operators on model spaces. We also discuss and give out a couple of interesting open problems in model spaces where any possible solution of the problems can be used to study a number of other operator theoretic related problems in the spaces.


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## Body

## 1. Introduction

We denote byH the space of holomorphic functions. We define the Volterra type integral operator on H induced by a homomorphic symbol g by
(1)Vgf(z)=∫0z‍fwg′(w)dw.
Questions about various operator theoretic properties of Vg expressed in terms of function theoretic conditions on g have been a subject of high interest mainly after the works of Pommerenke [1] and later by Aleman et al. [2–4]. In [3], it was proved that Vg on the Hardy space Hp(D), 1≤p<∞, is bounded if and only if g belongs to the space of BMOA, while compactness is characterized in terms of the corresponding VMOA spaces.In this paper we plan to study the bounded, compact, and Hilbert–Schmidt properties of the Volterra type integral operators and composition operators acting between the model spacesKI2 and the Hardy space H2(C+) of the upper-half plane. We recall that the model spaces KI2, where I is an inner function, are the subspace of H2(C+) defined by KI2=H2(C+)⊖IH2(C+). The spaces KI2 are reproducing kernel Hilbert spaces with kernel function
(2)Kw(z)=i2π·1-I(w)¯I(z)z-w¯,kw(z)=Kw(z)Kw2.
Each inner function I has either one or infinitely many components. We call it one-component if there exists a δ in (0,1) for which the level set z∈C+:|I(z)|<δ is connected. For model spaces generated by such class of inner functions, we prove the following.Theorem 1.
LetI be a one-component inner function on C+ and let g be a holomorphic function on C+. Then Vg:KI2→H2(C+) is(i)
bounded if and only if(3)sup⁡w∈C+∫C+‍kwz2g′z2IzdA(z)<∞;(ii)
compact if and only if(4)lim⁡|w|→∞∫C+‍kwz2g′z2IzdA(z)=0;(iii)
in the Hilbert–Schmidt classS2(KI2,H2(C+)) if and only if
(5)∫C+‍g′z21-Iz2dA(z)<∞,wheredA denotes the usual Lebesgue area measure.The criteria found above classify the bounded and compactVg on model spaces in terms of their action on the reproducing kernels. As will be seen in Section 4, by converting the boundary integral in Hardy spaces to an area integral in terms of derivatives, we could reduce the boundedness and compactness problems of Vg into questions of bounded and compact Carleson embedding maps (measures) for KI2. Such measures have been identified when I belongs to the class of one-component inner functions; for example, see [5–10]. Despite several attempts made by many experts in the subject, a complete identification of the measures remains open when I belongs to the class of infinitely many component inner functions. For a special class of model spaces which admit reproducing kernel Riesz bases corresponding to sparse sequence of points, which includes the de Branges spaces, the author together with Belov et al. [11] has described the measures in terms of a condition analogous to the A2 weight condition. In what follows we will use those results in [11] to study the bounded and compact properties of Vg. We note in passing that those spaces constitute model spaces generated by the class of infinitely many component inner functions.


### 1.1. The Volterra Type Integral Operators onH(Γ,v)

LetΓ=(γn) be a sequence of distinct complex numbers and let v=(vn) be a weight sequence that satisfies the admissibility condition
(6)∑n=1∞‍vn1+γn2<∞.
Any such pair (Γ,v) parameterizes the space H(Γ,v) which consists of all functions
(7)f(z)=∑n=1∞anvnz-γn
for which
(8)fH(Γ,v)=(an)lv2<∞,lv2=an:av2=∑n=1∞an2vn<∞
and z belongs to the set
(9)Γ,v*=z∈C:∑n=1∞vnz-γn2<∞.
The equation in (7) means that we obtain the value of a function f in H(Γ,v) at a point z in (Γ,v)* by computing the weighted discrete Hilbert transform:
(10)(an)⟼∑n=1∞‍anvnz-γn,
which is well defined whenever (an) belongs to lv2. This follows from an application of Cauchy–Schwarz inequality along with (8) and (9). The de Branges spaces, model subspaces of the Hardy space H2 which admit Riesz bases of reproducing kernels, and the Fock-type spaces studied in [12] are all examples of spaces of the kind H(Γ,v). We refer to [11, 13, 14] for detailed accounts. The Carleson measures on H(Γ,v) have been described in [11] when Γ grows at least exponentially, that is, when
(11)inf⁡n≥1|γn+1||γn|>1.
When we consider such sparse sequence Γ, it is natural to partition C+ in the following way. We set Ω1=z∈C+:|z|<(|γ1|+|γ2|)/2 and then for n≥2(12)Ωn=z∈C+:γn-1+γn2≤z<γn+γn+12.
We now state our first result on H(Γ,v).Theorem 2.
Suppose that the sequenceΓ satisfies the sparseness condition (11) and g:(Γ,v)*→C+ is homomorphic. Then Vg:H(Γ,v)→H2(C+) is(i)
bounded if and only if(13)sup⁡n≥1∫Ωn‍vng′z2IzdA(z)z-γn2<∞,(14)sup⁡n≥1∑l=1nvl∑m=n+1∞∫Ωm‍g′z2IzdA(z)z2lllllllll+∑l=n+1∞‍vlγl2∑m=1n∫Ωm‍g′z2IzdAz∑l=1nvl∑m=n+1∞∫Ωm‍g′z2IzdA(z)z2<∞;(ii)
compact if and only if the “little oh” counterparts of (13) and (14) hold;(iii)
in the Hilbert–Schmidt classS2(H(Γ,v),H2(C+)) if and only if
(15)∑n=1∞‍∫Ωnvng′z2Izz-γn2dAz<∞,∑k=1∞vk∑k=n+1∞∫Ωkg′z2IzdA(z)z2+∑k=n+1∞vnγn2∑k=1n∫Ωng′z2IzdA(z)<∞.Condition (13) of the theorem is a condition about the local behavior of the symbol g, while condition (14) deals with its global behavior. Combining the two conditions, we see that (13) may be replaced by the stronger global condition
(16)sup⁡n≥1∫C+vng′z2IzdA(z)z-γn2<∞.
In some special cases, this condition or the original one (13) automatically gives condition (14), for example, if the weight sequence vn grows at least exponentially and the numbers vn/|γn|2 decay at least exponentially with n. The same conclusion follows if the weight sequence vn is also summable.


## 1.1. The Volterra Type Integral Operators onH(Γ,v)

LetΓ=(γn) be a sequence of distinct complex numbers and let v=(vn) be a weight sequence that satisfies the admissibility condition
(6)∑n=1∞‍vn1+γn2<∞.
Any such pair (Γ,v) parameterizes the space H(Γ,v) which consists of all functions
(7)f(z)=∑n=1∞anvnz-γn
for which
(8)fH(Γ,v)=(an)lv2<∞,lv2=an:av2=∑n=1∞an2vn<∞
and z belongs to the set
(9)Γ,v*=z∈C:∑n=1∞vnz-γn2<∞.
The equation in (7) means that we obtain the value of a function f in H(Γ,v) at a point z in (Γ,v)* by computing the weighted discrete Hilbert transform:
(10)(an)⟼∑n=1∞‍anvnz-γn,
which is well defined whenever (an) belongs to lv2. This follows from an application of Cauchy–Schwarz inequality along with (8) and (9). The de Branges spaces, model subspaces of the Hardy space H2 which admit Riesz bases of reproducing kernels, and the Fock-type spaces studied in [12] are all examples of spaces of the kind H(Γ,v). We refer to [11, 13, 14] for detailed accounts. The Carleson measures on H(Γ,v) have been described in [11] when Γ grows at least exponentially, that is, when
(11)inf⁡n≥1|γn+1||γn|>1.
When we consider such sparse sequence Γ, it is natural to partition C+ in the following way. We set Ω1=z∈C+:|z|<(|γ1|+|γ2|)/2 and then for n≥2(12)Ωn=z∈C+:γn-1+γn2≤z<γn+γn+12.
We now state our first result on H(Γ,v).Theorem 2.
Suppose that the sequenceΓ satisfies the sparseness condition (11) and g:(Γ,v)*→C+ is homomorphic. Then Vg:H(Γ,v)→H2(C+) is(i)
bounded if and only if(13)sup⁡n≥1∫Ωn‍vng′z2IzdA(z)z-γn2<∞,(14)sup⁡n≥1∑l=1nvl∑m=n+1∞∫Ωm‍g′z2IzdA(z)z2lllllllll+∑l=n+1∞‍vlγl2∑m=1n∫Ωm‍g′z2IzdAz∑l=1nvl∑m=n+1∞∫Ωm‍g′z2IzdA(z)z2<∞;(ii)
compact if and only if the “little oh” counterparts of (13) and (14) hold;(iii)
in the Hilbert–Schmidt classS2(H(Γ,v),H2(C+)) if and only if
(15)∑n=1∞‍∫Ωnvng′z2Izz-γn2dAz<∞,∑k=1∞vk∑k=n+1∞∫Ωkg′z2IzdA(z)z2+∑k=n+1∞vnγn2∑k=1n∫Ωng′z2IzdA(z)<∞.Condition (13) of the theorem is a condition about the local behavior of the symbol g, while condition (14) deals with its global behavior. Combining the two conditions, we see that (13) may be replaced by the stronger global condition
(16)sup⁡n≥1∫C+vng′z2IzdA(z)z-γn2<∞.
In some special cases, this condition or the original one (13) automatically gives condition (14), for example, if the weight sequence vn grows at least exponentially and the numbers vn/|γn|2 decay at least exponentially with n. The same conclusion follows if the weight sequence vn is also summable.


## 2. The Composition Operator onH(Γ,v)

Letψ be a homomorphic function in a given domain. We denote by Cψ the composition operator f↦f∘ψ. As a consequence of Littlewood’s Subordination Theorem [15], it has long been known that all composition operators are bounded on all the Hardy spaces Hp of the unit disc, where 0<p≤∞. Although corresponding Hardy spaces of the disc and half-plane are isomorphic, composition operators act differently in the two domains. Unlike the case of Hp(D), not all composition operators are bounded on Hp(C+).The study of compact composition operators onH2(D) first appeared in Schwartz [16] thesis in the late sixties though a complete function theoretic characterization, in terms of the inducing map’s Nevanlinna counting function, was obtained by Shapiro later in [17]. Unlike Hp(D) again, it was proved in [18] that there exists no compact composition operator on the Hardy space Hp(C+). The work in [17] was continued by Lyubarskii and Malinnikova [19] for Cψ:KI2(D)→H2(D) and a complete extension was made when the generating inner function I is one-component. More general trace ideal criteria for Cψ were obtained later in [20]. The problem remains open when I has infinitely many components.In this section, we study the bounded, compact, and Hilbert–Schmidt properties of composition operators on model spaces when the spaces admit normalized reproducing kernel Riesz bases associated with sparse sequences of points inC+. We note that such spaces represent some classes of model spaces generated by infinitely many component inner functions.Theorem 3.
Suppose that the sequenceΓ satisfies the sparseness condition (11) and ψ is a nonconstant analytic function on C+, and m refers to the usual Lebesgue measure on the real line. Then Cψ:H(Γ,v)→H2(C+) is(i)
bounded if and only if(17)sup⁡n≥1sup⁡y≥0∫ψΩnvndxψx+iy-γn2<∞,sup⁡n≥1⁡∑l=1nvl∑k=n+1∞sup⁡y≥0∫ψ(Ωk)‍dxψx+iy2llllllll+∑l=n+1∞vlγl2sup⁡y≥0∑k=1n‍mψΩk<∞,whereψ(Ωk)={ψ(x+iy):x+iy∈Ωk};(ii)
compact if and only if the “little oh” counterparts of conditions (17) hold;(iii)
in the Hilbert–Schmidt classS2(H(Γ,v),H2(C+)) if and only if
(18)∑k=1∞‍sup⁡y>0∫ψΩk‍vkdxψx+iy-γk2<∞,∑k=1∞vk∑j=k+1∞sup⁡y>0∫ψ(Ωj)‍dxψx+iy2+∑k=1∞‍vkγk2sup⁡y>0∑j=1k-1‍mψΩj<∞.


## 3. Open Problems

### 3.1. Open Problem 1

As will be seen in the next section, the proof of our boundedness and compactness results for bothCψ and Vg relies mainly on previously obtained Carleson measure results on H(Γ,v). When the sequence of points γn=n and the weight sequences vn≃1, the space H(Γ,v) becomes the classical Paley–Wiener space for which its Carleson measures have long been identified. When the sequence γn grows at least exponentially, the measures are completely described in terms of A2 type condition in [11]. The case when γn grows between these two extreme cases remains an interesting open problem. A solution of this will settle the long-standing open problem of identifying the Carleson measures for model spaces, of course modulo the existence of reproducing kernel Riesz basis in model spaces. The novelty and the core of the approach in [11] has been in turning a number of problems including the Carleson measure problem into questions about different mapping properties of Hilbert transforms in weighted spaces of functions and sequences. To extend the approach, one may need to look into all possible interplays between the smoothness (regularity) of the weight sequence (vn) and the growth of the sequence Γ=(γn).


### 3.2. Open Problem 2

Both of our results in parts (iii) of Theorems2 and 3 deal with the Hilbert–Schmidt properties of the operators Vg and Cψ on H(Γ,v). A more natural and interesting question is to ask the Schatten Sp class membership of these maps from H(Γ,v) to H2(C+) for all p in the range 0<p<∞. Dealing with these problems would have been easier had we known a complete characterization of the Sp class membership of the imbedding maps induced by Carleson measures for the spaces, which itself is still another open problem. A good starting point for all of these problems could be to consider first the case when sequence γn grows at least exponentially. A very special case of the Sp membership problem pertaining to the embedding maps may be seen in [21].A word on notation: the notationU(z)≲V(z) (or equivalently V(z)≳U(z)) means that there is a constant C such that U(z)≤CV(z) holds for all z’s in the set of a question. We write U(z)≃V(z) if both U(z)≲V(z) and V(z)≲U(z).


## 3.1. Open Problem 1

As will be seen in the next section, the proof of our boundedness and compactness results for bothCψ and Vg relies mainly on previously obtained Carleson measure results on H(Γ,v). When the sequence of points γn=n and the weight sequences vn≃1, the space H(Γ,v) becomes the classical Paley–Wiener space for which its Carleson measures have long been identified. When the sequence γn grows at least exponentially, the measures are completely described in terms of A2 type condition in [11]. The case when γn grows between these two extreme cases remains an interesting open problem. A solution of this will settle the long-standing open problem of identifying the Carleson measures for model spaces, of course modulo the existence of reproducing kernel Riesz basis in model spaces. The novelty and the core of the approach in [11] has been in turning a number of problems including the Carleson measure problem into questions about different mapping properties of Hilbert transforms in weighted spaces of functions and sequences. To extend the approach, one may need to look into all possible interplays between the smoothness (regularity) of the weight sequence (vn) and the growth of the sequence Γ=(γn).


## 3.2. Open Problem 2

Both of our results in parts (iii) of Theorems2 and 3 deal with the Hilbert–Schmidt properties of the operators Vg and Cψ on H(Γ,v). A more natural and interesting question is to ask the Schatten Sp class membership of these maps from H(Γ,v) to H2(C+) for all p in the range 0<p<∞. Dealing with these problems would have been easier had we known a complete characterization of the Sp class membership of the imbedding maps induced by Carleson measures for the spaces, which itself is still another open problem. A good starting point for all of these problems could be to consider first the case when sequence γn grows at least exponentially. A very special case of the Sp membership problem pertaining to the embedding maps may be seen in [21].A word on notation: the notationU(z)≲V(z) (or equivalently V(z)≳U(z)) means that there is a constant C such that U(z)≤CV(z) holds for all z’s in the set of a question. We write U(z)≃V(z) if both U(z)≲V(z) and V(z)≲U(z).


## 4. Proofs of the Results

Proof of Theorem1.
The key tool in the proofs of the first two results is the Littlewood–Paley description of Hardy spaces in terms of derivative(19)f22=2∫C+f′z2IzdA(z)
for each f in H2(C+). The formula helps to convert a boundary integral to an area integral in terms of derivatives. Since Vgf′(z)=fzg′(z), an application of the above identity ensures that for each f in KI2 we have
(20)Vgf22=2∫C+fz2g′z2IzdA(z).
It follows from this that the boundedness of the operator Vg:KI2→H2(C+) is equivalent to saying that the measure μg where
(21)dμg(z)=g′z2IzdA(z)
is a Carleson measure for KI2. Assuming that I is a one-component inner function, a result of Cohn [8] ensures that the measure property holds if and only if the normalized reproducing kernels of KI2 are uniformly embedded into L2(C+,dμg) from which the boundedness part of the result follows.
To prove (ii), first from a standard normal family argument, we have thatVg is compact if and only if Vgfn→0 in H2(C+) for each sequence of functions fn in KI2 such that the sequence is uniformly bounded in norm and converges uniformly to zero on compact subsets of C+. It means that the compactness of Vg is equivalent to saying that the embedding map from KI2 to L2(μg) is compact. Since I is one-component, then the desired conclusion follows from a result of Cohn [8] again.
To prove part (iii), recall that the operatorVg belongs to the Hilbert–Schmidt class if and only if, for any orthonormal basis (en) of KI2, the sequence (Vgen22) is summable. It follows by (20) that
(22)∑n=1∞Vgen22≃∑n=1∞∫C+enz2g′z2IzdA(z)=∫C+‍∑n=1∞enz2g′z2IzdA(z).
On the other hand, because of the reproducing property of the kernel and Parseval’s identity, we have
(23)Kwz=∑n=1∞Kw,enenz=∑n=1∞enzenw¯,Kw22=∑n=1∞enw2.
The desired conclusion easily follows since
(24)∑n=1∞enz2=Kz22≃1-Iz2Iz.Proof of Theorem2.
Applying the Littlewood–Paley identity again, the estimate(25)Vgf22=2∫C+fz2g′z2IzdA(z)≲fH(Γ,v)2
holds if and only if the embedding map from H(Γ,v) to L2(C,μg) is bounded, where μg is the nonnegative measure defined by (21). Then, the boundedness part of the result immediately follows from an application of Theorem 1.1 in [11].
By a result of Nordgren’s [22], a sequence of functions fn in H(Γ,v) converges weekly to zero if and only if it converges pointwise to zero and sup⁡nfnH(Γ,v)<∞. It follows that Vψ is compact if and only if the measure μg is a vanishing Carleson measure for H(Γ,v). Then an application of Theorem 1.2 in [14] finishes the proof of the compactness part.
It remains to prove the statement about Hilbert–Schmidt membership. First, we may observe that the reproducing kernel ofH(Γ,v) at a point λ in (Γ,v)* is explicitly given by
(26)kλ(z)=∑n=1∞‍vnλ¯-γn¯z-γn;
this is a direct consequence of the definition of the space H(Γ,v). Furthermore, the sequence en(z)=vn/(z-γn) constitutes an orthonormal basis to H(Γ,v). Then
(27)VgS22=∑n=1∞Vgen22≃∑n=1∞∫C+enz2g′z2IzdA(z)=∫C+KzH(Γ,v)2g′z2IzdA(z)=∫C+‍∑n=1∞vnz-γn2g′z2IzdA(z).
Using the growth condition (11), we observe that if z∈Ωm, then
(28)∑n=1∞vnz-γn2≃∑n=1m-1vnz2+vmz-γm2+∑n=m+1∞vnγn2.
Plugging this estimate in (27), we find that VgS2 is finite if and only if both equations of  (15) hold.Proof of Theorem3.
Suppose now thatψ is analytic on C+. Then, for each fixed y>0, we define ψy(x)=ψ(x+iy) from R to C+. We denote by mψy-1 the pullback measure
(29)mψy-1(E)=mψy-1E
for each Borel subset E⊂C+, where m refers to the usual Lebesgue measure on the real line.
Applying change of variables, we observe(30)∫RCψfx+iy2dx=∫C+fz2dmψy-1z,
which holds because of a classical result in measure theory ([23], page 163). This implies that the estimate
(31)Cψf22=sup⁡y>0∫C+f∘ψx+iy2dx≲fH(Γ,v)2
holds if and only if (mψy-1)y>0 constitute a family of Carleson measures for H(Γ,v) with a uniform bound on y in the sense that
(32)sup⁡y>0∫C+‍fz2dmψy-1(z)fH(Γ,v)2<∞.
Then an application of Theorem 1.1 of [11] gives the first part of the result in the theorem.The compactness part of our result follows easily by similar arguments used to prove the compactness part in Theorem2 along with Theorem 1.2 of [14]. Thus, we remain to prove part (iii) of the result. Proceeding as in the proof of part (iii) of Theorem 2 and applying (31) we have
(33)CψS22=∑n=1∞Cψen22=∑n=1∞sup⁡y>0∫C+en∘ψx+iy2dx=∑n=1∞sup⁡y>0∫C+enz2dmψy-1(z)=∑n=1∞sup⁡y>0∫C+vnz-γn2dmψy-1(z)=∑n=1∞‍sup⁡y>0∑j=1∞‍∫Ωj‍vnz-γn2dmψy-1z.
Applying the growth condition in (11), we observe that if z∈Ωn, then
(34)∑n=1∞vnsup⁡y>0∑j=1∞∫Ωjdmψy-1zz-γn2≃∑n=1∞vnsup⁡y>0∑j=1n-1‍∫Ωjdmψy-1zz2(35)+∑n=1∞‍sup⁡y>0∫Ωnvndmψy-1zz-γn2+∑j=n+1∞vn∫Ωjdmψy-1zγn2.
The sum on the right-hand side of (34) can be further rewritten as
(36)∑n=1∞vnsup⁡y>0∑j=1n-1∫Ωjdmψy-1zz2=∑n=1∞vnsup⁡y>0∑j=1n-1∫ψΩjdxψx+iy2.
On the other hand, the sum on the right-hand side of (35) becomes
(37)∑n=1∞vnsup⁡y>0∫Ωndmψy-1zz-γn2+∑j=n+1∞‍∫Ωjvnγn2dmψy-1z≃∑n=1∞vnsup⁡y>0∫ψ(Ωn)‍dxψx+iy-γn2+∑n=1∞vnγn2sup⁡y>0∑j=n+1∞mψΩj,
where m(ψ(Ωj))=m({ψ(x+iy):x+iy∈Ωj}). From the above series of estimates, we see that CψS2 is finite if and only if the sums in (36) and (37) are finite and from which our assertion follows.



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