# Beurling-Type Theorem for a Class of Reproducing Kernel Hilbert Spaces

Authors: Zhi-jie Wang; Jie-sheng Xiao; Xian-min Xu
Journal: Journal of Mathematics
 (2022)

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## Abstract

In this paper, it is proved that the Beurling-type theorem holds for the shift operator on a class of reproducing analytic Hilbert spaces.


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## Body

## 1. Introduction

LetT be a bounded linear operator on a Hilbert space H. A closed subspace M of H is said to be invariant for T if TM⊂M. In the present paper, we denote by LatT the lattice of invariant subspaces of T on H. A basic problem in functional analysis is to describe the invariant subspaces of T. We say that T is analytic (pure) on H if H∞=∩n≥1TnH=0. If T is an isometric Tx=x,x∈H and pure operator on H, the Wold–Kolmogorov decomposition theorem implies that(1)M=M⊖TM,∀M∈LatT,where M⊖TM is the orthogonal complement of TM in M (it is also called the wandering subspace of T on M), and(2)M⊖TM=M⊖TM⊕TM⊖T2M⊕⋅⋅⋅,is the smallest invariant subspace of T containing M⊖TM. The most famous example for the isometric and pure operator on a Hilbert space is the following:Letℓ2 be the space which consists of the collection of square-summable sequences of complex numbers. That is,(3)ℓ2=ann=0∞:∑n=0∞an2<∞.The norm of the vectorann=0∞ is(4)ann=0∞ℓ2=∑n=0∞an21/2.LetD be the open unit disk in the complex plane ℂ. The Hardy space H2 consists of all analytic functions on D having power series representations with ℓ2-complex coefficients sequence. That is(5)H2=f∈HD:fz=∑n=0∞anzn,ann=0∞∈ℓ2,where HD is the space of analytic functions in D. The norm of the vector fz=∑n=0∞anzn of H2 is defined as the ℓ2-norm of ann=0∞. The following mapping(6)ann=0∞↦∑n=0∞anzn,is clearly an isomorphism from ℓ2 onto H2, then H2 is a Hilbert space. Let T be the boundary of the unit disk D. Let L2=L2T be the Hilbert space of square-integrable functions on T with respect to Lebesgue measure, normalized so that the measure of the entire circle is 1. It is well known that there is an isomorphism between H2 and the closed subspace H2˜ of L2 which consists of the L2-functions with negative Fourier coefficients vanishing.It is clear that the unilateral shiftTz on H2 is an isometric and pure operator, then (1) holds for Tz on H2. On the other hand, the famous Beurling theorem [1] gives a complete characterization of the invariant subspaces of Tz on H2, that is, every invariant subspace of Tz on H2 other than 0 has the form ϕH2, where ϕz is an inner function. The Beurling theorem also implies that (1) holds for Tz on H2, so in this case, the Beurling theorem is a special case of the Wold-Kolmogorov decomposition theorem. To generalize the Beurling theorem, for a bounded linear operator T on a Hilbert space H, we say that the Beurling-type theorem for T holds on H if (1) holds. Hence, the Beurling-type theorem holds for all isometric and pure operators. Another basic problem in functional analysis is to find whether the Beurling-type theorem holds for a nonisometric operator on a Hilbert space. The most famous example for the nonisometric operator is the shift operator (it is also called the Bergman shift) on the Bergman space A2D which is the Hilbert space consisting of the square-integrable analytic functions on D with Lebesgue area measure.The study of the wandering subspaces of invariant subspaces for the Bergman shift tells that the dimension of wandering subspace ranges from 1 to∞ (see [2]). Nevertheless, Aleman, Richter, and Sundberg (see [3]) discovered that all invariant subspaces of the Bergman shift are also generated by their wandering subspaces. This reveals the internal structure of invariant subspaces of the Bergman space and becomes a fundamental theorem on the Bergman space (see [4]). Later, the Beurling-type theorem was studied by many mathematicians (see [5–9]). In [8], Shimorin studied the Beurling-type theorem for a nonisometric operator T which is close to an isometry in some sense (in particular, we can assume T is left invertible), and proved the following theorem.Theorem 1 (Shimorin’s theorem).
LetT be a linear operator on a Hilbert space H with the properties.(i)
Tx+y2≤2x2+Ty2for allx,y∈H, and(ii)
∩n=1∞TnH=0,ThenH=H⊖TH.Definition 1.
We say that Shimorin’s condition forT holds on H if T satisfies the above conditions (i) and (ii) (see [10]).
Shimorin’s theorem gives a simpler proof for the Beurling type theorem of Bergman shift obtained by Aleman, Richter, and Sundberg. The first step to finding whether the Beurling-type theorem holds for a nonisometric operatorT on a Hilbert space is to verify whether Shimorin’s condition for T holds on the invariant subspaces. It is always difficult to verify directly according to Definition 1, for example, the reproducing kernel Hilbert spaces with the complicated kernel functions (see [11]). Therefore, we hope to give a convenient and feasible criterion for judging whether the Beurling-type theorem holds for the shift operator on a class of reproducing kernel Hilbert spaces.
In [12], the authors proved the Beurling-type theorem for the shift operator on H2γ which is the Hilbert space over the bidisk D2 generated by a positive sequence γnmn,m≥0. In this paper, we mainly study the Beurling-type theorem for reproducing kernel Hilbert spaces. Let H2β be the space generated by a positive sequence β=βnn≥0; that is, the space consists of all formal power series fz=∑n=0∞anzn satisfying(7)fβ2=∑n=0∞βn2an2<∞.
We define the map on H2β⊗H2β as(8)f,g=∑n=0∞βn2anbn¯,where f,g∈H2β, and(9)fz=∑n=0∞anzn,gz=∑n=0∞bnzn.
Note thatf=g in H2β if and only an=bn for any n≥0, then it is easy to check that  is an inner product on H2β, so H2β is a Hilbert space. Let(10)Kwz=Kβz,w=∑n=0∞w¯nznβn2∈H2β,and it is easy to see that fw=f,Kw for any f∈H2β. Hence, we call H2β the reproducing kernel Hilbert spaces generated by the positive sequence β=βnn≥0.
A power seriesfz=∑n=0∞anzn in H2β will satisfy(11)limn⟶∞βnan=0.
Hence, forn sufficiently large, we have that an≤βn−1. Thus, the radius of convergence Rf of f satisfying(12)Rf−1=limsupn⟶∞an1/n≤limsupn⟶∞βn−1/n=liminfn⟶∞βn1/n−1.
Therefore,f∈H2β will have a radius of convergence greater than(13)R≔liminfn⟶∞βn1/n.
ProvidedR>0, then every function in H2β defines an analytic function on the disk of radius R and thus H2β can be viewed as a space of analytic functions on this disk.
The classical examples ofH2β is as follows.Example 1.
If(1)
Ifβn≡1,n=0,1,…, then R=1, Kwz=1/1−w¯z. In this case, H2β=H2D, the Hardy space.(2)
Ifβn=n+1,n=0,1,…, then R=1, Kwz=1/w¯zlog1/1−w¯z. In this case, H2β=D, the Dirichlet space.(3)
Ifβn=n!Γγ/Γn+γ,n=0,1,⋯, where γ>1, then R=1, Kwz=1/1−w¯zγ. In this case, H2β=Aγ−22D, the weighted Bergman space. In particular, when γ=2, A02D=A2D is the classical Bergman space.(4)
Ifβn=n!,n=0,1,…, then R=+∞, Kwz=ew¯z. In this case, H2β=F2ℂ, the Fock space.
LetTz be the shift operator on H2β defined by Tzfz=zfz. Note that Tz is a compression operator on H2β when(14)βn≤βn−1,n≥1,hence, in the present paper, we assume that Tz is bounded on H2β. We give a criterion for judging the Beurling-type theorem holds for Tz on H2β (Theorem 2), and as an application, we find that the Beurling-type theorem holds for the shift operator on a family of classical reproducing kernel Hilbert spaces.
In what follows, a nontrivial subspaceM of H2β is said to be invariant if zM⊂M, that is, M is invariant for the shift operator Tz. The following is our main result of the paper.Theorem 2.
LetH2β be the reproducing kernel Hilbert spaces generated by the sequence β=βnn≥0 with βn>0 which satisfies(15)β02≤2β12,βn+12+βn−122≤βn+12βn−12βn2,n=1,2,…,then for every nontrivial invariant subspace M of H2β, M=M⊖zM.We note that the above condition is equivalent to the following condition:(16)1βn−12+1βn+12≤21βn2.Namely, the sequence1/βn2n≥0 satisfies some property of “convex up,” and it is much convenient to verify the Beurling type theorem for the reproducing kernel Hilbert spaces H2β satisfying the above condition.


## 2. Proof of Main Result and Its Applications

In this section, we first prove our main result and then give several useful applications to a class of classical reproducing kernel Hilbert spaces.Proof. (Proof of Theorem 2) Let M be a nontrivial invariant subspace of the reproducing kernel Hilbert space H2β. Since it is closed, M itself is a Hilbert space under the inner product of H2β. It is easy to see that ∩n=1∞TznM=0, so the condition (ii) of Theorem 1 is satisfied. Given fz=∑n=0∞anzn and gz=∑n=0∞bnzn in H2β. Since(17)0≤an−1βn+12−bnβn−122=|an−1|2βn+14+|bn|2βn−14−2βn+12βn−12Rebnan−1¯,we obtain(18)2Rebnan−1¯≤an−12βn−12βn+12+bn2βn+12βn−12.So, we have(19)an−1+bn2βn2=an−12+bn2+2Rebnan−1¯βn2≤βn21+βn−12βn+12an−12+1+βn+12βn−12bn2=βn2βn−12+βn2βn+12an−12βn−12+βn2βn+12+βn2βn−12bn2βn+12.By the assumption of the theorem, we have(20)βn+12+βn−122≤βn+12βn−12βn2,for every n=1,2,3,⋯, which yields(21)βn2βn−12+βn2βn+12≤2.It follows from the above that (19) becomes(22)an−1+bn2βn2≤2an−12βn−12+2bn2βn+12.Hence, for anyf,g∈H2β, we have(23)zf+gH2β2=b0+∑n=1∞an−1+bnzn2=b02β02+∑n=1∞an−1+bn2βn2≤b02β02+2∑n=1∞an−12βn−12+bn2βn+12.On the other hand, we have(24)2fH2β2+zgH2β2=2∑n=0∞an2βn2+∑n=1∞bn−12βn2=2∑n=1∞an−12βn−12+∑n=0∞bn2βn+12=2b02β12+2∑n=1∞an−12βn−12+bn2βn+12.Combining the above with (23), and the assumption β02≤2β12 we get(25)zf+gH2β2≤fH2β2+zgH2β2,which implies that condition (i) of Theorem 1 is also satisfied. Hence, applying Shimorin’s theorem we finish the proof of the main result.We point out that the condition of Theorem2 is only sufficient for the Beurling-type theorem to be true. For example, the classical Dirichlet space D is a reproducing kernel Hilbert space H2β generated by the sequence β=n+1n≥0. It is well known that if M is a nontrivial invariant subspace of D, then M=M⊖zM and dimM⊖zM=1 (see [13]). However, we easily see that, for n=1,2,⋯,(26)βn+12+βn−122=n+1>n+1−1n+1=nn+2n+1=βn+12βn−12βn2.However, for the Hardy space or some weighted Bergman space case, we have the following two corollaries which are direct applications of the main result.Corollary 1.
Any nontrivial invariant subspaceM of the Hardy space H2D has the property M=M⊖zM.Proof. Since the classical Hardy space H2D is the space H2β generated by sequence βn2≡1 for n=0,1,⋯, we have(27)βn+12+βn−122=βn+12βn−12βn2,and β02=1<2=2β12. Thus, the corollary follows from Theorem 2.Corollary 2.
Any nontrivial invariant subspaceM of the weighted Bergman space Aγ−22D1<γ≤2 has the property M=M⊖zM.Proof. When γ>1, the weighted Bergman space Aγ−22D=H2β generated by the sequence β=βn:n=0,1,2,⋯, where βn=n!Γγ/Γn+γ. Thus, we have that for n=1,2,⋯,(28)βn+12+βn−122=12n+1!ΓγΓn+1+γ+n−1!ΓγΓn−1+γ=12nn+1n+γn−1+γ+1n−1!ΓγΓn−1+γ,(29)βn+12βn−12βn2=n+1!ΓγΓn+1+γ⋅n−1!ΓγΓn−1+γ⋅Γn+γn!Γγ=n+1n+γ⋅n−1!ΓγΓn−1+γ.Since1<γ≤2, then γ2−3γ+2≤0, it is equivalent to(30)n+γn−1+γ+nn+1≤2n+1n−1+γ,(31)ornn+1n+γn−1+γ+1≤2n+1n+γ,and hence we have(32)12nn+1n+γn−1+γ+1n−1!ΓγΓn−1+γ≤n+1n+γ⋅n−1!ΓγΓn−1+γ,which combining with (28) and (29) we obtain(33)βn+12+βn−122≤βn+12βn−12βn2,n=1,2,⋯.In addition, it is easy to see thatβ02=1 and β12=1/γ, thus we have β02≤2β12 since 1<γ≤2. Applying Theorem 2 we then get the desired conclusion. The proof is complete.We remark here that Hedenmalm and Zhu (see [14]) showed that the Beurling-type theorem can fail in certain weighted Bergman spaces Aγ−22D for some γ>2.Letgz=∑n=0∞cnzn, where cn>0 for all integer n≥0. Then Kwz=Kz,w=gw¯z=∑n=0∞cnw¯nzn is a kernel function. Accordingly, we denote the corresponding reproducing kernel Hilbert space generated by g as Hg2. Without loss of the generality, we may assume the generating function g has convergence radius 1. Thus, Hg2 is a functional Hilbert space consisting of analytic functions on the unit disk D. It is clear that cnznn≥0 is an orthonormal basis and zn2=1/cn. If f∈Hg2 and fz=∑n=0∞anzn, then its norm is defined by f2=∑n=0∞an2/cn.In the frameworkHg2 of reproducing kernel Hilbert spaces, if we put cn=1/βn2, then Hg2=H2β where β=βn:n=0,1,2,⋯. Then Theorem 2 can be restated as follows.Theorem 3.
LetHg2 be the reproducing kernel Hilbert spaces generated by gz=∑n=0∞cnzn. If cn satisfies(34)c1≤2c0,cn+1+cn−1≤2cn,n=1,2,⋯,then for each nontrivial invariant subspace M of Hg2, M=M⊖zM.If we restrict ourselves to theH2β generated by β=βn where βn=n+1ν for some real number ν. These spaces are also known in the literature as weighted Dirichlet spaces or Dν spaces. For f∈Dν, if fz=∑n=0∞anzn, then the norm is simply written as(35)f2=∑n=0∞n+12νan2.Let us considerφx=1+xα, where α is a real number and x>0. It is easy to see that φ is a convex-up function on −1,+∞ when 0≤α≤1. Setting cn=n+1α=1/βn2, we have(36)1βn+12+1βn−12=cn+1+cn−1≤2cn=21βn2,for n=1,2,⋯, and since β02=1/c0=1,β12=1/c1=1/2α, we have(37)β02=1≤21−α=2β12,when 0≤α≤1. From the discussion above, we obtain the following result.Corollary 3.
IfM is a nontrivial invariant subspace of weighted Dirichlet space Dν−1/2≤ν≤0, then M=M⊖zM.Using Theorem3, it is convenient to construct a more reproducing kernel Hilbert space for which the nontrivial invariant subspace has a Beurling-type theorem. As an example, we consider a positive function(38)φx=x+ae2lnx+ae2,for x≥0, where a>1 is a constant. Note that(39)φ″x=2−lnx+ae2x+ae2ln3x+ae2<0,since a>1, then φx is a convex-up function on 0,+∞. If we write(40)cn=n+ae2lnn+ae2,βn2=1cn,then for n=1,2,⋯, we have(41)1βn+12+1βn−12=cn+1+cn−1≤2cn=21βn2,and β02≤2β12. The application of our main result to this case, we can get the following result.Corollary 4.
LetH2β be reproducing kernel Hilbert space generated by the sequence β=βn where(42)βn=lnn+ae2n+ae2,n=0,1,2,⋯,and M a nontrivial invariant subspace of H2β. If a>1, then we have M=M⊖zM.



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