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

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.

1. Introduction

Let be a bounded linear operator on a Hilbert space . A closed subspace of is said to be invariant for if . In the present paper, we denote by the lattice of invariant subspaces of on . A basic problem in functional analysis is to describe the invariant subspaces of . We say that is analytic (pure) on if . If is an isometric and pure operator on , the Wold–Kolmogorov decomposition theorem implies thatwhere is the orthogonal complement of in (it is also called the wandering subspace of on ), andis the smallest invariant subspace of containing . The most famous example for the isometric and pure operator on a Hilbert space is the following:

Let be the space which consists of the collection of square-summable sequences of complex numbers. That is,

The norm of the vector is

Let be the open unit disk in the complex plane . The Hardy space consists of all analytic functions on having power series representations with -complex coefficients sequence. That iswhere is the space of analytic functions in . The norm of the vector of is defined as the -norm of . The following mappingis clearly an isomorphism from onto , then is a Hilbert space. Let be the boundary of the unit disk . Let be the Hilbert space of square-integrable functions on 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 and the closed subspace of which consists of the -functions with negative Fourier coefficients vanishing.

It is clear that the unilateral shift on is an isometric and pure operator, then (1) holds for on . On the other hand, the famous Beurling theorem [1] gives a complete characterization of the invariant subspaces of on , that is, every invariant subspace of on other than has the form , where is an inner function. The Beurling theorem also implies that (1) holds for on , 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 on a Hilbert space , we say that the Beurling-type theorem for holds on 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 which is the Hilbert space consisting of the square-integrable analytic functions on 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 which is close to an isometry in some sense (in particular, we can assume is left invertible), and proved the following theorem.

Theorem 1 (Shimorin’s theorem). Let T be a linear operator on a Hilbert space with the properties.(i), and(ii), Then .

Definition 1. We say that Shimorin’s condition for holds on if 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 operator on a Hilbert space is to verify whether Shimorin’s condition for 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 which is the Hilbert space over the bidisk generated by a positive sequence . In this paper, we mainly study the Beurling-type theorem for reproducing kernel Hilbert spaces. Let be the space generated by a positive sequence ; that is, the space consists of all formal power series satisfyingWe define the map on aswhere andNote that in if and only for any , then it is easy to check that is an inner product on , so is a Hilbert space. Letand it is easy to see that for any . Hence, we call the reproducing kernel Hilbert spaces generated by the positive sequence .
A power series in will satisfyHence, for sufficiently large, we have that . Thus, the radius of convergence of satisfyingTherefore, will have a radius of convergence greater thanProvided , then every function in defines an analytic function on the disk of radius and thus can be viewed as a space of analytic functions on this disk.
The classical examples of is as follows.

Example 1. If(1)If , then , . In this case, , the Hardy space.(2)If , then , . In this case, , the Dirichlet space.(3)If , where , then , . In this case, , the weighted Bergman space. In particular, when , is the classical Bergman space.(4)If , then , . In this case, , the Fock space.Let be the shift operator on defined by . Note that is a compression operator on whenhence, in the present paper, we assume that is bounded on . We give a criterion for judging the Beurling-type theorem holds for on (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 subspace of is said to be invariant if , that is, is invariant for the shift operator . The following is our main result of the paper.

Theorem 2. Let be the reproducing kernel Hilbert spaces generated by the sequence with which satisfiesthen for every nontrivial invariant subspace of , .

We note that the above condition is equivalent to the following condition:

Namely, the sequence satisfies some property of “convex up,” and it is much convenient to verify the Beurling type theorem for the reproducing kernel Hilbert spaces 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 be a nontrivial invariant subspace of the reproducing kernel Hilbert space . Since it is closed, itself is a Hilbert space under the inner product of . It is easy to see that , so the condition (ii) of Theorem 1 is satisfied. Given and in . Sincewe obtain

So, we have

By the assumption of the theorem, we havefor every , which yields

It follows from the above that (19) becomes

Hence, for any , we have

On the other hand, we have

Combining the above with (23), and the assumption we getwhich 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 Theorem 2 is only sufficient for the Beurling-type theorem to be true. For example, the classical Dirichlet space is a reproducing kernel Hilbert space generated by the sequence . It is well known that if is a nontrivial invariant subspace of , then and (see [13]). However, we easily see that, for ,

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 subspace of the Hardy space has the property .

Proof. Since the classical Hardy space is the space generated by sequence for , we haveand . Thus, the corollary follows from Theorem 2.

Corollary 2. Any nontrivial invariant subspace of the weighted Bergman space has the property .

Proof. When , the weighted Bergman space generated by the sequence , where . Thus, we have that for ,

Since , then , it is equivalent toand hence we havewhich combining with (28) and (29) we obtain

In addition, it is easy to see that and , thus we have since . 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 for some .

Let , where for all integer . Then is a kernel function. Accordingly, we denote the corresponding reproducing kernel Hilbert space generated by as . Without loss of the generality, we may assume the generating function has convergence radius 1. Thus, is a functional Hilbert space consisting of analytic functions on the unit disk . It is clear that is an orthonormal basis and . If and , then its norm is defined by .

In the framework of reproducing kernel Hilbert spaces, if we put , then where . Then Theorem 2 can be restated as follows.

Theorem 3. Let be the reproducing kernel Hilbert spaces generated by . If satisfiesthen for each nontrivial invariant subspace of , .

If we restrict ourselves to the generated by where for some real number . These spaces are also known in the literature as weighted Dirichlet spaces or spaces. For , if , then the norm is simply written as

Let us consider , where is a real number and . It is easy to see that is a convex-up function on when . Setting , we havefor , and since , we havewhen . From the discussion above, we obtain the following result.

Corollary 3. If is a nontrivial invariant subspace of weighted Dirichlet space , then .

Using Theorem 3, 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 functionfor , where is a constant. Note thatsince , then is a convex-up function on . If we writethen for , we haveand . The application of our main result to this case, we can get the following result.

Corollary 4. Let be reproducing kernel Hilbert space generated by the sequence whereand a nontrivial invariant subspace of . If , then we have .

Data Availability

Some or all data, models, or code generated or used during the study are available from the corresponding author by request.

Conflicts of Interest

The authors declare that they have no conflicts of interest.