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Journal of Function Spaces
Volume 2018, Article ID 1571365, 7 pages
https://doi.org/10.1155/2018/1571365
Research Article

Sarason’s Conjecture of Toeplitz Operators on Fock-Sobolev Type Spaces

1School of Mathematics and Information Science and Key Laboratory of Mathematics and Interdisciplinary Sciences of the Guangdong Higher Education Institute, Guangzhou University, Guangzhou 510006, China
2School of Mathematics and Statistics, Zhaoqing University, Zhaoqing 526061, China

Correspondence should be addressed to Jianjun Chen; moc.liamxof@ymranehc

Received 1 October 2017; Accepted 23 November 2017; Published 2 May 2018

Academic Editor: Aurelian Gheondea

Copyright © 2018 Xiaofeng Wang et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

In this note, we will solve Sarason’s conjecture on the Fock-Sobolev type spaces and give a well solution that if Toeplitz product , with entire symbols and , is bounded if and only if , , where is a linear complex polynomial and is a nonzero constant.

1. Introduction

Let denote the complex -space and be the ordinary volume measure on that is normalized so that . If given any two points and in , we denote For every , , we denote by the space of measurable functions such that Let be the set of entire functions on . Then for a given , the Fock-Sobolev type space with the norm is defined as

Obviously, the Fock-Sobolev type space equipped with the natural inner product defined by is a reproducing kernel Hilbert space for every real . As stated in [1], with respect to the above inner product, it is difficult to compute the reproducing kernel of explicitly. So we use the equivalent norm with respect to a new measure . In more detail, for , we will let and for we let where is the Taylor expansion of up to order and . Now we can bravely make sure that the inner product generates a new Hilbert space norm on that is equivalent to the norm . In particular, if we define the norm on by, when , and when , and then we have that both and are equivalent norms.

As is well known, is indeed a reproducing kernel Hilbert space (see Lemma 2.1 of [1] for more details). Therefore its reproducing kernel is where is any orthonormal basis for with respect to . Note that polynomials form a dense subset of (see Proposition   in [2]). Also note that monomials are mutually orthogonal, which means that is an orthonormal basis for . The arguments that are identical to the ones in the proof of Theorem   in [2] then give us that Here is the fractional integration operator defined as where each is a polynomial of degree . Moreover, for , is the tail part of the Taylor expansion of of degree higher than given by and we let (see [2] for more information on fractional differentiation and integration).

Now it is easy to see that if , is a closed subspace of with respect to . In this case, let denote the orthogonal projection, so that for any . Unfortunately, the inner product does not make sense on when . That means we can not define the Toeplitz operator on in the usual way in terms of this inner product. However according to the ideas of  [1], it makes sense to define the Toeplitz operator with the symbols in by the following formula: if and if , for any . In the sequel, we can reasonably define the Berezin transform of Toeplitz operator on by if , and if for any , where is the normalization of the kernel , that is, .

The original product problem, owed to Sarason firstly in [3], describes the pairs of outer functions and in the Hardy space such that the operator is bounded on the Hardy space. Sequentially, this problem was partially researched for the Hardy space in [4] and for the Bergman space in [58]. Unluckily it turns out that the Sarason’s conjecture is not true for both Hardy space and Bergman space of unit disk. See [9, 10] for counterexamples.

We will, in this note, give the equivalent conditions about the Sarason’s conjecture of Toeplitz product on Fock-Sobolev type spaces . Our main result will be the following.

Main Theorem. Suppose that and are two nonzero functions in Fock-Sobolev type spaces . Then the following conditions are equivalent:(1)The Toeplitz product is bounded on Fock-Sobolev type spaces . (2)There exists a complex linear polynomial on such that and , where is a nonzero complex constant. (3)The product is a bounded function on the complex space .

In 2014, Cho et al. studied the products of Toeplitz operators on the classical Fock space (see [11]). In the case of the Fock-Sobolev space, Chen et al. (see [12]) had already proven the same topics and obtained the similar results. What is more, they claimed that if and are two nonzero functions in the Fock(-Sobolev) space, then the Toeplitz product is bounded if and only if and , where is a nonzero constant and is a linear polynomial. More properties about Toeplitz operators on Fock-Sobolev spaces are referred to in [13]. Sequentially, Bommier-Hato et al. in [14] continued to research Cho’s results on the general Fock-type space with the weight functions . They took full advantage of the exact form of the reproducing kernel of the general Fock-type space and concluded that if and are two nonzero functions, then the Toeplitz product is bounded if and only if and , where is a nonzero constant and is a polynomial of degree at most . The similar techniques are founded in [15, 16]. However, the translations appearing to the classical Fock spaces are not suitable to the generalized Fock space. To tackle the main theorem, we have to use the main ideas of [14], that is, making good use of the explicit properties of the reproducing kernel in Fock-Sobolev type spaces instead of the Weyl operators defined by translations on the complex plane.

At last, it is remarked that, as stated in [1], the Fock-Sobolev type spaces are in fact very natural generalization of the Fock-Sobolev spaces and the Fock-Sobolev spaces of fractional order. For example, when , is the classical Fock space . Thus in this paper, we always omit discussing the case of and the similar result of this case is obtained in [11, 14].

Throughout this paper we write or for nonnegative quantities and whenever there is a constant independent of and such that . Similarly we write if and .

2. Proof of the Main Result

We begin with some properties of the Fock-Sobolev spaces . See [1] for more information.

Lemma 1. Suppose that belongs to the Fock-Sobolev type space for any real . Then for any , we have and when ,when ,More specifically, for any and there is a such that for any

A consequence of the first estimate in Lemma 1 is that, for any function , the Toeplitz operators and are both densely defined on .

Lemma 2. If the function belongs to the Fock-Sobolev type space , we then have .

Proof. In views of the Lemma  3.4 in [1], we can calculate that, for any polynomial and (see [1] for the definition of ), if , and if , Lastly the fact that the set of all holomorphic polynomials is dense in completes the proof.

Lemma 3. For given , if is bounded on , then for any .

Proof. When , in view of reproducing properties of , Lemma 3.4, the claim of Lemma   in [1], and Lemma 2, we see that On the other side, when , we have to use the Lemma 3.4, the claim of Lemma   in [1] to achieve that, if , is bounded, Together with in [1], Fubini’s theorem, and the reproducing property, we can see that Similarly, we can achieve that Therefore,

Theorem 4 ((1) ⇒ (2)). If we give that and are two nonzero functions in the Fock-Sobolev type space such that Toeplitz product is bounded on ; then there is a complex linear polynomial on such that and , where is a nonzero complex constant.

Proof. This proof is similar to Theorem   in [12] and here we only give its brief illustration.
If the condition holds that the Toeplitz product is bounded on , by Lemmas 2 and 3 and the Cauchy-Schwarz inequality, we can see that Sequentially, the local property of reproducing kernel shows us that the module of function is equivalent to when . It implies that is bounded in that situation.
On the other side, we give the representation of quadratic polynomial in the case of real inner product as follows: , where , is linear, is a homogeneous polynomial of degree 2, and is a complex matrix symmetric in the real sense. After we choose and , where is any real positive number, we achieve that is not bounded as . This contradiction finishes the proof.

Theorem 5 ((2) ⇒ (1)). If and where is a complex linear polynomial on , then is bounded on the Fock-Sobolev type space .

Proof. To prove the boundedness of , we will sufficiently obtain that is bounded by means of the idea of [1]. In fact we only discuss the case of because the other case is the same as the proof of Theorem   in [12]. Using the similar ways, our goal is to obtain that is bounded for any in view of the definition of the norm . To the end, we focus our attention on the integrands in it. By formulae    in [1] and the definition of Toeplitz operator, the integrands in the norm are By the reproducing property, the estimations of their module are, respectively, coming from and then, similarly, Therefore, using the Cauchy-Schwarz inequality, we have the estimation of the first term of the norm as follows: where −  . Similarly the estimation of the second norm has been achieved that where .
If we can affirm both we would finish the proof because To the finish, in terms of Lemma 1 and transformation, we can assert that This implies that is bounded and completes the proof.

Theorem 6 ((1) ⇒ (3)). If and are two functions in the Fock-Sobolev type space , not identically zero, such that the operator is bounded on , then is a bounded function on the complex space.

Proof. We omit the proof here for it is analogous to Theorem   in [12].

Theorem 7 ((3) ⇒ (2)). Suppose that and are two functions in the Fock-Sobolev type space , not identically zero, such that is bounded on . Then there is a complex linear polynomial on satisfying and , where is a nonzero complex constant.

Proof. Now we only consider the case of while the other case would be referred to in Theorem   in [12].
It is easy to see that, for any , When , we use the triangle inequality and Hölder’s inequality to calculate Because is a unit element, that is, we can see that From the above inequations, the estimate of turns into If is a bounded function on , and are both bounded on . By Liouville’s theorem, the boundedness of implies that there exists a constant such that . Since neither nor is identically zero, we have . That is, both and are nonvanishing. By Lemma 1, there exists a complex polynomial on with such that and .
On the other side, by the definition of Berezin transformation in this case, Now giving a sufficiently small , we further obtain that When choosing a constant satisfying Lemma 1, we can see that Using the similar method like the case , we can get the desired results and the proof is finished at this moment.

3. Conclusions

In this content, we deal with the Sarason’s problem on the Fock-Sobolev type spaces and have a complete solution that , where is a linear complex polynomial and is a nonzero constant. As stated in [1], we know that the Fock-Sobolev type space clearly does not fall under the class of weighted Fock spaces . Therefore the Sarason’s problem of weighted Fock spaces is still open. We will focus on this open problem in the future study.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Authors’ Contributions

All authors contributed equally. All authors read and approved the final manuscript.

Acknowledgments

This paper is supported by National Natural Science Foundation of China (Grants nos. 11471084 and 11301101), Young Innovative Talent Project of Department of Education of Guangdong Province (no. 2017KQNCX220), and the Natural Research Project of Zhaoqing University (nos. 201732 and 221622).

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