Fredholm and Schatten Class Operators on Bergman Spaces with Exponential Weights

Ma, Xuedi; Wang, Xiaofeng; Xia, Jin

doi:https://doi.org/10.1155/2023/9436300

Journal of Function Spaces

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Abstract Introduction Preliminaries Data Availability Conflicts of Interest Acknowledgments References Copyright Related Articles

Research Article | Open Access

Volume 2023 | Article ID 9436300 | https://doi.org/10.1155/2023/9436300

Fredholm and Schatten Class Operators on Bergman Spaces with Exponential Weights

Xuedi Ma,¹Xiaofeng Wang,¹and Jin Xia¹

Academic Editor: Calogero Vetro

Received22 May 2023

Revised16 Sept 2023

Accepted21 Sept 2023

Published11 Oct 2023

Abstract

In this paper, we give a characterization of Fredholmness of the Toeplitz operators on the Bergman spaces with exponential weights in when . Also, we obtain the sufficient and necessary conditions which the Toeplitz and Hankel operators on belong to -Schatten class, where is a continuous increasing convex function.

1. Introduction

Let denote the complex plane, be the real line, and be the space of analytic functions in the unit disc For and a subharmonic function on , the weighted Lebesgue space consists of measurable functions such that where denotes the normalized Lebesgue area measure on . Then, the weighted Bergman space is defined as

Let be the space of all continuous functions on satisfying . The class is set to be

is defined to be the family of those with a property that for each there exists a compact set such that where .

We say that belongs to the weight class if satisfies the following statement: (a), (b)there exists such that where denotes the Laplacian operator and notation indicates that there exists nonessential positive constant such that .

In what follows, we are focused on with , and the collection contains nonradial weights, two classes of weights related closely to . One was introduced by Oleinik and Pavlov [1], denoted by , and the other was introduced by Borichev et al. [2], denoted by . As stated by Hu et al. [3], the weight covers , but there is no inclusion relationship between and .

It is easy to know that is a Banach space with norm if , whereas is a Fréchet space with metric if . In particular, is a Hilbert space. Let be the Bergman kernel of and for short, and it is obvious that . For and , set to be the normalized Bergman kernel for , and for short. The Bergman projection can be represented as

Moreover, is bounded from and for any when . Let . For and we know that is dense in under the -norm . Then, for , we define densely the Toeplitz operator and Hankel operator, respectively, with symbol as where is the identity operator.

In this paper, we will study the Fredholm properties of the Toeplitz operators on the Bergman spaces with exponential weights. Berger and Coburn [4] were the first to study the Fredholm Toeplitz operators on the Fock spaces ; recently, the Fredholm theory was extended to the doubling Fock spaces in Hu and Virtanen’s study [5]. Hagger [6] studied the Fredholm properties of the Toeplitz operators acting on the weighted Bergman spaces Zorboska [7] determined the Fredholm Toeplitz operators with symbols when the Berezin transforms of the operators are bounded and of vanishing oscillation. Taskinen and Virtanen [8] combined some of the best known results on the compactness of the Toeplitz and Hankel operators in order to generalize the results on the Fredholm properties of the Toeplitz operators. Our goals of the present paper is to characterize the Fredholm properties of the Toeplitz operators with vanishing mean oscillation symbols on exponential Bergman spaces with . In addition, as we know, the Toeplitz operators belong to the Schatten -class when was first considered by Luecking [9], and later, Arroussi et al. [10] considered the same problem and described the membership in Schatten -class . Recently Zhang et al. [11] described Schatten -class Toeplitz operators on . Also, Luecking [12] characterized the Schatten -class of the Hankel operators on the Bergman spaces, and Zhang et al. [11] characterized the Schatten -class Hankel operators with general symbols on the Bergman spaces with exponential weights when . The definition of was first introduced by EI-Falla et al. [13], and Arroussi et al. [14] characterized the -Schatten class Toeplitz operators on large Fock spaces, where is a continuous increasing convex function. At present, we will characterize -Schatten class Toeplitz and Hankel operators on . It is worth mentioning that the result of -Schatten class Hankel operators has not been studied before. And the details of our characterizations are shown in section 5.

This paper is organized as follows. In section 2, we will give some useful results which contain mainly Bergman kernel estimates, etc. Section 3 provides the proofs of boundedness and compactness of the Topelitz and Hankel operators. In section 4, we characterize the Fredholm properties of the Topelitz operators on . Section 5 contains the characterization of -Schatten class Toeplitz and Hankel operators on

2. Preliminaries

Let , we define the distance as where the infimum is taken over all piecewise curves with and . In fact, is equivalent to the Bergman distance induced by the Bergman metric .

For and , define the disks , , and and for more information, refer to [3].

Lemma 1. Let be positive. Then, there exists with the following properties: (a)There exists constants and such thatfor and . (b)There exists a constant such thatfor and . (c)There exist positive constants and such thatfor , and .

Proof. See Lemma 3.1 in Hu et al.’s study [3].

Lemma 2. If is positive, then there exist positive constants and , depending only on such that for , there exist a sequence satisfying (a)=(b) for (c) is a covering of of finite multiplicity

Proof. See Lemma 2.1 in Zhang et al.’s study [11].

A sequence satisfying of Lemma 2 will be called a -lattice. The set of -lattices will be denoted by . The statement (c) of Lemma 2 says that for , there exists an integer such that

Lemma 3. Let . There are positive constants , , and such that

Proof. See Theorem 12 in Hu et al.’s study [3].

Lemma 4. Let . Then

Proof. See Corollary 3.2 in Hu et al.’s study [3].

Lemma 5. Suppose , , . Then

Proof. See Corollary 3.1 in Hu et al.’s study [3].

Lemma 6. Suppose , . Then, there is a constant such that

Proof. Since , it is obvious that for , there is a constant such that for any . Then, for , we have

Lemma 7. For and , the following statements hold (a)(b)

Proof. (a)Lemma 3 implies thatIt follows from Lemma 6 that (b)Lemma 2.3 and Theorem 3.3 in Hu et al.’s study [3] show thatNote that , and then, for sufficient lager number , we have Since we obtain that . Then Moreover, Lemma 6 implies that Thus

For let be the set of all -th locally Lebesgue integrable function on . For and , the averaging function is defined as where is the volume of . Suppose and the -th mean oscillation of given by

The space consists of the functions such that , where and . When , we say .

For a continuous function on , and , let be the oscillation of .

For , denote the space of all continuous functions on such that . The space consists of functions in satisfying .

Suppose , and , let denote the space of all locally -th integrable functions on such that . The space consists of the functions for which .

For and for any , we define the Berezin transform of by for . When , it follows from Theorem 11 and Theorem 12 in Zhang et al.’s study [11] with that

Theorem 8. Let and . Then, the following statements are all equivalent: (a);(b) admits a decomposition , where and , moreover(c)The function is bounded, where and

Proof. . For with as in Lemma 1(B), we have Set and . Suppose with , and by the triangle inequality and Hölder’s inequality, one can get Thus it is easy to know that and .
By the triangle inequality, we have If the Hölder’s inequality implies that Then, for , there holds Thus, and .
Suppose where and , by the triangle inequality, there holds Lemma 3 and Lemma 4 imply that Note that for some constant , and By Lemma 6, we have For , by the same reason, we have It is obvious that for . Next, we prove the part with respect to . By triangle inequality, there holds Thus By Lemma 3, for Hence Note that Thus,

3. Boundedness and Compactness of Operators

For , we define an integral operator by

Lemma 9. Suppose with . Let , and there exist positive constants and such that for and , there holds (a),(b).

Proof. See Lemma 13 in Hu et al.’s study [3].

It follows from Lemma 9 that where ,

Lemma 10. The operator is bounded on with ; meanwhile, is bounded from to with .

Proof. First, we consider the case . Lemma 7 implies that Similarly, By interpolation theorem, is bounded on with .
Next, we consider the case. We know thatis equivalent to the Bergman distance, and then, by Lemma 1 in Hu and Virtanen’s study [5], we have for . For , we choose an -lattice as in Lemma 2, and then by Lemma 2 and Lemma 9 for we obtain Then, Lemma 7 implies that The proof is completed.

Theorem 11. Let . If has compact support, the Hankel operator is compact from .

Proof. For , refer to Theorem 4.3 in Hu and Pau [15]. Next, we prove the case . Without loss of generality, we assume that the support of is contained in some , . Write ; there is an so that when .
Then, for any bounded sequence in converging to 0 uniformly on any compact subset of , we get Since , when , for any , there exists such that when . Consider the -lattice with . Since whenever , it follows from (B) in Lemma 1 that And then, we have Thus, when . Therefore, is compact.

Theorem 12. Suppose and , the following statements hold. (a)If then is bounded, moreover and (b)If then is compact

Proof. (a)Because of Theorem 4.2 in Hu and Pau [15], bounded and when . Next, we prove the case , and it is easy to know thatThen Lemma 10 implies that result hold. (b)Because of Theorem 4.3 in Hu and Pau [15], is compact when . Suppose , , for such that when . Moreover, for , andthere exists such that , and . Hence therefore, there is an such that with . Define a function Set ; it is obvious that when and . Then By (55), . The compactness of follows from the compactness of .

Lemma 13. If with . Then, .

Proof. Lemma 6 and Lemma 7 imply that as . Then, for any , there exists a positive number such that for all , Note that , there is such that when . Thus

Notation 14. We use to represent the integral operator which is defined by Lemma 10 implies that is a bounded operator from with , and from with .

Lemma 15. If , , and , for satisfying , then

Proof. When and , by the same proof in Lemma 7, for there exists such that for all . Moreover, for the and , there exists if such that when . Thus, when , by Lemma 1 and proof in Theorem 11, there hold Therefore, for , by Lemma 7 and Fubini’s theorem, we obtain It is obvious that By Notation 14, we have When , By Lemma 10, we have Now, the results follow as in the case .

Lemma 16. If and is compact on , then , where is called Berezin transform of Toeplitz operator for.

Proof. When , the results follow from the fact when . Recall that is relatively compact if and only if for every , there exists such that If is compact on , then Note that Thus when .

Theorem 17. If , and . Then (a) is compact on and (b) is compact on if and only if

Proof. (a)Lemma 3 implies that for , we haveThen, Lebesgue’s dominated convergence theorem shows that It shows that is continuous on . Hence, , by Notation 14, we know that . Theorem 4.2 Zorboska’s study [7] implies that is compact on for . Note when , then is compact, and thus is compact. (b)If is compact on Lemma 13 shows that . If it is easy to know thatsince and , the conclusion holds.

4. Fredholm Theory

A linear mapping on a topological vector space is called to be Fredholm if

When is a Banach space, it is well known that is Fredholm if and only if is invertible in the Calkin algebra , where and represent, respectively, the spaces of bounded and compact operators. It shows that an operator on a Banach space is Fredholm if and only if there are bounded operators and on , such that for some compact operators and on .

A pair is said to be a quasi-Banach space if satisfies all the properties of a norm except for the triangle inequality and if there is a constant such that

Note that Bergman space are quasi-Banach spaces.

Theorem 18. Let , for , under the pairing

Proof. Refer to Hu et al. [3].

Theorem 19. A bounded linear operator on a dual rich quasi-Banach space is Fredholm if and only if it has a regular; that is, there is a bounded linear operator on such that and are both compact on .

Proof. See Section 3.5.1 in Runst and Sickel’s study [16].

Theorem 20. Let , , and . Then, the Toeplitz operator is Fredholm on if and only if

Proof. Because of the decomposition , there are functions and such that . Set . Then, we have which shows that is the vanishing Carleson measure. By Theorem 12 and Theorem 4.2 in Zhang et al.’s study [11], we have and is compact on for . Therefore, is Fredholm if and only if is Fredholm. (91) shows that Now, we just need to prove the conclusion for . If , and is Fredholm on , then is bounded on . Suppose that , note that then Hence, . When , and Hölder’s inequality show that . If are false, there are some sequence such that . By Lemma 15Then, for any bounded operator on , we have Because that where is the conjugate exponent of . We have It is obvious that is Fredholm on ; thus, there is a bounded operator such that where is identity and is the compact operator on . Hence, which contradicts (98).
Conversely, if and satisfies (89), then there are positive constants , , and such that By Lemma 13 and . Set function satisfying Then, , and when . Theorem 12 shows that is compact operator from . Note that Therefore, on . Thus It is obvious that and are compact. Similarly, , where is a compact operator on . Then, is Fredholm.

Corollary 21. Let and . If Theorem 20 holds, then and essential spectrum is connected.

5. Schatten Class Toeplitz and Hankel Operators

If is a bounded linear operator , where and are two Hilbert spaces, the singular values of are defined by where rank denotes the rank of operator . is compact if and only if as . For , is in the Schatten class , if

is a norm when , and a quasi-norm when . In fact, we have

In addition, if and only if .

Definition 22. Let be a compact operator from to and is a continuous increasing convex function, we say that if there is a positive constant such that

Theorem 23. Let are continuously increasing convex function. Let be a positive Borel measure on such that the Toeplitz operator is compact. Then, if and only if there is a constant such that where .

Proof. Suppose that that is for some constant . Let be an orthogonality basis of . Then where are also the eigenvalues of . It is obvious that Then, it follows from the convexity of , Jensen’s inequality, and Lemma 4 that Conversely, assume that for some . Then, by Lemma 4, we have Note that Jensen’s formula shows that Therefore, .

Let and

It follows from Hu et al. [3] that is dense in . Define where is a set of all analytic functions on .

Theorem 24. Suppose that is a continuous increasing convex function, , and . Then, the following statements are equivalent: (a)Hankel operator belongs to (b)For some (any) , there is a constant such that

Proof. (A)⇒(B). Let be orthogonality basis of . Define where is the -lattice of . It is obvious that , and then, is bounded. The convexity of shows that is also a convex function. Set and then, we have where , and are positive constants.
(B)⇒(A). Suppose that , we define the square mean of over by setting for and , decomposing as where is a partition of unity subordinate to and given for such that Then, and Therefore, Letto beor, considering the multiplication operators, and by the assumptionand Lemma 3.4 in Zeng et al.’s study [17], we have, andis bounded fromto. Since for any, there holds We know that on . Thus, if and only if . According to Theorem 23 and the condition which is a convex function, if and only if In proof of Theorem 23, we get . On the other side, we assert that In fact, by proof in proposition 2.5 of Zeng et al. [17], we have and together with Jensen’s inequality, there holds Therefore, we conclude that is equivalent to or equivalently hence . Since and , both and belong to which leads to . The proof is complete.

Data Availability

No data were used in this paper.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

The paper was supported by the National Natural Science Foundation of China (Grant No. 11971125).

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Copyright © 2023 Xuedi Ma 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.

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