Products of Toeplitz Operators on the 2-Analytic Bergman Space

Zhang, Bo; Yang, Yixin; Lu, Yufeng

doi:https://doi.org/10.1155/2021/6227981

Journal of Function Spaces

On this page

Abstract Introduction Data Availability Conflicts of Interest References Copyright Related Articles

Research Article | Open Access

Volume 2021 | Article ID 6227981 | https://doi.org/10.1155/2021/6227981

Products of Toeplitz Operators on the 2-Analytic Bergman Space

Bo Zhang,¹Yixin Yang,²and Yufeng Lu²

Academic Editor: Nikolai L. Vasilevski

Received24 May 2021

Revised06 Sept 2021

Accepted15 Sept 2021

Published04 Oct 2021

Abstract

Let and be bounded functions, and let and be Toeplitz operators on . We show that if the product equals zero and one of and is a radial function satisfying a Mellin transform condition, then the other function must be zero.

1. Introduction

Let be the open unit disk in equipped with the normalized Lebesgue area measure , and let denote the Lebesgue space on . For , let denote the -analytic Bergman space, that is, the subspaces of consisting of -differentiable functions such that , where

As we know, is a Hilbert subspace with the inner productwhere , .

The planar Beurling transform is the singular integral operator given by

It is well known that the Beurling transform is a unitary operator acting on (see [1], p. 364). For , the compression of the Beurling transform to is a bounded linear operator acting on defined by

The -analytic Bergman projection is defined to be the orthogonal projection of onto . The singular integral operator is related to , and it is known (see [2]) that

For a function , the Toeplitz operator with symbol on is defined by

-analytic functions play an important role in mathematical, and the space has been intensively studied. More details about the structure of these spaces can be found in paper [3–5] and Balk’s book [6].

Zero-product problem is a very important question in the operator theory. For Toeplitz operators, we have the general zero-product problem. Namely, if and are bounded functions such that , then must one of the functions be zero? Ahern and Cučković (see [7]) obtained an affirmative answer for Toeplitz operators on when one of the functions is radial. Le (see [8, 9]) generalized this result to more than two functions. Cučković and Le (see [10]) gave a positive answer when both functions are harmonic. While the general zero-product problem (even on ) is still far from being solved, it is known that Toeplitz operators with radial symbols are diagonal with respect to the standard orthonormal basis of . However, this is not the case on when . Then, Cučković and Le (see [10]) raised the following open question:

Question 1. Let and be bounded functions, one of which is radial. If on (or more generally, has finite rank), must one of these functions be zero?

In this paper, we give a partial answer to this question on the 2-analytic Bergman space . We show that if is a radial function satisfying a Mellin transform condition, then if and only if is a zero function.

2. Some Preliminary Results

We adopt the following boundary conditions for the binomial coefficients:

An orthogonal basis in the space is given by (see [3, 11])where . The orthogonal basis can also be written aswhere . For , we have the following lemma.

Lemma 2. An orthogonal basis in is given bywhere .

For each , since the point evaluation at is a bounded linear functional on , there exists a unique reproducing kernel function such that

for every . On -analytic Bergman space ,

The Mellin transform of a function is defined by

It is easy to see that is well defined and analytic on the right half-plane . Cučković and Rao (see [12]) first used the Mellin transform to study Toeplitz operators on the classical Bergman space.

For notational convenience, we define and . For some Toeplitz operators on 2-analytic Bergman space , we obtain the following lemmas.

Lemma 3. Let be a bounded radial function. Then, for each , we havewhere , , , and .

Proof. For each , since is a bounded radial function, thus

It is well known that radial Toeplitz operators acting on are diagonal, and radial Toeplitz operators acting on can be represented as matrix sequences (see [13]). In the following, we give the exact expression of radial Toeplitz operators on .

Lemma 4. Let be a bounded radial function. Then, for each ,

Proof. Since is a bounded radial function, for each , using Lemma 3, we get

Applying Lemma 4, we conclude that radial Toeplitz operators on are not diagonal. The following corollary is an immediate consequence of Lemma 4.

Corollary 5. Let be a bounded radial function. Then, for each ,

3. Products of Two Toeplitz Operators

A bounded function is said to be quasihomogeneous of degree ifwhere is a radial function (see [14]). For any function , it has the polar decomposition, i.e.,where are radial functions in (see [12]). A direct calculation gives the following lemma.

Lemma 6. Let be a bounded function. Then, for each ,

Proof. For all , it is easy to verify thatSimilarly, the rest of the lemma can be proved.

When considering the product of two Toeplitz operators, we often use the Mellin convolution. If , then their Mellin convolution is given by

The Mellin convolution theorem (see [15]) states that

and if and are bounded, then so is .

It is well known that the Mellin transform is uniquely determined by its value on an arithmetic sequence of integers. The following results (see [15], p. 102, [16]) will be needed later.

Theorem 7. Suppose is a bounded analytic function on which vanishes at the pairwise distinct points , , , whereThen, vanishes identically on .

Remark 8. Using this theorem, we can see that if and if there exists a sequence such thatthen, for all , by the Müntz-Szasz theorem (see [17], p. 312), .

For , we obtainthe numbers can also be called the moment Mellin sequence of . Let

is closed related to the moment Mellin sequence of , and we have the following lemma.

Lemma 9. Let be a fixed positive integer. Then, the following statements hold:(i) if and only if (ii) if and only if (iii) if and only if (iv) if and only if

Proof. From Lemma 3, it is easy to check that (i), (ii), and (iii) hold.
To prove (iv), in fact, for a fixed ,It follows that if and only ifUsing , , and Mellin convolution (24), we get the above equality is equivalent to

Lemma 10. Let be a bounded radial function. The function if and only if there exists a sequence ,

Proof. If the function , then , for each ,This proves the sufficient condition.
Next, we prove the necessary condition. Suppose ,Using Remark 8, we havefor all . Therefore, for each ,That is, is a geometric sequence. There exists a constant such thatThen,Since is a sequence and , by Remark 8, , which implies .

For each , let , , , and . Let

The first main result of this paper is the following theorem.

Theorem 11. Let be a bounded function and be a bounded radial function. Then, on if and only if for each , .

Proof. Using the fact that is a bounded function, we haveIf , then for each ,By Lemma 4,from which we conclude thatSince is arbitrary, it follows thatAnalogously, for each , it is easily verified thatthus, we getThe above equations are equivalent toThis completes the proof of the theorem.

For , firstly if , then , using Remark 8, we get . Now, if , then

Letting , we haveIt is easy to see that the function is a periodic function with a period . Using the same argument as the one at the end of Section 2 in [12], we conclude that must be a constant function. Hence,where is a constant and it is clear that is also a constant. Finally, if , thenthat is,

Similarly, we can also conclude that is a constant. Thus, if is a bounded radial function, it must be zero. Finally, we obtain the following lemma.

Lemma 12. Let be a bounded radial function. Then, the following statements hold:(i) for all if and only if (ii) for all if and only if is a constant(iii) for all if and only if

Remark 13. In Lemma 12, the condition “for all ” can also be replaced by “a sequence satisfying .”

In Theorem 11, if or , or , then , so it is clear that . If , then is a constant; it is also easy to see that if is not zero and , then must be zero. If , is not invertible. On the other hand, when , then is an invertible matrix. For a bounded radial function such that , if , is it necessary that ? The second main theorem of this paper answers this question by giving a sufficient and necessary condition.

Theorem 14. Let and be bounded functions and be a bounded radial function satisfyingfor each . Then, on if and only if .

Proof. If is a zero function, it is obvious that .
Now, we assume and we shall prove . If is a bounded radial function and for each ,then, by the Mellin convolution theorem (24), it follows thatFor each ,Applying (55), we get , that is, is an invertible matrix. If and for each , we getSince is invertible,Thus, , by Lemma 6, we haveThat is,where . Since and are arbitrary elements in , by Remark 8, we obtain for all in . It follows that . This completes the proof of the theorem.

Example 1. Let , where . Then, for each ,Obviously, . It is easy to see that on if and only if .

In the following, we discuss when condition (53) is not satisfied.

Case 1. If is a bounded radial function and for each ,Then, using the Mellin convolution theorem (24), we haveThat is, is a geometric sequence. Using Lemma 10, we get must be zero. It is clear that .

Case 2. If is a bounded radial function and for some ,(1)If there exists a sequence satisfying such thatthen, by using Lemma 10, we get that must be zero function.(2)If there exists a finite sequence , or an infinite sequence satisfying , such thatthen, the radial function may not be zero function. For example, if is finite sequence and , there exist some nonzero bounded radial functions such thatLet , where . Then,When , , a direct calculation shows that condition (67) is satisfied. In this case, we can prove that is not invertible. AsthenSince , it follows from (68) and (70) thatWhen , , a direct calculation shows that and is not invertible.

Remark 15. For a nonzero function whose related matrices are , , if there exist matrices , such that(i) are not all zero(ii)For each , then, we can construct a nonzero function , such that . The following example solves (i) and (ii) for a fixed . However, it is still unknown if (i) and (ii) hold for all , and we will study this question in the future work.

Example 2. Suppose . ThenAs is not invertible, there exist some nonzero matrix such that . For example,For each , analogous to Lemma 3, we defineand .

In Theorem 14, if and are all bounded radial function and there exists a sequence ,the conclusion is still valid; then, we have the following corollary.

Corollary 16. Let and be bounded radial functions. Suppose there exists a sequence ,If on , then .

Proof. For , defineBy the hypothesis, is a bounded radial function, it follows from Lemma 4 and Theorem 11 that if and only if for each ,Let and there exists a sequence ,Then, it follows that is an invertible matrix. Combining this with , we get . It follows thatThis implies that , combing withand using Remark 8, we get .
For , if and are bounded radial functions, it follows from Lemma 4 thatwhereIf has a finite rank, there exists , for all , such thatAs in Corollary 16, there exists a sequence meet the conditions, where and ; using properties of Mellin transform, we can obtain that has a finite rank if and only if .

Remark 17. As in Corollary 16, let and are bounded radial functions and there exists a sequence ,Then, has finite rank if and only if .

The following question is the general zero-product problem on when .

Question 18. Let be a bounded function and be a bounded radial function. Suppose that on when , can we obtain any similar conclusions?

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

References

E. M. Stein, Harmonic Analysis: Real Variable Methods, Orthogonality, and Oscillatory Integrals, Princeton University Press, Princeton, 1993.
Y. I. Karlovich and L. V. Pessoa, “Poly-Bergman projections and orthogonal decompositions of -spaces over bounded domains,” Operator Theory: Advances and Applications, vol. 181, pp. 263–282, 2008.
View at: Google Scholar
A. D. Košelev, “The kernel function of a Hilbert space of funcitions that are polyanalytic in the disc,” Doklady Akademii Nauk SSSR, vol. 232, pp. 277–279, 1977.
View at: Google Scholar
A. K. Ramazanov, “On the structure of spaces of polyanalytic functions,” Matematicheskie Zametki, vol. 72, no. 5, pp. 750–764, 2002.
View at: Publisher Site | Google Scholar
N. L. Vasilevski, “On the structure of Bergman and poly-Bergman spaces,” Integral Equations and Operator Theory, vol. 33, no. 4, pp. 471–488, 1999.
View at: Publisher Site | Google Scholar
M. B. Balk, Polyanalytic Functions, Akademie Verlag, Berlin, 1991.
P. Ahern and Ž. Cučković, “Some examples related to the Brown-Halmos theorem for the Bergman space,” Acta Scientiarum Mathematicarum, vol. 70, pp. 373–378, 2004.
View at: Google Scholar
T. Le, “Finite-rank products of Toeplitz operators in several complex variables,” Integral Equations and Operator Theory, vol. 63, no. 4, pp. 547–555, 2009.
View at: Publisher Site | Google Scholar
T. Le, “A refined Luecking’s theorem and finite rank products of Toeplitz operators,” Complex Analysis and Operator Theory, vol. 4, no. 2, pp. 391–399, 2010.
View at: Publisher Site | Google Scholar
Ž. Cučković and T. Le, “Toeplitz operators on Bergman spaces of polyanalytic functions,” The Bulletin of the London Mathematical Society, vol. 44, no. 5, pp. 961–973, 2012.
View at: Publisher Site | Google Scholar
L. V. Pessoa, “Planar Beurling transform and Bergman type spaces,” Complex Analysis and Operator Theory, vol. 8, no. 2, pp. 359–381, 2014.
View at: Publisher Site | Google Scholar
Ž. Cučković and N. V. Rao, “Mellin transform, monomial symbols, and commuting Toeplitz operators,” Journal of Functional Analysis, vol. 154, no. 1, pp. 195–214, 1998.
View at: Publisher Site | Google Scholar
R. M. Barrera-Castelán, E. A. Maximenko, and G. Ramos-Vazquez, “Radial operators on polyanalytic weighted Bergman spaces,” Boletín de la Sociedad Matemática Mexicana, vol. 27, no. 2, 2021.
View at: Publisher Site | Google Scholar
L. Zakariasy, “The rank of Hankel operators on harmonic Bergman spaces,” Proceedings of the American Mathematical Society, vol. 131, no. 4, pp. 1177–1180, 2003.
View at: Publisher Site | Google Scholar
R. Remmert, “Classical topics in complex function theory,” in Graduate Texts in Mathematics, vol. 172, Springer, New York, 1998.
View at: Google Scholar
X. T. Dong and Z. H. Zhou, “Products of Toeplitz operators on the harmonic Bergman space,” Proceedings of the American Mathematical Society, vol. 138, no. 5, pp. 1765–1773, 2010.
View at: Publisher Site | Google Scholar
W. Rudin, Real and Complex Analysis, McGraw-Hill Book Co., New York, Third edition, 1987.

Copyright

Copyright © 2021 Bo Zhang 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.

PDF Download Citation

Download other formats

Order printed copies