Abstract

We describe certain -algebras generated by Toeplitz operators with nilpotent symbols and acting on a poly-Bergman type space of the Siegel domain . Bounded measurable functions of the form are called nilpotent symbols. In this work, we consider symbols of the form , where both limits and exist, and belongs to the set of piecewise continuous functions on and having one-side limit values at each point of a finite set . We prove that the -algebra generated by all Toeplitz operators is isomorphic to , where and .

1. Introduction

In the study of Toeplitz operators, one of the common strategies consists in selecting a set of symbols in such a way that the algebra generated by Toeplitz operators with symbols in can be described up to isomorphism, say, with an algebra of continuous functions or finding its spectrum. In this paper, we study Toeplitz operators with nilpotent symbols and acting on a poly-Bergman type space of the Siegel domain . In [1–3], the authors have fully described all commutative -algebras generated by Toeplitz operators with symbols invariant under the action of a maximal abelian subgroup of biholomorphisms and acting on the Bergman spaces of both the unit disk and the Siegel domain . For the unit disk, they discovered three families of symbols associated to commutative -algebras of Toeplitz operators, while for the Siegel domain, they found classes of symbols. Each class of symbols is invariant under the action of a maximal abelian group of biholomorphism. Certainly, one can use these classes of symbols to study Toeplitz operators acting on poly-Bergman type spaces of the unit disk or the Siegel domain.

Let be the upper half-plane. Toeplitz operators with vertical symbols, which depend on , and acting on Bergman type spaces have been studied. In [4–9], the authors proved that the algebra generated by Toeplitz operators with vertical symbols and acting on the weighted Bergman space is isometrically isomorphic to the algebra of all bounded functions that are very slowly oscillating on . Taking vertical symbols having limit values at and , in [10, 11], the authors found that is the spectrum of the algebra generated by all Toeplitz operators on the true-poly Bergman space . Similar research was made for Toeplitz operators on poly-Bergman spaces with homogeneous symbols ([12, 13]). Other works about it were made in [14–16], where the authors studied Toeplitz operators acting on from the point of view of wavelet spaces. On the other hand, in [17, 18], the authors studied Toeplitz operators on the Fock space with radial and bounded horizontal symbols; they found that spectral functions are uniformly continuous with respect to an adequate metric. Taking horizontal symbols having one-side limits at , in [19, 20], the authors studied Toeplitz operators acting on poly-Fock spaces ; they found the spectrum of the -algebra generated by such Toeplitz operators. Even though the authors described the -algebras generated by all spectral functions, the spectrum of the algebras is not fully understood in some cases; for this reason, additional conditions on the symbols are imposed.

In [1, 2], the authors made remarkable research on the study of Toeplitz operators acting on the Bergman space of the Siegel domain . In particular, they studied the -algebra generated by all Toeplitz operators with bounded nilpotent symbols, which are functions of the form , where . Let us denote this kind of symbols by . Although is commutative, it is too large, so it is impossible to figure out what its spectrum is. In particular, in [21, 22], the authors described the algebra generated by Toeplitz operators acting on the weighted Bergman space over three-dimensional Siegel domain using nilpotent symbols of the form .

The main purpose of the paper is to find the spectrum of the algebra generated by Toeplitz operators acting on the true-poly-Bergman type space over two-dimensional Siegel domain by selecting a particular set of nilpotent symbols. In this sense, we just consider nilpotent symbols of the form and . This paper is organized as follows. In Section 2, we recall how poly-Bergman type spaces are defined for the Siegel domain, and how they can be identified with a -space through a Bargmann type transform. In Section 3, we introduce Toeplitz operators acting on with nilpotent symbols; we show that such Toeplitz operators are unitary equivalent to multiplication operators. In Section 4, we take symbols of the form for which both limits and exist; it is proved that the -algebra generated by all Toeplitz operators is isomorphic to , where is the one-point compactification of . In Section 5, we take nilpotent symbols of the form , where , and is the two-point compactification of ; we prove that the -algebra generated by all Toeplitz operators is isomorphic to , where and . In Section 6, we describe the -algebra generated by all Toeplitz operators , where , and is the set of all piecewise continuous functions on having one-side limit values at each point of a finite set . Finally, in Section 7, we describe the -algebra generated by all Toeplitz operators .

2. Poly-Bergman Type Spaces of the Siegel Domain

In this section, we recall some results obtained in [23], which are needed in our research about Toeplitz operators. Each will be represented as an ordered pair , where . Besides, the Euclidean norm function will be denoted by . The Siegel domain is defined by

We will study Toeplitz operators acting on certain poly-Bergman type subspaces of , where , with , and is the usual Lebesgue measure. Once and for all, means , where is any subset of a Euclidean space and is the Lebesgue area measure on .

For each multi-index , the poly-Bergman type space is the closed subspace of consisting of all -analytic functions, that is, all functions satisfying the equations

In particular, for , is just the Bergman space. Likewise, the anti-poly-Bergman type space is defined to be the complex conjugate of Thus, we introduce true-poly-Bergman type spaces as follows: where and the 1 is placed at the -entry. We assume that whenever .

In [23], the authors proved that equals to the direct sum of all the true-poly-Bergman type spaces:

The authors also proved that is isomorphic and isometric to the tensor product where . Both and are one-dimensional spaces defined below. Recall the Hermite and Laguerre polynomials: for Recall also the Hermite and Laguerre functions where and is the usual Gamma function. It is well known that and are orthonormal bases for and , respectively. Finally, and .

In this work, we restrict ourselves to the study of Toeplitz operators acting on the true-poly-Bergman type spaces over two-dimensional Siegel domain with the Lebesgue measure (). Henceforth, the space will be simply denoted by ; similarly, and stand for and , respectively. The true-poly-Bergman type space can be identified with through a Bargmann type transform ([23]), such identification fits to the study of Toeplitz operators with nilpotent symbols. Several operators are needed to define such identification. To begin with, we introduce the flat domain , where . Then, can be identified with using the mapping

Thus, we have the unitary operator given by where . Take , with and . We identify with . Then

Introduce where is the Fourier transform acting on by the rule

Consider now the following two mappings acting on : where , , , and . Both functions and lead to the following unitary operators acting on :

Henceforth, .

Theorem 1 (see [23]). The operator is unitary and maps onto the space For each , the operator restricted to is an isometric isomorphism onto the space

Introduce the isometric linear embedding defined by

Of course, is the range of , and it is also the image of under . Thus, the operator isometrically maps the true-poly-Bergman type space onto . Therefore, and , where is the orthogonal projection from onto . In addition, the operator plays the role of the Segal-Bargmann transform for the true-poly-Bergman type space , where the adjoint operator is given by with .

3. Toeplitz Operators with Nilpotent Symbols

In this section, we study Toeplitz operators with nilpotent symbols and acting on the true-poly-Bergman type space . In [3], the author has widely developed the theory of Toeplitz operators on the Bergman spaces, and the author’s techniques can be applied to the study of Toeplitz operators acting on . To begin with, a function is said to be a nilpotent symbol if it has the form . Then, the Toeplitz operator acting on , with nilpotent symbol , is defined by where is the orthogonal projection from onto . The Bargmann-type operator identifies the space with , and it fits properly in the study of the Toeplitz operator .

Theorem 2. Let be a nilpotent symbol. Then, the Toeplitz operator is unitary equivalent to the multiplication operator , where is given by

Proof. We have Recall that , where and . For , That is, , where . It is easy to see that Thus where is given in (21).

By Theorem 2, the -algebra generated by all Toeplitz operators is commutative (see [1, 3]), but its spectrum is difficult to figure out what it is. For this reason, we assume certain continuity conditions on the nilpotent symbols in order to describe the spectrum of the subalgebra generated by the Toeplitz operators. We will split our research into two cases concerning the symbols. Firstly, we study Toeplitz operators with symbols of the form , for which

Secondly, we analyze Toeplitz operators with symbols of the form , for which

As mentioned above, the -algebra generated by all Toeplitz operators is still complicated to be fully described despite its commutative property. Fortunately, the -algebra generated by all operators can be described when the symbols are taken to be continuous on , where is the two-point compactification of . Even more, the -algebra of Toeplitz operators can be still described for symbols having finitely many jump discontinuities, as shown in Section 6. Finally, we analyze Toeplitz operators with symbols of the form .

4. Toeplitz Operators with Symbols

In this section, we study the -algebra generated by all Toeplitz operators with symbols of the form , where has limit values at . Under this continuity condition, we will see that is continuous on , where is the two-point compactification of . Apply the change of variable in the integral representation of , then

Actually, depends only on the variable and is continuous on because of the continuity of and the Lebesgue dominated convergence theorem.

Let denote the subspace of consisting of all functions having limit values at and . For , define

It is worth mentioning that was obtained in [10] as the spectral function of a Toeplitz operator acting on a true-poly-Bergman space of the upper half-plane. Thus, we have at least two scenarios in which appears as a spectral function.

Lemma 3 (see [10]). Let . Then, the spectral function satisfies

According to Lemma 3 and Theorem 4.8 in [10], we have the following.

Theorem 4. For , the spectral function is continuous on . The -algebra generated by all functions , with , is isomorphic and isometric to the algebra . That is, the -algebra generated by all Toeplitz operators , with , is isomorphic to , where the isomorphism is defined on the generators by

Obviously, the spectral function is defined and continuous on , but it is constant along each horizontal straight line. Thus, is identified with a continuous function on the quotient space , which is homeomorphic to .

5. Toeplitz Operators with Continuous Symbols

In this section, we study the -algebra generated by all Toeplitz operators , where symbols are taken to be continuous on . Once again, such a -algebra can be identified with the algebra of all continuous functions on a quotient space of . Henceforth, will denote points in instead of intervals.

It is fairly simple to see that is continuous on . Take the change of variable in the integral representation of , then

The function is continuous at each point because of the continuity of and the Lebesgue dominated convergence theorem. Next, we will prove that has one-side limit value at each point of . For , we introduce the notation if such limits exist.

Lemma 5. Let , and suppose that converges at . Then, for each , the spectral function satisfies

Proof. Let denote the right-hand side of equality (34). Take . We will prove that there exist such that whenever and . Note that . Since , there exists such that Then We have assumed that converges at ; then, there exists such that and for . Let . Then, we have if , , and . Thus, Finally, we conclude that whenever and .

In general, does not converge at each point ; however, has limit values along the parabolas , with . We will define a bijective mapping so that will be a continuous mapping on with the usual topology.

5.1. Modified Spectral Function for

Let be the mapping

It is easy to see that . Concerning the spectral properties of , the function is as important as , but behaves much better than , at least for continuous on . From now on, we take as the spectral function for the Toeplitz operator . A direct computation shows that

Both and are continuous on . Besides, the spectral function is continuous on because is. Since , we have . By Lemma 5, is also continuous on .

Theorem 6. For , the spectral function can be extended continuously to .

Proof. Follows from Lemmas 5 and 7–9.

For any domain and a function , we write to mean the limit value of at , even if does not belong to . For example, means .

Lemma 7. Let , and suppose that converges at the points . Then, satisfies That is, for , there exists and such that whenever and . Analogously,

Proof. Suppose that . Let . Since , there exists such that Take into account in the following computation Since converges to zero at , there exists such that for . Take . Then, we have for . On the other hand, assume and . Then The right-hand side of this inequality converges to when tends to ; thus, there exists such that for . Consequently, For and , we have We define in the case , where is a constant. Note that converges to zero at , and for any nilpotent symbols and . Then Finally, the limit of at can be proved analogously.

Lemma 8. Let . If is continuous at , then the spectral function satisfies Analogously, if is continuous at , then

Proof. Suppose that converges to zero at . Let . Since , there exists such that Take into account in the following computation Because of the continuity of at , there exists such that for . Let us estimate the value of the argument of : Choose in such a way for . Pick such that whenever . Now assume that and . Then, . Thus, converges to when tends to . Therefore, there exists such that for . The additional condition implies Hence, if and .
If does not converge to zero at , then take the function and proceed as in the proof of Lemma 7, where .
Finally, the justification of the limit of at can be done analogously.

Lemma 9. Let be continuous at . For , the spectral function satisfies Actually, we have uniform convergence of at , that is, for , there exists such that for all and for all .

Proof. Suppose that . Let , and choose such that equations (50) hold. Then By the continuity of at , there exists such that for . For , we have Take . The inequality implies . Thus, if , , and , then Consequently, for all and .
Finally, in the case , the proof can be carry out by considering the symbol .

For each nilpotent symbol , the spectral function is continuous on and is constant along . For this reason, the -algebra generated by all spectral functions is not , but it coincides with the algebra of continuous functions on a quotient space of .

5.2. Toeplitz Operators with Continuous Symbols

Introduce the quotient space . By Lemma 9, the spectral function can be identified with a continuous function on , which will be also denoted by . We establish now one of our main results in this work.

Theorem 10. The -algebra generated by all spectral functions , with , is isomorphic and isometric to the algebra . That is, the -algebra generated by all Toeplitz operators is isomorphic to , where the isomorphism is defined on the generators by the rule

Proof. The functions separate the points of according to Lemmas 11–13 below. The Stone-Weierstrass theorem completes the proof.

Lemma 11. Let and be distinct points of , where they do not belong simultaneously to . Then, there exists such that .

Proof. First consider the nilpotent symbol , which is continuous on . Note that (i)for,(ii),(iii) for ,(iv) for .From , we get This formula also says that . On the other hand, the Hermite function is continuous, and it has just a finitely many roots. Hence, is monotonically decreasing with respect to . Thus, points in are separated by .
Recall that is identified with one point in so take as representative point of the equivalence class . For , the three points , , and are separated by because of the injective property of .
Consider now the nilpotent symbol , which is continuous on . We have (i) for ,(ii) for ,(iii) for all .Thus, separates each point from the points and .

Now our aim is to separate the points of . Consider the following family of continuous functions:

Then, , where

Lemma 12. If and are distinct points in , then there exists such that , where is defined in (60).

Proof. At first suppose that and satisfy . Introduce . It is easy to see that can be written as This integral representation allows us to prove easily that Take small enough that . Then Now suppose that and . Let . Then Since , we have . Besides, the following inequality holds for . Hence, . Finally, from , we get .

Lemma 13. Let and . Then, there exists such that , where is given in (60).

Proof. Take , , and . It is easy to see that . Then, we have for . Now suppose that . Let . The inequality holds for small enough. For such , This proves that all points of can be separated from points of . On the other hand, For , we have ; hence,
Finally, consider the function . Then, , and consequently, .

6. Toeplitz Operators with Piecewise Continuous Symbols

Take a finite subset and let be the set of functions continuous on and having one-side limit values at each point of . In this section, we study the -algebra generated by all Toeplitz operators , where . Obviously, we have to study the algebra generated by the spectral functions . To begin with, take the indicator function . Then, the spectral function is continuous on according to Lemmas 5, 7, and 8. Actually, where is defined in (61). Hence, is continuous on because is. Of course, we have now a spectral function which is not constant along anymore.

For any , we have where and are the one-side limits of at , and . This function has a removable discontinuity at ; thus, is continuous on .

Theorem 14. The -algebra generated by all Toeplitz operators , with , is isomorphic and isometric to . The isomorphism is defined on the generators by the rule

Proof. If , the function separates any two points and . By Lemma 12, two points in can be separated by , where is the nilpotent symbol given in (60). Further,

We continue our study by introducing another point of discontinuity. Take the indicator function , where . We have

According to Lemmas 5 and 7–9, the spectral function is continuous on , except at the point . For , takes the constant value along the curve . From this equation, we get

The horizontal line is an asymptote of the graph of , and of course, for Thus, the level curves of converge to the point ; our aim is to separate them at through a mapping in such a way that is continuous on .

Lemma 15. Let be any bijective, smooth, and increasing function, with . Take , and let be the function on defined by the rule where . Then, is an homeomorphism from onto itself, which can be continuously extended to with range , where .

Proof. The function has range contained in . Hence, . Now suppose that . Then, and . The function is strictly increasing with respect to . Thus, implies that . Consequently, is injective. Let be a point in . Consider the equation , which is equivalent to the system of equations , . Thus, we have to prove that is solvable. The function is bijective from into itself with respect to . Therefore, has a unique solution. That is, is surjective. On the other hand, is smooth and so is because of the Inverse Function Theorem.
The correspondence (76) also defines on by . Actually, can be defined on according to the following limits: We will justify just the last limit. For , If and is close enough to , then tends to when tends to . Thus, . Consequently, that is, converges to when tends to . A local analysis proves that is continuous. This completes the proof.

Lemma 16. The function is continuous on , and it separates the points in the line segment . For each continuous function , is also continuous on and has constant value along .

With a discontinuity , we define . Introduce another point of discontinuity , with . Let . The function has a continuous extension to , and its level curves , , converge to . As in Lemma 15, we can construct a mapping that separates all these level curves, and is continuous on . Adding more discontinuity points , with , we can construct mappings in such a way that is continuous on .

Let denote the -algebra generated by all Toeplitz operators with nilpotent symbols , where . For simplicity in our explanation, we assume that and . We will explain how the Toeplitz algebra increases as does. By Theorem 14, is isomorphic to , where the isomorphism is given on the generators by the rule

Consider now the algebra , where . By Lemma 16, is continuous on for every . Then, the algebra is also isomorphic and isometric to , where the isomorphism is given by

At first sight, both algebras and seem to have the same spectrum , but they do not; they are identified with through different isomorphisms. Of course, is a subalgebra of . If , then and has constant value along the line segment . According to the isomorphism , we can say that the spectrum of equals ; meanwhile, the spectrum of is the quotient space . This phenomenon persists as long as the set grows.

Theorem 17. Let . Then, there exist bijective continuous functions , , such that admits a continuous extension to for each . The -algebra generated by all Toeplitz operators is isomorphic and isometric to . The isomorphism is defined on the generators by the rule

Note that for each piecewise continuous symbol , in general, the spectral function does not admit a continuous extension to , but does, which means that is uniformly continuous with respect to a new metric on ; this metric is the pushforward of the usual metric using the mapping .

7. Toeplitz Operators with Symbols

In this section, we describe the -algebra generated by all Toeplitz operators with symbols of the form , where , and has limit values at . For such a symbol , we have that , which means that . Although belongs to , the spectral function is continuous on , where and . Since the level curves of are the horizontal lines , the level curves of are given by the equations , with .

Lemma 18. Let be any bijective, smooth, and increasing function. Then, the function is an homeomorphism from onto itself, which can be continuously extended to with range , where . We have for and acts like the identity mapping at the rest of points in .

Proof. Similar to the proof of Lemma 15.

The image of the level curve under is the curve

This means that the level curves of do not converge to a single point anymore.

Lemma 19. The function is continuous on , and for each continuous function , is also continuous on and has constant value along each component of .

Theorem 20. The -algebra generated by all Toeplitz operators , with and having limits values at , is isomorphic and isometric to . The isomorphism is defined on the generators by the rule

For the Toeplitz operator with symbol , there exists a mapping such that admits a continuous extension to . The construction of is similar to the construction of given in Lemma 15, where one has to take into account the level curves of the spectral function , which converge to the point .

Implicitly, we have considered several compactifications of associated to the -algebras studied herein; each compactification depends on the kind of symbols. Take and , let us explain the situation in the case of the algebra generated by the Toeplitz operators with symbols . Essentially, the corresponding compactification of is obtained from by gluing a line segment at each corner and . Each spectral function is continuous on and has limit values when moves along the parabolas and tends to . For a net tending to , converges if is eventually in gaps between two parabolas close enough from each other with respect to the parameter .

Data Availability

The data used to support the findings of this study are available from the corresponding author upon request.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

This work was supported by Universidad Veracruzana under P/PROFEXCE-2020-30MSU0940B-22 project, México.