Journal of Function Spaces
Volume 2018 (2018), Article ID 8031259, 8 pages
https://doi.org/10.1155/2018/8031259

Research Article

Toeplitz Operators with Horizontal Symbols Acting on the Poly-Fock Spaces

Armando Sánchez-Nungaray,¹ Carlos González-Flores,² Raquiel Rufino López-Martínez,¹ and Jorge Luis Arroyo-Neri¹

¹Universidad Veracruzana, Facultad de Matemáticas, Lomas del Estadio S/N, 91000 Veracruz, Mexico
²Instituto Politécnico Nacional, Escuela Superior de Ingeniería Mecánica y Eléctrica, Unidad Profesional Adolfo López Mateos, Av. Instituto Politécnico Nacional s/n, Col. Zacatenco, Del. Gustavo A. Madero, 07738 Ciudad de México, Mexico

Correspondence should be addressed to Armando Sánchez-Nungaray; armsanchez@uv.mx

Received 12 April 2018; Accepted 13 June 2018; Published 16 July 2018

Academic Editor: Mitsuru Sugimoto

Copyright © 2018 Armando Sánchez-Nungaray 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.

We describe the -algebra generated by the Toeplitz operators acting on each poly-Fock space of the complex plane with the Gaussian measure, where the symbols are bounded functions depending only on and have limit values at and . The algebra generated with this kind of symbols is isomorphic to the -algebra functions on extended reals with values on the matrices of dimension , and the limits at and are scalar multiples of the identity matrix.

1. Introduction

Recall that the poly-Fock space is denoted by and consists of the -analytic functions which satisfy the equation

The true poly-Fock space is denote by , which consists of the all true--analytic functions; i.e.,

for , and .

It is clear that is the classical Fock space on the complex plane , which is also denoted by .

In [1], N. Vasilevski proved that has a decomposition as a direct sum of the true poly-Fock and true anti-poly-Fock spaces:

Moreover, they proved that the spaces are isomorphic and isometric to , where is the one-dimensional space generated by Hermite function of order . Finally, they found the explicit expressions for the reproduction kernels of all these function spaces.

In [2], K. Esmeral and N. Vasilevski introduced the so-called horizontal Toeplitz operators acting on the Fock space and give an explicit description of the -algebra generated by them. They showed that any Toeplitz operator with -symbol, which is invariant under imaginary translations, is unitarily equivalent to the multiplication operator by its “spectral function”. They stated that the corresponding spectral functions form a dense subset in the -algebra of bounded uniformly continuous functions with respect to the standard metric on .

The Toeplitz operators acting on spaces of polyanalytic functions have been object of study of several authors in different direction. For example, in [3], Sánchez-Nungaray and Vasilevski studied Toeplitz operators with pseudodifferential symbols acting on poly-Bergman spaces upper half plane. A different approach by Hutník, Maximenko, and Mišková in [4] considers Toeplitz Localization operators on the space of Wavelet transform or the space of short-time Fourier transform. They studied these operators with symbols that just depend on the first coordinate in the phase space, which are unitary equivalent to multiplication operators of certain specific functions “spectral functions”. In particular, the poly-Bergman spaces are spaces of Wavelet transform which is related to Laguerre functions, and the poly-Fock spaces are spaces of short-time Fourier transform which is related to Hermite functions.

In [5], J. Ramírez-Ortega and A. Sánchez-Nungaray described the -algebra generated by the Toeplitz operators with bounded vertical symbols and acting over each poly-Bergman space in the upper plane . They considered bounded vertical symbols that have limit values at and prove that the -algebra generated by the Toeplitz operator acting on with this kind of symbols is isomorphic and isometric to the -algebra of matrix-valued functions of the compact . Similar result can be found in [6], where M. Loaiza and J. Ramírez-Ortega gave an analogous description to the above for the -algebra generated by the Toeplitz operators with bounded homogeneous symbols acting over each poly-Bergman space in the upper plane.

The main result of this paper is the classified -algebra generated by the Toeplitz operators with bounded vertical symbols with limits at and acting over poly-Fock space in the complex plane.

This paper is organized as follows. In Section 2 we introduce preliminary results about the -polyanalytic function spaces and their relationship with the Hermite polynomials. In Section 3 we prove that every Toeplitz operator with bounded horizontal symbol acting on Fock space is unitary equivalent to a multiplication operator acting on , where is a continuous matrix-valued function on . Finally, in Section 4, we describe the pure states of the algebra

We prove that the algebra generated by Toeplitz operator with bounded vertical symbols that have limit values at acting on Fock space is isomorphic and isometric to the -algebra .

2. Poly-Fock Space on the Complex Plane

In this work we use the following standard notation: , with the usual complex conjugation ; thus . The Gaussian measure on is given by

where is the usual Euclidean measure on .

The Hilbert space of square integrable functions on with the inner product

The closed subspace of consisting of all analytic functions is called the Fock or Segal-Bargmann space. Also, the Fock space can be defined as the closure of the set of all smooth functions satisfying the equation . Similarly, given a natural number , the poly-Fock space is the closure of the set of all smooth functions in satisfying .

Recall that the Hermite polynomial of degree is defined by

and the system of Hermite functions

form an orthonormal basis for . By abuse of notation we also denote to the one-dimensional space generated by for . Further, define

The one-dimensional projection from onto is given by . Thus, is the orthogonal projection from onto , and

On the other hand, we consider the unitary operator defined by that transforms the space into , the set of all functions in , which satisfy the following equation:

The image of the space under the unitary transform is the closure of the set of all smooth functions in which satisfy the equation

where is the Fourier transform.

Finally, we take the isomorphism defined by

to transform the space onto the space , which is the closure of the set of smooth functions satisfying the equation

In summary, the unitary operator provides an isometric isomorphism from the space into the space , under which the poly-Fock space is mapped into . We denote by and the orthogonal projections from onto and , respectively. The true poly-Fock spaces are defined as follows:

Thus, is true -Bargmann projection from into .

The above construction is due to Vasilevski in [1], using the unitary operator , they obtain the following characterizations:

(1)The true-poly-Fock space is mapped onto .(2)The true-poly-Fock projection is unitary equivalent to the following one:

(3)The poly-Fock space is mapped onto .(4)The poly-Fock projection is unitary equivalent to the following one:

We introduce the isometric embedding by the rule Clearly, the adjoint operator is given by

The previous operators satisfy the following relations:

On the other hand, we introduce the operator from onto , and its restriction to is an isometric isomorphism. Thus, the adjoint operator is an isometric isomorphism from onto the subspace . Hence, these operators satisfy the following relations:

Similarly, introduce the isometric embedding by the rule

where and

and the superscript means that we are taking the transpose matrix.

Further, the adjoint operator is given by

Since the image of is the space , hence these operators satisfy the following relations:

Now the operator from onto , and its restriction to is an isometric isomorphism. Furthermore, the adjoint operator is an isometric isomorphism from onto the space . Hence, these operators satisfy the following relations:

3. Toeplitz Operators with Horizontal Symbol

In this section we introduce a certain class of Toeplitz operators acting on the poly-Fock spaces, and we prove that they are unitarily equivalent to multiplication operators by continuous matrix-valued functions on . Let be a function in depending only on and we called this function a horizontal symbol.

Definition 1. Let be a function in . The Toeplitz operator with symbol acting on true-poly-Fock space (or poly-Fock space) is defined as

or

where and are the orthogonal projections for true-poly-Fock space and poly-Fock space, respectively.

In [2], K. Esmeral and N. Vasilevski show that every Toeplitz operator with horizontal symbol acting on is unitary equivalent to the multiplication operator acting on , where is defined in Section 2. The function is given by

The following theorem is a generalization of above result for Toeplitz operators with horizontal symbols acting on true-poly-Fock space.

Theorem 2. For any , the Toeplitz operator acting on is unitary equivalent to the multiplication operator acting on , where the function is given by

Proof. We know that the operator is unitary and using (20), we obtain that the Toeplitz operator is unitary equivalent to the following operators:

Now calculate the explicit expression of the above operator

where and is the Hermite function of degree .

We called the spectral function for the Toeplitz operator with vertical symbol in the true-poly-Fock space.

Remark 3. Notice that we obtain (28) from (29) taking .

The following result is an extension of above theorem for Toeplitz operators with horizontal symbols acting on poly-Fock space.

Theorem 4. For any , the Toeplitz operator acting on is unitary equivalent to the matrix multiplication operator acting on , where the matrix-valued function is given by

That is,

for .

Proof. We have that the operator is unitary and using (25), we obtain that the Toeplitz operator is unitary equivalent to the following operators:

Now calculate the explicit expression of the above operator

where and is given by (22).
Therefore we obtain that each component of is given by (33), which proves the theorem.

Remark 5. The component function (33) is equal to , where denotes the convolution in , and . From [7][p.283, 32.45] it is guaranteed that the function belongs to where is the set of uniformly continuous functions in .

4. Description of the -Algebra Generated by Toeplitz Operators with Extended Horizontal Symbols

Denote by the closed subspace of which consists of all functions having limit values at the “endpoints” and ; i.e., for each the following limits exist

We will identify the functions with their extensions to the complex plane , where . We shall say that is an extended horizontal symbol.

In this section we study the -algebra generated by all the Toeplitz operators on with extended horizontal symbols.

Definition 6. We define some -algebras that we will use in this paper.

(i)Denote by the set of horizontal spectral functions given by

(ii)Denote by the set of horizontal spectral matrix-valued functions given by

(iii)Denote by the -algebra generated by all the Toeplitz operators acting on the true-poly-Fock space , with .(iv)Denote by the -algebra generated by all the Toeplitz operators acting on the poly-Fock space , with .

Corollary 7. The -algebra is isometrically isomorphic to the -algebra generated by .

Corollary 8. The -algebra is isometrically isomorphic to the -algebra generated by .

The next lemma is important to describe the behavior at the infinity of the spectral matrix-valued-function related to Toeplitz operators with extended horizontal symbols on poly-Fock space.

Lemma 9. We consider a horizontal function , and let

Then the matrix-valued function satisfies

Proof. First we consider the case when (analogously we have ). We proceed to show that the limit value at of each entry of is equal to zero. We know that ; then

Let be a fixed number; hence there exists such that

Moreover, there exist such that for . Under the above assumptions, we estimate the value of each entry of as follows:

If and , then . By the above, it is clear that

which implied the result for the limit value at when . By similar argument the result is valid for the limit value at when .
Now we consider the case when . If , from the previous case we have that

By similar argument, we obtain . This completes the proof.

Recall that is the set of continuous functions that vanishes at infinity and is the set of continuous functions belonging to .

Remark 10. Let be an extended horizontal function; then

where and are given by (29) and (32), for , respectively. In particular, it is clear that for each .

We denote the algebra of all matrices with complex entries by and we define the -algebra that consists of the algebra of all matrices with entries in . We introduce the algebra , which is a -subalgebra of defined by

It is clear that is a -subalgebra of . We want to prove that , where is an algebra of type I. Thus, using a Stone-Weierstrass theorem [8], we just need to show that separates all the pure states of . Now, we proceed to describe the pure states of .

Notice that is a -bundle and the fibers are given by

Moreover, the set of its pure states of is determined by the pure states on the fibers; i.e., each pure state of has the form

where , and is a pure state of ; see [9] for more details.

In [10], T. K. Lee characterizes the set of all states of the matrix algebra . The author shows that each pure state of is given by a functional defined as

where and denotes the usual inner product on . Moreover, if such that then , where and .

In consequence, we have that the set of pure states of consists of all functionals of the form

where and .

In the case when or , we just have one pure state which can be realized by for each ; i.e.,

for all .

Now, we will show that separates all the pure states of . For this task we are going to use horizontal symbols of the form , where is the characteristic function of the set . Thus for this function the spectral matrix-valued-function has the form

To simplify the notation we made the following convention: , where and .

Let ; we define the function , whose explicit form is given by

In particular, from Lemma 9 we have that and . Moreover, it is clear that for all . Hence we separated the unique pure state at (or ) from every other pure state of the -algebra .

Now we define the function for . Notice that this function can be expressed as , where

is a polynomial of degree at most nonnegative valued.

The following lemma provides us a tool to show that the -algebra separates the pure states of of the form , where with and .

Lemma 11. We assume that , , and , where . If we have that for all , then and for all

Proof. By hypothesis we have that , which is equivalent to the following:

where is given by (55). Taking the derivative with respect to of the above equation yields

for all . We can rewrite the above equation as

We know that are polynomials, which implies that the above equation is valid if and only if the exponential part is constant with respect to . Hence we obtain that ; using this fact it is clear that for all . Therefore we have that and which is equivalent to for all .

Remark 12. A consequence of the above lemma is that if and , then there exist such that ; i.e., the spectral matrix-valued-function separated the pure states and .

Let be real numbers different from each other and recall that . We define the matrix where the row is equal to . Thus

where is the diagonal matrix given by . Notice that is the leading coefficient of ; thus

Now, we calculate the determinant of using the properties of multilineality, alternativity, and Vandermonde’s formula; we have the following relations:

To complete the proof of the fact that the -algebra separates all the pure states of , only missing step is to separate the pure states of the forms and , where and .

Lemma 13. Given and being fixed, consider the spectral matrix-valued-functions for all , which are given by (32). If for all , then , where and .

Proof. From Lemma 11, we have for all , which implies that there exist a function such that

For each , we define the function given by

Without loss of generality we can assume that , just to simplify the calculations. Thus we calculate the second derivative of with respect to and ; we obtain

By hypothesis, we have that which implies that ; using this fact and (62) and (64) we obtain

for all .
It is clear that there exist fixed such that . Thus, turn out the above equation as follows:

for all .
Notice that is nonzero polynomial with respect to ; thus the above equation implies that the function is constant. From (62), we obtain that for all . Using (61) it is clear that , which implies the result.

Remark 14. The above result completes the proof that the algebra separates to all pure states of .

The noncommutative Stone-Weierstrass conjecture: let be a -subalgebra of a -algebra , and suppose that separates all the pure states of (and if is nonunital). Then .

This conjecture for a -algebra type I was proved by I. Kaplansky in [8]. In consequence, we have proved that the algebra is equal to . From Corollary 8 we have that the algebra of Toeplitz operators is isometric and isomorphic to algebra . In summary, we have the following result.

Theorem 15. The -algebra is isomorphic and isometric to the -algebra . The isomorphism is given by

where is given in (32).

Corollary 16. The -algebra is isomorphic and isometric to the commutative -algebra . The isomorphism is given by

where is given in (29).

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that there are no conflicts of interest regarding the publication of this article.

Acknowledgments

This work was partially supported by the Conacyt Project, México.

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