Abstract
We introduce the so-called extended Lagrangian symbols, and we prove that the -algebra generated by Toeplitz operators with these kind of symbols acting on the homogeneously poly-Fock space of the complex space is isomorphic and isometric to the -algebra of matrix-valued functions on a certain compactification of obtained by adding a sphere at the infinity; moreover, the matrix values at the infinity points are equal to some scalar multiples of the identity matrix.
1. Introduction
Let , the one-dimensional poly-Fock space consists of all -analytic functions which satisfy where is the Gaussian measure in and is the Euclidian measure in . Further, the one-dimensional true poly-Fock space of order is given by
In the case of several variables, for , the -dimensional Gaussian measure in is given by , where is the Euclidian measure in . We have that the space is the tensorial product of components and the Fock space is
Given a multi-index , the poly-Fock space of order is given by
Similarly, the true poly-Fock space is
In [1], Vasilevski introduced the poly-Fock spaces over and he obtained the following decomposition formula:
Moreover, he showed that the true poly-Fock space is isomorphic and isometric to , where is the one-dimensional space generated by the function and each is a Hermite’s function in .
Another treatment of the poly-Fock spaces can be found in [2], where the author characterized all lattice sampling and interpolation sequences in the poly-Fock spaces. He introduced the polyanalytic Bargmann transform from vector-valued Hilbert spaces to poly-Fock spaces, and he showed the duality between sampling and interpolation in polyanalytic spaces and multiple interpolation and sampling in analytic spaces.
The Toeplitz operators acting on the Fock space have been investigated by several authors. For example, in [3], the authors studied Toeplitz operators acting on the one-dimensional Fock space and on true poly-Fock space whose symbols are bounded radial functions that have a finite limit at the infinity. They considered an orthonormal basis of normalized complex Hermite polynomials to prove that the radial operators are diagonal. In [4], the authors studied Toeplitz operators acting on the one-dimensional poly-Fock space with horizontal symbols such that the limit values at and exist. They proved that the -algebra generated with this class of symbols is isomorphic to the -algebra of functions on with values on the matrices, whose limit value at and are equal to some scalar multiples of the identity matrix. In [5], the authors introduced the Toeplitz operators with -invariant symbols over the Fock space for a Lagrangian plane , and they proved that the corresponding -algebra generated is isometric to the -algebra generated by Toeplitz operators with horizontal symbols.
On the other hand, the spaces of homogeneously polyanalytic functions have been studied recently. For example, in [6], the authors computed the reproducing kernel of the Bergman space of homogeneously polyanalytic functions on the unit ball in and on the Siegel domain.
The main result of this paper is the following: the -algebra generated by Toeplitz operators with extended Lagrangian symbols acting on the homogeneously poly-Fock space over is isomorphic and isometric to the -algebra of matrix-valued functions on a certain compactification of with the sphere at the infinity; moreover, the values at the infinity points are scalar multiplies of the identity matrix.
This paper is organized as follows. In Section 2, we define the so-called homogeneously poly-Fock space and study some of its properties. In Section 3, we prove that every Toeplitz operator with a horizontal symbol acting on the poly-Fock (or homogeneously poly-Fock) space is unitary equivalent to a multiplication operator by a matrix-valued function. In Section 4, we introduce the concept of extended horizontal symbol and we describe the -algebra generated by Toeplitz operators with this kind of symbols acting on both the poly-Fock space and the homogeneously poly-Fock space. Finally, in Section 5, we define the extended Lagrangian symbols and we prove that the -algebra generate by Toeplitz operators with these symbols acting on the homogeneously poly-Fock space is isomorphic to the -algebra generated by Toeplitz operators with horizontal symbols acting on the same space.
2. Poly-Fock Spaces over
In this section, we define the homogeneously poly-Fock space and we review some facts about the classic poly-Fock spaces.
For , we denote , , where . The Fock space is given by
We have that is a Hilbert space with the usual inner product of functions:
Moreover, the Fock space is a reproducing kernel Hilbert Space, whose reproducing kernel is given by
For the multi-index , we recall the operations , , and , with Also, for , we have . We consider the set with the lexicographic order. Finally, for , we write . For the multi-index , sometimes, we write instead of as long as no there is no confusion.
Let be a multi-index and consider the poly-Fock space defined in (5), since every one-dimensional poly-Fock space is a direct sum of true poly-Fock spaces whose order is less than or equal to , see [1], p. 5-6, we have
Note that the number of components in (12) is equal to .
Now, let be a natural number such that .
Definition 1. The homogeneously poly-Fock space of order over is given by The number of multi-indices whose absolute value is exactly is equal to .
Definition 2. The poly-Fock space of order in is given by The number of multi-indices whose absolute value is less than or equal to is equal to .
Remark 3 (see [6], Proposition 2.7). The authors introduced the concept of homogeneously polyanalytic function; this concept is very important in the development of this paper. Also, they proved that homogeneously polyanalytic spaces are invariant under linear change of variables.
In [1], Vasilevski applied the “creation” and “annihilation” operators in the Fock spaces and he proved the following results: (1)All true poly-Fock spaces are isomorphic one to each other(2)The explicit expression of the functions in the true poly-Fock space is given bywhere and (3)The reproducing kernel of the true poly-Fock space can be obtained applying the “creation” operator to the reproducing kernel of the Fock space
Remark 4. Using the creation operator defined in [1], we have that the homogeneously poly-Fock space and the poly-Fock space are reproducing kernel Hilbert spaces.
On the other hand, for and , consider the function introduced in (8): where and each is Hermite’s function in . The set , with forms an ortonormal base in the space . We denote to the one-dimensional space generated by the function , whose orthogonal projection is given by
Now, we consider the operators with for , , and . See [1], the operator from the space onto satisfies that the image of the true poly-Fock space is and the orthogonal projection of onto is unitary equivalent to the operator
Analogously, for and , we define the spaces with respective orthogonal projections
If we denote by , , and the orthogonal projections of onto , and , respectively; thus,
Consider the isometric inmersion , defined by Whose adjoint operator is given by These operators satisfy the following relations:
Now, we define the operator from onto , and the adjoint operator , which satisfy
For the multi-index , remember that , so we introduce the isometric inmersion , defined by where the -tuples , and are taken over all multi-indices whose entries are less than or equal to each of the corresponding entries of . The adjoint operator is given by
We have
The operator from onto and its adjoint satisfy
Similarly to (27), we define the operators
The arrangement is formed by the functions over all multi-indices such that Analogously, the functions in are indexed with the multi-indices whose absolute value are less or equal to . Remember that and . The adjoint operators and are defined similarly to (28). Finally, the operators , , and its adjoint operators satisfy
3. Toeplitz Operators with Horizontal Symbols
In this section, we define Toeplitz operators with certain class of symbols acting on the poly-Fock, true poly-Fock, and homogeneously poly-Fock spaces over . And, we prove that this operators are unitary equivalent to certain multiplication operators. Let be a function in depending only on , we call to this kind of functions horizontal symbols. Henceforth, denote a fixed multi-index and a fixed natural number.
Definition 5. Let be a horizontal symbol. The Toeplitz operator with symbol , acting on the true poly-Fock space (or poly-Fock space) of order is defined as Similarly, the Toeplitz operator with symbol , acting on the homogeneously poly-Fock space (or poly-Fock space) of order is defined as
The following theorem characterizes the Toeplitz operators with horizontal symbols acting on the true poly-Fock space .
Theorem 6. Let be a horizontal symbol, then the Toeplitz operator acting on is unitary equivalent to the multiplication operator acting on where the function is given by and is defined in (8).
Proof. Remember that and using (25) and (26), we obtain
Explicitly for a function , we have
We call to the function the spectral function for the Toeplitz operator with horizontal symbol acting on the true poly-Fock space .
Naturally, we can extend the above result to the case of the Toeplitz operator with horizontal symbols acting on the poly-Fock space .
Theorem 7. Let be a horizontal symbol; thus, the Toeplitz operator acting on is unitary equivalent to the multiplication operator , acting on , where the matrix is given by That is, each component function is equal to with such that and is defined in (27).
Proof. Since and using (30) and (31), we obtain Calculating for a function We have the next two theorems, whose proofs are analogous to the above one.
Theorem 8. For a horizontal symbol the Toeplitz operator acting on the homogeneously poly-Fock space is unitary equivalent to the multiplication operator , acting on , where
Theorem 9. For a horizontal symbol , the Toeplitz operator acting on the poly-Fock space is unitary equivalent to the multiplication operator , acting on , where
We call to the matrices , , and the spectral matrices correspondent to the Toeplitz operator with horizontal symbol acting on the poly-Fock space , on the homogeneously poly-Fock space , and on the poly-Fock space respectively.
Remark 10. The components of the spectral matrices , and can be expressed as a convolution of functions , where . Since , from [7], p. 283, we guaranteed that they belong in the set of uniformily continuous functions .
4. The -Algebras Generated by Toeplitz Operators with Extended Horizontal Symbols
In this section, we introduce the concept of extended horizontal symbol and we describe the -algebras generated by Toeplitz operators with these symbols acting on the poly-Fock spaces and on the homogeneously poly-Fock spaces. Following the terminology and the notation introduced in [8], Section 3, we have the following.
Definition 11. Let be a horizontal symbol. We say that is an extended horizontal symbol if there exists a function such that We denote by the set of extended horizontal symbols. We note that equipped with the supremum norm is a -subalgebra of .
The compact of maximal ideals of the -algebra coincides with the compactification of , denoted by , obtained by adding an “infinitely far” -sphere . This compact space is isomorphic to We can identify the elements with the points as follows. For every extended horizontal symbol , we have
We identify the extended horizontal symbols with its extensions to the complex space , where .
The following lemma shows that the different spectral matrices , , and , corresponding to Toeplitz operators with extended horizontal symbol , posses a limit value to infinity in any direction. We write to refer to any of this spectral matrices.
Lemma 12. Let be an extended horizontal symbol and let . Then the spectral matrix satisfies
Proof. We apply the dominated convergence theorem. Let be two multi-indices corresponding to some entry of the spectral matrix. For each , we consider the function defined by Since , and we have . Note that the integrable function limits to for all Since the function is continuous, we have and using (48) We can take this limit along the line ; thus, and when Therefore, Let be a fixed multi-index and be a fixed natural number.
Definition 13. We introduce the following algebras, which are very useful to our study (i)Denote to the set of all horizontal spectral functions(ii)Denote to the set of all horizontal spectral matrices of order (iii)Denote to the set of all horizontal spectral matrices of order exactly (iv)Denote to the set of all horizontal spectral matrices of order at most (v)Denote by the -algebra generated by the set of Toeplitz operators acting on the true poly-Fock space , with (vi)Denote by the -algebra generated by the set of Toeplitz operators acting in the poly-Fock space , with .(vii)Denote by the -algebra generated by the set of Toeplitz operators acting on the homogeneously poly-Fock space , with (viii)Denote by the -algebra generated by the set of Toeplitz operators acting in the poly-Fock space , with
We have the following results.
Corollary 14. The -algebra is isometrically isomorphic to the -algebra generated by .
Corollary 15. The -algebra is isometrically isomorphic to the -algebra generated by .
Corollary 16. The -algebra is isometrically isomorphic to the -algebra generated by .
Corollary 17. The -algebra is isometrically isomorphic to the -algebra generated by .
Now, we describe the -algebras , , and , generated by the different spectral matrices. First, we start with . Consider the -algebra defined by , which consists of the algebra of all matrices with entries in , where . Now, we introduce the -subalgebra given by
We note that is a -subalgebra of . In fact, we prove that . For this, we use a Stone-Weirstrass theorem. We need to show that separates the pure states of .
Since is a -bundle, the set of all its pure states is completely determined by the pure states on the fibers:
So each pure state of has the form where and is a pure state of . Every pure state in the matrix algebra is given by a functional defined as with . Moreover, if such that ; thus, where and , see [9] for more details.
In consequence, the set of all pure states of consists of all functional of the form with and .
In the cases of the -algebras and , we consider the -algebras and . And their corresponding -subalgebras and . For this two -subalgebras, the pure states are determined in a similar way to (59). Remember that and .
Now to fixing ideas, we return to the previous case ; the other ones are totally analogous, and we analyzed them at the end of this section.
For each element , we have only one pure state for any , that is, . To separate the pure states corresponding to two different elements and in , using the identification given by (48), we note that the corresponding elements differ at least one coordinate. Suppose that the coordinate of this vectors is different, that is, Thus, we consider the horizontal extended symbol defined by
For with , we have Clearly, is a continuous function. Now, consider the spectral matrix , for every , we have
Hence, for , we have
Thus, the spectral matrix separates the corresponding pure states.
In the case when we have the pure states corresponding to the points and , we consider the set with , and the function . We have . We write instead of . Notice that
For , we have
Note that because , except in a set of measure zero. On the other hand, if is the corresponding element of , we have
Therefore, the spectral matrix separates the pure states of the points and .
To separate the pure states corresponding to two points , we consider again the extended horizontal symbol From (64), we define the function We can express this function as with
where is the product of the one-dimensional Hermite polynomials , so is a nonnegative-valued polynomial of degree at most .
The following lemma provides us a tool to prove that the -algebra separates the pure states of of the form , where and
Lemma 18. We assume that and with . If for all vector , then . Moreover, for all.
Proof. The hypothesis is equivalent to the following equation:
Taking the partial derivative , we obtain
thus,
Since and are polynomials, this fact implies that the exponential part in the above equation is constant for all . Hence, for all ; therefore, . Using this fact, it is clear that for all , that is, for all .
As consequence of the above lemma, if and , then there exists such that ; hence, the spectral matrix separates the pure states and .
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 . For this, we need to deduce some useful facts before.
For the multi-index , we consider . From (8), we have , where is a one-dimensional Hermite function. We can consider that all correspondent one-dimensional Hermite’s polynomials , whose degrees are equal to , are monic polynomials. Hence, for every multi-index such that , we can write
Now, for , we construct the vector with the form
Evaluating this vector in Hermite’s polynomials corresponding to the multi-index we obtain a polynomial dependents on only one variable, whose degree we can calculate with the equation:
Consequently, evaluating in , we can write
From (73), we notice that for two different multi-indices and , the corresponding degrees satisfy . Moreover, for , we have . And for ,
Therefore, the multi-indices such that for every generate different polynomials of degrees between and . Now, for each of these multi-indices , we consider vectors defined by (71), and we define the matrix whose dimension is and the row is equal to . Since the components of are sorted ascending by the lexicographic order, we claim that the matrix has the form: where
We can calculate the determinant of using multilineality and the VandermondeÂ’s formula; we obtain
Example 1. Consider and the multi-index . We have and the multi-indices, arranged with the lexicographic order, whose coordinates are less or equal to the corresponding coordinates of are In this case, the vector . From (73), we can calculate the different degrees, for example for , And for the rest of multi-indices, we have Now, we can continue with the separation of the pure states.
Lemma 19. Given and being fixed, consider the spectral matrices . If for all then , with and .
Proof. From Lemma 18, we have for all ; thus, there exists a function such that For , we define the function given by Without loss of generality, we can assume that , just to simplify the calculations. Taking the derivative of order of the function , with respect to and , we obtain The hypothesis implies that . Using (82) and (84), for all Clearly, there exists such that Thus, Notice that is a nonzero polynomial with respect to ; thus, the above equation implies that the function is constant. From (82), we obtain that for all . Using (78), we have and
Finally, we consider the case of the -algebras and . Since the proof of the Lemma 19 is independent of the dimension and the nature of the multi-indices and , we can obtain the following analogous results.
Lemma 20. For and with . If for all vector , then . Also, for all .
Lemma 21. For and with . If for all vector , then . Also, for all .
On the other hand, we consider the -algebra and the multi-index
If is a multi-index such that then all its components satisfy . So, if we construct the invertible matrix as in (76), corresponding to the multi-index , and apply it to the vector where the coordinates of occupy the same positions that the multi-indices whose absolute value is equal to in the matrix . Thus, we can prove the following.
Lemma 22. Given and being fixed, consider the spectral matrices . If for all then , with and .
Analogously, for the case of the -algebra and the set of multi-indices with absolute value is less than or equal to , we have the following.
Lemma 23. Given and being fixed, consider the spectral matrices . If for all then , with and .
For 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, .
In [10], Kaplansky proved this conjecture for a -algebra type I. In consequence, we prove that the algebra is equal to . From Corollary 15, we have that the algebra of Toeplitz operators is isometric and isomorphic to the algebra . Analogously, applying the Corollary 16 and the Corollary 17, we have that and are isometric and isomorphic to and , respectively. In summary, we have the following results.
Theorem 24. The -algebra is isomorphic and isometric to the -algebra . The isomorphism is given by where is given in (41).
Theorem 25. The -algebra is isomorphic and isometric to the -algebra . The isomorphism is given by where is given in (45).
Theorem 26. The -algebra is isomorphic and isometric to the -algebra . The isomorphism is given by where is given in (46).
Corollary 27. The -algebra is isomorphic and isometric to the commutative -algebra . The isomorphism is given by where is given in (38).
5. Toeplitz Operators with -Invariant Symbols
In this section, we introduce the extended Lagrangian symbols, and we prove that the -algebra generated by Toeplitz operators with this kind of symbols acting on the homogeneously poly-Fock space is isomorphic and isometric to the -algebra generated by Toeplitz operators with extended horizontal symbols acting on this same space.
We consider the standard symplectic form of given by where
Recall that a -dimensional subspace is called a Lagrangian plane if for every it satisfy Clearly, is a Lagrangian plane. We denote by Lag the set of all Lagrangian planes in . If we consider the transitive group action of onto Lag defined by we have that for every Lagrangian plane there is an unitary matrix such that For more details, see [11], Proposition 43. Since the unitary group is isomorphic to , each Lagrangian plane can be identified with a subspace of ; abusing the notation, we denote this subspace with too.
Let be a Lagrangian plane, we say that a function is -invariant or Lagrangian invariant if for every it satisfies so we can consider it like a function depending only on the elements of .
In [5], Esmeral and Vasilevski introduced the concept of -invariant functions and they provided the following criterion for a function to be so.
Lemma 28. Consider a Lagrangian plane and such that . Then, a function is -invariant if and only if there exists such that Moreover, they established the following result.
Proposition 29. The -algebra generated by Toeplitz operators with horizontal symbols acting on the Fock space is unitary equivalent to the -algebra generated by Toeplitz operators with -invariant symbols.
For this, they introduced the operator defined by
Since , this operator is unitary and . It too satisfies
where is the reproducing kernel of in the point .
In the case of the poly-Fock space and the true poly-Fock space, the above result could fail, because for some multi-index and some unitary matrix ; the spaces and might not be invariant under the operator .
Example 1. Consider , . Using (15), we have however, if with is an unitary matrix, it is clear that where , .
This is the main motivation for which we consider the homogeneously poly-Fock space and the poly-Fock space for .
Note that for Lag, such that , and defined by (97), using the explicit form of the elements in the true poly-Fock space given by (15), from [6], Proposition 2.7, we have that the homogeneously poly-Fock space and the poly-Fock space are invariant under
Now, we define the extended Lagrangian symbols; these kind of symbols is related with the extended horizontal symbols as follows.
Definition 30. For and such that , equivalently and , consider the following diagram.
We say that the -invariant symbol is an extended Lagrangian symbol or an extended -invariant symbol if its pullback by is an extended horizontal symbol. In other words, if the function given in (96), it is an extended horizontal symbol.
According to the above diagram, for , there exists such that and . If is an extended Lagrangian symbol, using (96) and (48), we have
If we define the continuous function given by we have for
This function is invariant under translations by Lagrangians elements whose norm is equal to 1. Let such that , so and ; thus, for , we have
Lemma 31. Consider the unitary operator for. Ifdenotes the reproducing kernel of the true homogeneously poly-Fock spacethen.
Proof. Let . Using the reproducing property, we can express Apply since is unitary and taking , we have On the other hand, for the uniqueness of the reproducing kernel, we have ; therefore
Corollary 32. The reproducing kernel of the poly-Fock space satisfies .
Now, consider and . Using the Lemma 31, we obtain that the Toeplitz operator acting on the true homogeneously poly-Fock space is unitary equivalent to
where . Analogously, by Corollary 32, the Toeplitz operator acting in the poly-Fock space is unitary equivalent to .
Using the above results, we obtain the following generalizations of Proposition 29.
Theorem 33. The -algebra generated by Toeplitz operators with extended horizontal symbols acting on the true homogeneously poly-Fock space is unitary equivalent to the -algebra generated by Toeplitz operators with extended -invariant symbols.
Theorem 34. The -algebra generated by Toeplitz operators with extended horizontal symbols acting in the poly-Fock space is unitary equivalent to the -algebra generated by Toeplitz operators with extended -invariant symbols.
Finally, using Theorems (25) and (26), we have the following.
Corollary 35. The -algebra generated by Toeplitz operators with extended -invariant symbols acting on the homogeneously poly-Fock space is isomorphic and isometric to the -algebra .
Corollary 36. The -algebra generated by Toeplitz operators with extended -invariant symbols acting on the poly-Fock space is isomorphic and isometric to the -algebra .
Data Availability
Data sharing is not applicable to this article as no datasets were generated or analyzed during the current study.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Acknowledgments
The authors have been partially supported by the Proyecto CONACYT 280732, by Consejo Nacional de Ciencia y Tecnología (CONACYT) (México) scholarships, and by Universidad Veracruzana.