Toeplitz Operators with Lagrangian Invariant Symbols Acting on the Poly-Fock Space of Cn

We introduce the so-called extended Lagrangian symbols, and we prove that the C∗-algebra generated by Toeplitz operators with these kind of symbols acting on the homogeneously poly-Fock space of the complex space Cn is isomorphic and isometric to the C∗-algebra of matrix-valued functions on a certain compactification of Rn 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.


Introduction
Let m ∈ ℕ, the one-dimensional m poly-Fock space F 2 m ðℂÞ ⊂ L 2 ðℂ, dμÞ consists of all m-analytic functions φ which satisfy where dμ = π −1 e −z· z dxdy is the Gaussian measure in ℂ and d xdy is the Euclidian measure in ℝ 2 = ℂ. Further, the onedimensional true poly-Fock space of order m is given by In the case of several variables, for n ∈ ℕ, the n-dimensional Gaussian measure in ℂ n is given by dμ n ðzÞ = π −n e −jzj 2 dxdy, where dxdy is the Euclidian measure in ℝ 2n . We have that the space L 2 ðℂ n , dμ n Þ is the tensorial product of n components and the Fock space F 2 ðℂ n Þ is Given a multi-index α = ðα 1 , ⋯, α n Þ ∈ ℤ n + , the poly-Fock space F 2 α ðℂ n Þ of order α is given by Similarly, the true poly-Fock space F 2 ðαÞ ðℂ n Þ is In [1], Vasilevski introduced the poly-Fock spaces over ℂ n and he obtained the following decomposition formula: Moreover, he showed that the true poly-Fock space F 2 ðαÞ ðℂ n Þ is isomorphic and isometric to L 2 ðℝ n , dxÞ ⊗H α−1 , whereH α−1 is the one-dimensional space generated by the function and each h α j −1 ðy j Þ is a Hermite's function in ℝ.
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 onedimensional 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 x = ∞ and x = −∞ exist. They proved that the C * -algebra generated with this class of symbols is isomorphic to the C * -algebra of functions on ℝ with values on the m × m matrices, whose limit value at x = ∞ and x = −∞ are equal to some scalar multiples of the identity matrix. In [5], the authors introduced the Toeplitz operators with L-invariant symbols over the Fock space F 2 ðℂ n Þ for a Lagrangian plane L, and they proved that the corresponding C * -algebra generated is isometric to the C * -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 ℂ n and on the Siegel domain.
The main result of this paper is the following: the C * -algebra generated by Toeplitz operators with extended Lagrangian symbols acting on the homogeneously poly-Fock space over ℂ n is isomorphic and isometric to the C * -algebra of matrix-valued functions on a certain compactification of ℝ n 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 matrixvalued function. In Section 4, we introduce the concept of extended horizontal symbol and we describe the C * -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 C * -algebra generate by Toeplitz operators with these symbols acting on the homogeneously poly-Fock space is isomorphic to the C * -algebra generated by Toeplitz operators with horizontal symbols acting on the same space.

Poly-Fock Spaces over ℂ n
In this section, we define the homogeneously poly-Fock space and we review some facts about the classic poly-Fock spaces.
Let α ∈ ℤ n + be a multi-index and consider the poly-Fock space F 2 α ðℂ n Þ defined in (5), since every one-dimensional poly-Fock space F 2 α j ðℂÞ is a direct sum of true poly-Fock spaces whose order is less than or equal to α j , see [1], p. 5-6, we have Note that the number of components in (12) is equal to PðαÞ = α 1 ⋯ α n . Now, let k ∈ ℕ be a natural number such that k ≥ n.
Definition 1. The homogeneously poly-Fock space of order k over ℂ n is given by Journal of Function Spaces The number of multi-indices whose absolute value is exactly k is equal to s ðkÞ = k − 1 Definition 2. The poly-Fock space of order k in ℂ n is given by The number of multi-indices whose absolute value is less than or equal to k 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 ψðzÞ in the true poly-Fock space F 2 ðαÞ ðℂ n Þ is given by where φðzÞ ∈ F 2 ðℂ n Þ and ∂ λ φ = ∂ jλj φ/∂z (3) The reproducing kernel of the true poly-Fock space F 2 ðαÞ ðℂ n Þ can be obtained applying the "creation" operator to the reproducing kernel of the Fock space F 2 ðℂ n Þ Remark 4. Using the creation operator defined in [1], we have that the homogeneously poly-Fock space F 2 ðkÞ ðℂ n Þ and the poly-Fock space F 2 k ðℂ n Þ are reproducing kernel Hilbert spaces.

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 ℂ n . And, we prove that this operators are unitary equivalent to certain multiplication operators. Let aðzÞ = aðx 1 , ⋯, x n Þ be a function in L ∞ ðℝ n , dxÞ depending only on x = ðRe z 1 , ⋯, Re z n Þ, we call to this kind of functions horizontal symbols. Henceforth, α ∈ ℤ n + denote a fixed multi-index and k ∈ ℕ a fixed natural number.
Definition 5. Let aðxÞ be a horizontal symbol. The Toeplitz operator with symbol aðxÞ, acting on the true poly-Fock space (or poly-Fock space) of order α is defined as Journal of Function Spaces Similarly, the Toeplitz operator with symbol aðxÞ, acting on the homogeneously poly-Fock space (or poly-Fock space) of order k is defined as The following theorem characterizes the Toeplitz operators with horizontal symbols acting on the true poly-Fock space F 2 ðαÞ ðℂ n Þ.

Theorem 6.
Let aðxÞ ∈ L ∞ ðℝ n , dxÞ be a horizontal symbol, then the Toeplitz operator T ðαÞ,a acting on F 2 ðαÞ ðℂ n Þ is unitary equivalent to the multiplication operator γ ðαÞ,a I =R ðαÞ T ðαÞ,ã R * ðαÞ acting on L 2 ðℝ n , dxÞ where the function γ ðαÞ,a is given by andh α−1 ðyÞ is defined in (8).
Proof. Remember thatR ðαÞ =R * 0,ðαÞ U and using (25) and (26), we obtaiñ Explicitly for a function f ∈ L 2 ðℝ n Þ, we havẽ We call to the function γ ðαÞ,a ðxÞ the α th spectral function for the Toeplitz operator with horizontal symbol a acting on the true poly-Fock space F 2 ðαÞ ðℂ n Þ. Naturally, we can extend the above result to the case of the Toeplitz operator with horizontal symbols acting on the poly-Fock space F 2 α ðℂ n Þ.

Theorem 7.
Let aðxÞ ∈ L ∞ ðℝ n , dxÞ be a horizontal symbol; thus, the Toeplitz operator T α,a acting on F 2 α ðℂ n Þ is unitary equivalent to the multiplication operator γ α,a ðxÞI =R α T α,ã R * α , acting on ðL 2 ðℝ n , dxÞÞ PðαÞ , where the matrix γ α,a is given by That is, each component function is equal to with λ, μ ∈ ℤ n + such that λ i , μ i ≤ α i and N α ðyÞ is defined in (27).
Proof. SinceR α =R * 0,α U and using (30) and (31), we obtaiñ Calculating for a function f ∈ ðL 2 ðℝ n , dxÞÞ PðαÞ We have the next two theorems, whose proofs are analogous to the above one.
Theorem 8. For a horizontal symbol aðxÞ ∈ L ∞ ðℝ n , dxÞ, the Toeplitz operator T ðkÞ,a acting on the homogeneously poly-Fock space F 2 ðkÞ ðℂ n Þ is unitary equivalent to the multiplication operator γ ðkÞ,a ðxÞI =R ðkÞ T ðkÞ,aR * ðkÞ , acting on ðL 2 ðℝ n , dxÞÞ s ðkÞ , where Theorem 9. For a horizontal symbol aðxÞ ∈ L ∞ ðℝ n , dxÞ, the Toeplitz operator T k,a acting on the poly-Fock space F 2 k ðℂ n Þ is unitary equivalent to the multiplication operator γ k,a ðxÞI =R k T k,aR * k , acting on ðL 2 ðℝ n , dxÞÞ s k , where We call to the matrices γ α,a ðxÞ, γ ðkÞ,a ðxÞ, and γ k,a ðxÞ the spectral matrices correspondent to the Toeplitz operator with horizontal symbol aðxÞ, acting on the poly-Fock space 5 Journal of Function Spaces F 2 α ðℂ n Þ, on the homogeneously poly-Fock space F 2 ðkÞ ðℂ n Þ, and on the poly-Fock space F 2 k ðℂ n Þ, respectively.

The C * -Algebras Generated by Toeplitz Operators with Extended Horizontal Symbols
In this section, we introduce the concept of extended horizontal symbol and we describe the C * -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 aðxÞ ∈ L ∞ ðℝ n Þ be a horizontal symbol. We say that aðxÞ is an extended horizontal symbol if there exists a function a ∞ ðxÞ ∈ CðS n−1 Þ such that We denote by HSðℝ n Þ the set of extended horizontal symbols. We note that HSðℝ n Þ equipped with the supremum norm is a C * -subalgebra of L ∞ ðℝ n Þ.
The compact of maximal ideals of the C * -algebra HS ðℝ n Þ coincides with the compactification of ℝ n , denoted by e ℝ n = ℝ n ∪ S n−1 ∞ , obtained by adding an "infinitely far" n-sphere S n−1 ∞ . This compact space is isomorphic to D n : We can identify the elements s ∞ ∈ S n−1 ∞ with the points s ∈ S n−1 as follows. For every extended horizontal symbol a ∈ H Sðℝ n Þ, we have We identify the extended horizontal symbols aðxÞ with its extensions to the complex space ℂ n , where x = Re z.
The following lemma shows that the different spectral matrices γ α,a ðxÞ, γ ðkÞ,a ðxÞ, and γ k,a ðxÞ, corresponding to Toeplitz operators with extended horizontal symbol aðxÞ, posses a limit value to infinity in any direction. We write γ □,a ðxÞ to refer to any of this spectral matrices.
Lemma 12. Let aðxÞ be an extended horizontal symbol and let x 0 ∈ S n−1 . Then the spectral matrix γ □,a ðxÞ satisfies Proof. We apply the dominated convergence theorem. Let λ, μ ∈ ℤ n + be two multi-indices corresponding to some entry of the spectral matrix. For each m ∈ ℕ, we consider the function F m : ℝ n ⟶ ℝ defined by Sinceh λ−1 ðyÞ,h μ−1 ðyÞ ∈ L 2 ðℝ n Þ, and aðxÞ ∈ L ∞ ðℝ n Þ we have F m ðyÞ ∈ L 1 ðℝ n Þ. Note that the integrable function kak ∞ kh λ−1 ðyÞh μ−1 ðyÞk limits to F m ðyÞ for all m ∈ ℕ: Since the function a ∞ is continuous, we have and using (48) We can take this limit along the line t = m; thus, and F m ðyÞ ⟶ a ∞ ðxÞh λ−1 ðyÞh μ−1 ðyÞ when m ⟶ ∞: Therefore, Let α = ðα 1 , ⋯, α n Þ ∈ ℤ n + be a fixed multi-index and k ∈ ℕ be a fixed natural number.
Definition 13. We introduce the following C * algebras, which are very useful to our study (i) Denote G H ðαÞ = fγ ðαÞ,a : aðxÞ ∈ HSðℝ n Þg to the set of all horizontal spectral functions (ii) Denote G H α = fγ α,a : aðxÞ ∈ HSðℝ n Þg to the set of all horizontal spectral matrices of order α (iii) Denote G H ðkÞ = fγ ðkÞ,a : aðxÞ ∈ HSðℝ n Þg to the set of all horizontal spectral matrices of order exactly k (iv) Denote G H k = fγ k,a : aðxÞ ∈ HSðℝ n Þg to the set of all horizontal spectral matrices of order at most k (v) Denote by T ðαÞ ∞ the C * -algebra generated by the set of Toeplitz operators T ðαÞ,a acting on the true poly-Fock space F 2 ðαÞ ðℂ n Þ, with aðxÞ ∈ HSðℝ n Þ 6 Journal of Function Spaces (vi) Denote by T α ∞ the C * -algebra generated by the set of Toeplitz operators T α,a acting in the poly-Fock space F 2 α ðℂ n Þ, with aðxÞ ∈ HSðℝ n Þ.
(vii) Denote by T ðkÞ ∞ the C * -algebra generated by the set of Toeplitz operators T ðkÞ,a acting on the homogeneously poly-Fock space F 2 ðkÞ ðℂ n Þ, with aðxÞ ∈ HSðℝ n Þ (viii) Denote by T k ∞ the C * -algebra generated by the set of Toeplitz operators T k,a acting in the poly-Fock space F 2 k ðℂ n Þ, with aðxÞ ∈ HSðℝ n Þ We have the following results.

Corollary 14.
The C * -algebra T ðαÞ ∞ is isometrically isomorphic to the C * -algebra G H ðαÞ generated by G H ðαÞ .
Corollary 15. The C * -algebra T α ∞ is isometrically isomorphic to the C * -algebra G H α generated by G H α .
Corollary 16. The C * -algebra T ðkÞ ∞ is isometrically isomorphic to the C * -algebra G H ðkÞ generated by G H ðkÞ .
Now, we describe the C * -algebras G H α , G H ðkÞ , and G H k , generated by the different spectral matrices. First, we start with G H α . Consider the C * -algebra defined by C α = M PðαÞ ðℂÞ ⊗ Cðℝ n ∪ S n−1 ∞ Þ, which consists of the algebra of all P ðαÞ × PðαÞ matrices with entries in Cðℝ n ∪ S n−1 ∞ Þ, where PðαÞ = α 1 ⋯ α n . Now, we introduce the C * -subalgebra D α given by We note that G H α is a C * -subalgebra of D α . In fact, we prove that G H α = D α . For this, we use a Stone-Weirstrass theorem. We need to show that G H α separates the pure states of D α .
Since D α is a C * -bundle, the set of all its pure states is completely determined by the pure states on the fibers: So each pure state of D α has the form where x ∈ ℝ n ∪ S n−1 ∞ and f x is a pure state of D α ðxÞ. Every pure state in the matrix algebra M PðαÞ ðℂÞ is given by a functional f v defined as with v ∈ S PðαÞ = fz ∈ ℂ PðαÞ : jzj = 1g. Moreover, if v, w ∈ S PðαÞ such that f v = f w ; thus, v = tw where t ∈ ℂ and jtj = 1, see [9] for more details.
In consequence, the set of all pure states of D α consists of all functional of the form with x ∈ ℝ n ∪ S n−1 ∞ and v ∈ S PðαÞ . In the cases of the C * -algebras G H ðkÞ and G H k , we consider the C * -algebras C ðkÞ = M s ðkÞ ðℂÞ ⊗ Cðℝ n ∪ S n−1 ∞ Þ and C k = M s k ðℂÞ ⊗ Cðℝ n ∪ S n−1 ∞ Þ. And their corresponding C * -subalgebras D ðkÞ and D k . For this two C * -subalgebras, the pure states are determined in a similar way to (59). Remember Now to fixing ideas, we return to the previous case G H α ; the other ones are totally analogous, and we analyzed them at the end of this section.
For each element x ∞ ∈ S n−1 ∞ , we have only one pure state for any v, w ∈ S PðαÞ , that is, f x ∞ ,v = f x ∞ ,w . To separate the pure states corresponding to two different elements x 0,∞ and x 1,∞ in S n−1 ∞ , using the identification given by (48), we note that the corresponding elements x 0 , x 1 ∈ S n−1 differ at least one coordinate. Suppose that the j th coordinate of this vectors is different, that is, x 0,j ≠ x 1,j : Thus, we consider the horizontal extended symbol C j : ℝ n ⟶ ℂ defined by , in otherwise: For x = ðx 1 , ⋯, x n Þ ∈ ℝ n with kxk = 1, we have C j ∞ ðxÞ = lim t⟶∞ C j ðtxÞ = x j : Clearly, C j ∞ is a continuous function. Now, consider the spectral matrix γ α,C j , for every x ∞ , we have Hence, for x 0,∞ ≠ x 1,∞ , we have Thus, the spectral matrix γ α,C j ðxÞ separates the corresponding pure states.
In the case when we have the pure states corresponding to the points x 0 ∈ ℝ n and x 1,∞ ∈ S n−1 ∞ , we consider the 7 Journal of Function Spaces set ½0, p = ½0, p 1 × ⋯× ½0, p n with p = ðp 1 , ⋯, p n Þ ∈ ðℝ + Þ n , and the function cðxÞ = χ ½0,p ðxÞ. We have c ∞ ðxÞ ≡ 0. We write γ α,p ðxÞ instead of γ α,c ðxÞ. Notice that For x 0 ∈ ℝ n , we have Note that f x 0 ,v ðγ α,p Þ > 0 because j<v, N α ðyÞ > j 2 ≥ 0, except in a set of measure zero. On the other hand, if x 1 ∈ S n−1 is the corresponding element of x 1,∞ ∈ S n−1 ∞ , we have Therefore, the spectral matrix γ α,p separates the pure states of the points x 0 and x 1,∞ .
The following lemma provides us a tool to prove that the C * -algebra G H α separates the pure states of D α of the form f x 0 ,v , f x 1 ,w , where x 0 ≠ x 1 and v, w ∈ S PðαÞ : Lemma 18. We assume that v, w ∈ S PðαÞ ,x 0 , x 1 ∈ ℝ n and γ α, Now, for y ∈ ℝ − f0g, we construct the vector y ∈ ℝ n with the form y = y α 2 ⋯α n , y α 3 ⋯α n , ⋯, y α n−1 ·α n , y α n , y ð Þ : Evaluating this vector in Hermite's polynomials corresponding to the multi-index λ = ðλ 1 , ⋯, λ n Þ we obtain a polynomial dependents on only one variable, whose degree we can calculate with the equation: Consequently, evaluating y inh λ−1 , we can writẽ From (73), we notice that for two different multiindices λ and μ, the corresponding degrees satisfy p λ ≠ p μ . Moreover, for 1 n = ð1, ⋯, 1Þ, we have p 1 n = 0. And for α = ðα 1 , ⋯, α n Þ, Therefore, the multi-indices λ ∈ ℤ n + such that λ i ≤ α i for every i generate different polynomials of degrees between 0 and PðαÞ − 1. Now, for each of these multi-indices λ, we consider vectors y λ defined by (71), and we define the matrix N whose dimension is PðαÞ × PðαÞ and the λ th row is equal to N α ðy λ Þ. Since the components of N α ðyÞ are sorted ascending by the lexicographic order, we claim that the matrix N has the form: ð78Þ Example 1. Consider n = 3 and the multi-index α = ð2, 4, 1Þ.
We have PðαÞ = 8 and the multi-indices, arranged with the lexicographic order, whose coordinates are less or equal to the corresponding coordinates of α are

Toeplitz Operators with L-Invariant Symbols
In this section, we introduce the extended Lagrangian symbols, and we prove that the C * -algebra generated by Toeplitz operators with this kind of symbols acting on the homogeneously poly-Fock space is isomorphic and isometric to the C * -algebra generated by Toeplitz operators with extended horizontal symbols acting on this same space. We consider the standard symplectic form ω 0 of ℂ n = ℝ 2n given by ω 0 ðz, wÞ = Jz · w, forallz, w, where Recall that a n-dimensional subspace L ⊂ ℝ 2n is called a Lagrangian plane if for every z, w ∈ L it satisfy ω 0 ðz, wÞ = 0: Clearly, iℝ n = f0g × ℝ n is a Lagrangian plane. We denote by Lagð2n, ℝÞ the set of all Lagrangian planes in ℝ 2n . If we consider the transitive group action of Uð2n, ℝÞ onto Lagð2n, ℝÞ defined by we have that for every Lagrangian plane L there is an unitary matrix X such that XL = iℝ n : For more details, see [11], Proposition 43. Since the unitary group Uð2n, ℝÞ is isomorphic to Uðn, ℂÞ, each Lagrangian plane L can be identified with a subspace of ℂ n ; abusing the notation, we denote this subspace with L too. Let L be a Lagrangian plane, we say that a function φ ∈ L ∞ ðℂ n Þ is L-invariant or Lagrangian invariant if for every h ∈ L it satisfies so we can consider it like a function depending only on the elements of L ∁ . In [5], Esmeral and Vasilevski introduced the concept of L-invariant functions and they provided the following criterion for a function to be so.
Lemma 28. Consider a Lagrangian plane L and X ∈ Uðn, ℂÞ such that XL = iℝ n . Then, a function φ ∈ L ∞ ðℂ n Þ is L -invariant if and only if there exists a ∈ L ∞ ðℝ n Þ such that φ X * z ð Þ= a Re z 1 , ⋯, Re z n ð Þ , for almost all z ∈ ℂ n : ð96Þ Moreover, they established the following result.
Proposition 29. The C * -algebra generated by Toeplitz operators with horizontal symbols acting on the Fock space F 2 ð ℂ n Þ is unitary equivalent to the C * -algebra generated by Toeplitz operators with L-invariant symbols.