Toeplitz operators on two poly-Bergman-type spaces of the Siegel domain $D_2 \subset \mathbb{C}^2$ with continuous nilpotent symbols

1.   Introduction

In recent years, the theory of Toeplitz operators has been generalized from Bergman spaces of square-integrable holomorphic functions to poly-Bergman spaces of square-integrable polyanalytic functions ^[1,2]. Bianalytic functions emerged in the mathematical theory of elasticity, but the mathematical relevance of more general polyanalytic functions was soon realized ^[3].

Similar to the study of Toeplitz operators on spaces of analytic functions, we select a set of symbols $E \subset L^{\infty}$ in such a way that the $C^*$-algebra generated by Toeplitz operators with symbols in $E$ can be explicitly described up to isomorphism, that is, as an algebra of matrix-valued functions. For the unit disk $\mathbb{D}$ and the Siegel domain $D_n \subset \mathbb{C}^n$, in ^[4,5,6] the authors considered the set $E_G$ of symbols invariant under the action of a maximal Abelian subgroup $G$ of biholomorphisms, they found that the $C^*$-algebra $\mathcal{T}_G$ generated by Toeplitz operators acting on the Bergman spaces with symbols in $E_G$ is commutative. The authors also proved that $\mathcal{T}_G$ is isomorphic and isometric to a $C^*$-subalgebra of continuous functions on a locally compact Hausdorff space $X$. Of course, $\mathcal{T}_G$ is isomorphic to $C(\sigma(\mathcal{T}_G))$, where $\sigma(\mathcal{T}_G)$ is the spectrum of $\mathcal{T}_G$. Unfortunately, $X$ is far away to be $\sigma(\mathcal{T}_G)$. In this sense, we can note that the full description of a commutative $C^*$-algebra depends on the spectrum of the algebra. It makes sense to select a smaller class $E \subset E_G$ in order to fully describe the $C^*$-algebra $\mathcal{T}_E$ generated by the Toeplitz operators with symbols in $E$. For the case of the Siegel domain $D_2$, in ^[7] the authors choose a family $E$ of piece-wise continuous symbols invariant under a nilpotent group and prove that the spectrum of the $C^*$-algebra $\mathcal{T}_E$ is a compactification of the upper half-plane $\Pi = \{z = x+iy \in \mathbb{C} \mid y > 0\}$. In this work, we extend the results in ^[7] by studying the noncommutative $C^*$-algebra generated by Toeplitz operators acting on two type-poly-Bergman spaces of the Siegel domain $D_2$, where it is possible to identify the space of all irreducible representation of such algebra. In general, the spectrum of a commutative $C^*$-algebra generated by Toeplitz operators is too large, and it is impossible to have a full description of it. In ^{[8,9,10,11,12]}, we can find outstanding contributions about the spectrum of commutative algebras and the spectrum of operators acting on Hilbert spaces with reproducing kernel, where the Berezin transform play a significant role.

In the case of the upper half-plane $\Pi$, a vertical symbol is a bounded measurable function $a(z)$ depending only on $y = {\rm Re}\, z$. Taking vertical symbols, in ^[13,14,15] the authors studied Toeplitz operators acting on the true-poly-Bergman space $\mathcal{A}^{2}_{(n)}(\Pi)$ from the point of view of wavelet spaces. Toeplitz operators with vertical symbols acting on poly-Bergman-type spaces have also be studied. Taking vertical symbols with limits at $y = 0$ and $y = \infty$, in ^[13,16] the authors described the $C^*$-algebra generated by all Toeplitz operators on the poly-Bergman space $\mathcal{A}^{2}_{n}(\Pi)$; this algebra is isomorphic to a subalgebra of $M_n(\mathbb{C}) \otimes C[0, +\infty]$. Similar research has studied Toeplitz operators on poly-Bergman spaces with homogeneous symbols (^[17,18]). Taking horizontal symbols having one-sided limits at $x = \pm \infty$, in ^[19,20] the authors studied Toeplitz operators acting on poly-Fock spaces $F_{k}^{2}(\mathbb{C})$ and they proved that the $C^*$-algebra generated by such Toeplitz operators is isomorphic to a subalgebra of $M_n(\mathbb{C}) \otimes C[-\infty, +\infty]$. In ^{[21,22,23,24]}, the authors studied the decomposition of the von Neumann algebra of radial operators acting on the poly-Fock spaces $F_{k}^{2}(\mathbb{C})$ and weighted poly-Bergman spaces $\mathcal{A}^{2}_{n}(\mathbb{D})$.

In ^[4,5], the authors found all classes of bounded measurable symbols associated to commutative algebras generated by Toeplitz operators acting on the Bergman space of the Siegel domain $D_{n} \subset \mathbb{C}^n$. In particular, they studied the $C^*$-algebra ${\cal T}_{{\cal N}_n}$ generated by all Toeplitz operators with bounded nilpotent symbols, which are functions of the form $\tilde{c}(\zeta) = c({\rm{Im}} \zeta_{1}, ..., {\rm{Im}} \zeta_{n-1}, {\rm{Im}} \zeta_{n} - |\zeta'|^{2})$, where $\zeta' = (\zeta_1, ..., \zeta_{n-1})$. Furthermore, in ^[7] the authors studied Toeplitz operators on the true-poly-Bergman-type space $\mathcal{A}^{2}_{(L)}(D_{2})$, with nilpotent symbols of the form $\tilde{a}(\zeta) = a({\rm{Im}} \zeta_1)$ and $\tilde{b}(z) = b({\rm{Im}} \zeta_2 -|\zeta_1|^{2})$. In this paper, we consider two poly-Bergman-type spaces of the Siegel domain $D_{2} \subset \mathbb{C}^{2}$, in which Toeplitz operators with continuous nilpotent symbols are studied. The main purpose of this work is to describe the $C^*$-algebra generated by all the Toeplitz operators acting on the poly-Bergman-type spaces $\mathcal{A}^{2}_{(1, n)}(D_{2})$ and $\mathcal{A}^{2}_{(n, 1)}(D_{2})$. We take nilpotent symbols of the form $\tilde{a}(\zeta) = a({\rm{Im}} \zeta_1)$ and $\tilde{b}(\zeta) = b({\rm{Im}} \zeta_2 -|\zeta_1|^{2})$.

In Section 2, we introduce poly-Bergman-type spaces for the Siegel domain and discuss how they are identified through a Bargmann-type transform. In Section 3, we define Toeplitz operators acting on $\mathcal{A}^{2}_{(1, n)}(D_{2})$ with nilpotent symbols; such Toeplitz operators are unitarily equivalent to multiplication operators.

In Section 3.1, we take symbols of the form $\tilde{b}(\zeta) = b({\rm{Im}} \zeta_2 -|\zeta_1|^{2})$ for which both limits $\lim\limits_{s\rightarrow 0^+} b(s)$ and $\lim\limits_{s\rightarrow +\infty} b(s)$ exist; the $C^*$-algebra generated by all Toeplitz operators $T_b$ is isomorphic to $\mathfrak{C} = \{M \in M_n(\mathbb{C})\otimes C[0, \infty]: M(0), M(+\infty) \in \mathbb{C} I\}$. Let $PC(\overline{ \mathbb{R}}, \{0\})$ denote the set of all functions continuous on $\overline{ \mathbb{R}}\setminus \{0\}$ and having one-side limit values at $0$, where $\overline{ \mathbb{R}}$ is the two-point compactification of $\mathbb{R}$. In Section 3.2, we take nilpotent symbols of the form $\tilde{a}(\zeta) = a({\rm{Im}} \zeta_1)$, where $a\in PC(\overline{ \mathbb{R}}, \{0\})$; the $C^*$-algebra generated by all Toeplitz operators $T_a$ is isomorphic to $C(\overline{\Pi})$, where $\overline{\Pi} = \overline{ \mathbb{R}} \times \overline{ \mathbb{R}}_+$.

In Section 4, we introduce Toeplitz operators acting on $\mathcal{A}^{2}_{(n, 1)}(D_{2})$ with nilpotent symbols $\tilde{c}$ and we show that such Toeplitz operators are unitarily equivalent to multiplication operators $\gamma^{c}I$ acting on $L^2 (\mathbb{R} \times \mathbb{R}_+)$, where $\gamma^{c}$ is a continuous matrix-valued function on $\Pi$. In this work, we take symbols of the form $\tilde{a}(\zeta) = a({\rm{Im}} \zeta_1)$, where $a \in PC(\overline{ \mathbb{R}}, \{0\})$. In Section 4.1, we prove that the matrix-valued function $\gamma^{a}$ can be continuously extended to $\overline{\Pi}$ under a change of variable, which is one of our main results. In Section 4.3, we prove that the $C^*$-algebra generated by all Toeplitz operators $T_a$ is isomorphic to a $C^*$-subalgebra of $M_n(\mathbb{C})\otimes C(\overline{\Pi})$, thus the spectrum of such algebra is fully described.

2.   Poly-Bergman type spaces of the Siegel domain

In this section, we recall some results obtained in ^[25] that are needed in this paper. We recall how the poly-Bergman-type spaces are defined and how they are identified with a tensor product of $L_2$-spaces. This allows us to study Toeplitz operators with nilpotent symbols through a Bargmann-type transform. We clarify that if $X$ is any positive-measure subset of a Euclidean space, then $L^2(X)$ refers to $L^2(X, d\mu)$, where $d\mu$ is Lebesgue measure restricted to $X$. We will study Toeplitz operators acting on certain poly-Bergman-type subspaces of $L^{2}(D_{2})$, where $D_2$ is the two-dimensional Siegel domain

For each pair $L = (j, k) \in \mathbb{N}^2$, the poly-Bergman-type space $\mathcal{A}^{2}_{L}(D_{2})$ is the closed subspace of $L^{2}(D_{2})$ consisting of all $L$-analytic functions, that is, all functions $f\in C^\infty(D_2)$ satisfying the equations

Note that $\mathcal{A}^{2}_{L}(D_{2})$ is just the Bergman space when $L = (1, 1)$. The anti-poly-Bergman type space $\widetilde{\mathcal{A}}^2_{L}(D_{2})$ is defined to be the complex conjugate of $\mathcal{A}^{2}_{L}(D_{2}).$ Now, true-poly-Bergman-type spaces are defined as follows:

where $e_{1} = (1, 0)$ and $e_2 = (0, 1)$. We assume that $\mathcal{A}^{2}_{L}(D_{2}) = \{0 \}$ whenever $L\in \mathbb{Z}^2 \setminus \mathbb{N}^2$. The Hilbert space $L^{2}(D_{2})$ is the direct sum of all the true-poly-Bergman-type spaces and the true-anti-poly-Bergman-type spaces; see ^[25] for details. Let us briefly recall how the authors also constructed a unitary map from $\mathcal{A}^{2}_{(L)}(D_{n})$ to the tensor product $L^{2}(\mathbb{R})\otimes {\cal H}_{j-1} \otimes L^{2}(\mathbb{R}_{+}) \otimes {\cal L}_{k-1},$ where ${\cal H}_{m}$ and ${\cal L}_{m}$ are the one-dimensional spaces generated by the Hermite and Laguerre functions, respectively, which are given by

and

for $m = 0, 1, 2, ...$ As usual, $\mathbb{R}_+ = \{x\in \mathbb{R}: \ x > 0\}$, and $\chi_+$ denotes the indicator function of $\mathbb{R}_+$. The Hermite and Laguerre polynomials are defined by the Rodrigues formulae as follows:

It is well known that $\{h_m \}_{m = 0}^{\infty}$ and $\{\ell_m \}_{m = 0}^{\infty}$ are orthonormal bases for $L^2(\mathbb{R})$ and $L^2(\mathbb{R}_+)$, respectively. Finally, ${\cal H}_m = \text{span}\{h_m\}$ and ${\cal L}_m = \text{span}\{\ell_m\}$.

Consider the flat domain $\mathcal{D} = \mathbb{C} \times \Pi$, where $\Pi = \mathbb{R} \times \mathbb{R}_{+} \subset \mathbb{C}$, then $\mathcal{D}$ can be identified with $D_{2}$ under the mapping $\kappa: \mathcal{D} \rightarrow D_2$ given by the rule

Thus we have the unitary operator $U_{0}: L^{2}(D_{2}) \longrightarrow L^{2}(\mathcal{D})$ given by

Take $w = (w_{1}, w_{2}) \in \mathbb{C} \times \Pi$, with $w_{m} = u_m +i v_m$ and $m = 1, 2$. We identify $w = (u_1+ iv_1, u_2+iv_2)$ with $(u_1, v_1, u_2, v_2)$, then

Introduce the unitary operator $U_{1} = F\otimes I \otimes F \otimes I: L^2(\mathcal{D})\rightarrow L^2(\mathcal{D}),$ where $F$ is the Fourier transform acting on $L^{2}(\mathbb{R})$ by the rule

Consider now the following two mappings $\psi_1, \ \psi_2: \mathcal{D} \rightarrow \mathcal{D}$ defined by

and

where $\xi_m = t_m+is_m$ and $z_m = x_m+iy_m$. Both functions $\psi_1$ and $\psi_2$ induce the unitary operators acting on $L^{2}(\mathcal{D})$ by

Theorem 2.1. ^[25] The operator $U = V_{2}V_{1}U_{1}U_{0}$ is unitary and maps $L^{2}(D_{2})$ onto the space

For each $L = (j, k) \in \mathbb{N}^{2}$, the operator $U$ maps the true-Bergman-type space $\mathcal{A}^{2}_{(L)}(D_{2})$ onto the space

We will study Toeplitz operators with nilpotent symbols acting on $\mathcal{A}^{2}_{L}(D_{2})$ in the cases $L = (1, n)$ and $L = (n, 1)$. In both cases, the poly-Bergman-type space can be identified with $(L^{2}(\mathbb{R} \times \mathbb{R}_{+}))^n$ through a Bargmann type transform ^[25]. Since $\mathcal{A}^{2}_{(1, n)}(D_{2}) = \bigoplus_{k = 1}^n \mathcal{A}^{2}_{((1, k))}(D_{2})$, the operator $U$ isometrically maps $\mathcal{A}^{2}_{(1, n)}(D_{2})$ onto the space

Analogously, the operator $U$ restricted to $\mathcal{A}^{2}_{(n, 1)}(D_{2})$ is an isometric isomorphism onto the space

We introduce the following linear isometric embeddings

defined by

where $g = (g_1, ...., g_n)^T \in(L^{2}(\mathbb{R} \times \mathbb{R}_{+}))^{n}$ and

Clearly, $\mathcal{H}_{(1, n)}^{+}$ and $\mathcal{H}_{(n, 1)}^{+}$ are the images of $R_{0(1, n)}$ and $R_{0(n, 1)}$, respectively. Consequently, the operators

isometrically map the poly-Bergman-type spaces $\mathcal{A}^{2}_{(1, n)}(D_{2})$ and $\mathcal{A}^{2}_{(n, 1)}(D_{2})$ onto $(L^{2}(\mathbb{R} \times \mathbb{R}_{+}))^n$. Therefore,

where $B_{(1, n)}$ and $B_{(n, 1)}$ are the orthogonal projections from $L^{2}(D_{2})$ onto $\mathcal{A}^{2}_{(1, n)}(D_{2})$ and $\mathcal{A}^{2}_{(n, 1)}(D_{2})$, respectively. Thus, $R_{(1, n)}^{*}$ and $R_{(n, 1)}^{*}$ play the role of the Segal-Bargmann transform for the poly-Bergman-type spaces $\mathcal{A}^{2}_{(1, n)}(D_{2})$ and $\mathcal{A}^{2}_{(n, 1)}(D_{2})$, where the adjoint operators $R_{0(1, n)}^{*}$ and $R_{0(n, 1)}^{*}$ are given by

where $(x_1, x_2)\in \mathbb{R} \times \mathbb{R}_+$.

3.   Toeplitz operators on the poly-Bergman space $\mathcal{A}^{2}_{(1, n)}(D_{2})$

Toeplitz operators with nilpotent symbols acting on the poly-Bergman-type space $\mathcal{A}^{2}_{(1, n)}(D_{2})$ are studied in this section. The Vasilevski's techniques, as in ^[6], allow us to identify Toeplitz operators with multiplication operators. Recall that $\tilde{c} \in L^{\infty}(D_{2}, d\mu)$ is said to be a nilpotent symbol if it has the form $\tilde{c}(\zeta_{1}, \zeta_{2}) = c({\rm{Im}} \zeta_{1}, {\rm{Im}} \zeta_{2}- |\zeta_{1}|^{2})$, where $c: \mathbb{R} \times \mathbb{R}_+\rightarrow \mathbb{C}$, then the Toeplitz operator acting on $\mathcal{A}^{2}_{(1, n)}(D_{2})$ with nilpotent symbol $\tilde{c}$ is defined by

where $B_{(1, n)}$ is the orthogonal projection from $L^{2}(D_{2})$ onto $\mathcal{A}^{2}_{(1, n)}(D_{2})$. If we define

then $T_{c}$ is equal to $B_{(1, n)}M_{\tilde{c}}$ restricted to $\mathcal{A}^{2}_{(1, n)}(D_{2})$.

The Bargmann-type operator $R_{(1, n)}$ identifies the space $\mathcal{A}^{2}_{(1, n)}(D_{2})$ with $(L^{2}(\mathbb{R}\times \mathbb{R}_{+}))^n$, and it fits properly in the study of the Toeplitz operators $T_{\widetilde{c}}$.

Theorem 3.1. Let $\tilde{c}$ be a nilpotent symbol, then the Toeplitz operator $T_{c}$ is unitary equivalent to the multiplication operator $M_{\gamma^c}$ and, in fact, $M_{\gamma^c} = R_{(1, n)}T_{c}R_{(1, n)}^*$, where the matrix-valued function $\gamma^{c}: \mathbb{R} \times \mathbb{R}_{+} \rightarrow M_n(\mathbb{C})$ is given by

Proof. We have

Recall that $\zeta = \kappa(w) = (w_{1}, w_2 + i |w_{1}|^{2})$, where $w = (w_{1}, w_2) \in \mathcal{D}$ and $w_m = u_m +iv_m$. For $f \in L^{2}(\mathcal{D})$,

That is, $U_{0} M_{\tilde{c}} U_{0}^{-1} = M_{\tilde{c}\circ\kappa}$. It is easy to see that $U_{1} M_{\tilde{c}\circ\kappa} U_{1}^{-1} = M_{\tilde{c}\circ\kappa}.$ Furthermore,

and

Denote $\tilde{c}\circ\kappa\circ\psi_1\circ\psi_2$ by $C$. It is easy see that

where $z = (x_1 +iy_1, x_2 +iy_2)\in \mathcal{D}$. Finally, let $g = (g_1, ..., g_n)^T \in(L^{2}(\mathbb{R} \times \mathbb{R}_{+}))^{n}$ and $A = (R_{(1, n)}T_cR_{(1, n)}^* g)(x_1, x_2)$, then

Thus, $R_{(1, n)}T_{c}R_{(1, n)}^* = M_{\gamma^{c}}$, where $\gamma^{c}(x_1, x_2)$ is given in (3.1). □

Studying the full $C^*$-algebra generated by all Toeplitz operators with nilpotent symbols is a difficult task due to the fact that its spectrum is too large. For this reason, we study Toeplitz operators in special cases. In particular, we consider two specific cases of nilpotent symbols. First, we study the Toeplitz operators with symbols of the form $\tilde{b}(\zeta) = b({\rm{Im}} \zeta_{2}- |\zeta_{1}|^{2})$, for which

Second, we analyze Toeplitz operators with symbols of the form $\tilde{a}(\zeta) = a({\rm{Im}} \zeta_{1})$, for which

3.1.   Toeplitz operators with symbols $\tilde{b}(\zeta)=b({\rm{Im}} \zeta_{2}- |\zeta_{1}|^{2})$

In this section, we study the $C^*$-algebra generated by all Toeplitz operators $T_b$ with symbols of the form $\tilde{b}(\zeta) = b({\rm{Im}} \zeta_{2}- |\zeta_{1}|^{2})$, where $b: \mathbb{R}_+\rightarrow \mathbb{C}$ has limits at $0$ and $+\infty$. Under this continuity condition, we will see that $\gamma^{b}$ is continuous on $\overline{\Pi}: = \overline{ \mathbb{R}} \times \overline{ \mathbb{R}}_+$, where $\overline{ \mathbb{R}} = [-\infty, +\infty]$ and $\overline{ \mathbb{R}}_+ = [0, +\infty]$ are the two-point compactification of $\mathbb{R} = (-\infty, +\infty)$ and $\mathbb{R}_+ = (0, +\infty)$, respectively. The spectral function $\gamma^{b} = (\gamma^b_{jk}): \mathbb{R}\times \mathbb{R}_+\rightarrow M_n(\mathbb{C})$ is continuous if all of its matrix entries are continuous. These matrix entries are given by

Let $L^{\infty}_{\{0, +\infty\}}(\mathbb{R}_{+})$ denote the subspace of $L^{\infty}(\mathbb{R}_{+})$ consisting of all functions having limit values at $0$ and $+\infty$. For $b \in L^{\infty}_{\{0, +\infty\}}(\mathbb{R}_{+})$, define

We sometimes think of $\gamma_{jk}^{b}$ as a function from $\mathbb{R}_+$ to $\mathbb{C}$, as it depends only on the variable $x_2$. The form of the matrix-valued function $\gamma^b$ was obtained in ^[16] as the spectral function of a Toeplitz operator acting on poly-Bergman spaces of the upper half-plane with vertical symbols, i.e., symbols depending only on the imaginary part of $z$. Thus, we have at least two scenarios in which $\gamma^b$ appears as a spectral matrix-valued function.

Lemma 3.2. ^[16] Let $b \in L^{\infty}_{\{0, +\infty\}}(\mathbb{R}_{+})$, then the spectral matrix-valued function $\gamma^{b}: \mathbb{R}_+\rightarrow M_n(\mathbb{C})$ is continuous and satisfies

Obviously, in this context, the spectral matrix-valued function $\gamma^b$ is defined and continuous on $\overline{\Pi}$, but it is constant along each horizontal straight line. Thus, $\gamma^b$ is identified with a continuous function on $\overline{ \mathbb{R}}_+$.

Let $M_n(C([0, \infty])) = M_n(\mathbb{C})\otimes C([0, \infty])$, where $M_n(\mathbb{C})$ is the algebra of all $n \times n$ matrices with complex entries. Let $\mathfrak{C}$ be the $C^*$-subalgebra of $M_n(C([0, \infty]))$ given by

By Lemma 3.2 and Theorem 4.8 in ^[16], we have the following

Theorem 3.3. For all $b\in L^{\infty}_{\{0, +\infty\}}(\mathbb{R}_{+})$, the spectral matrix-valued function $\gamma^{b}$ belongs to the $C^*$-algebra $\mathfrak{C}$. Moreover, the $C^{*}$-algebra generated by all the matrix-valued functions $\gamma^{b}:\overline{\Pi} \rightarrow M_n(\mathbb{C})$, with $b\in L^{\infty}_{\{0, +\infty\}}(\mathbb{R}_{+})$, is equal to $\mathfrak{C}$. That is, the $C^*$-algebra generated by all the Toeplitz operators $T_{b}$, with $b \in L^{\infty}_{\{0, +\infty\}}(\mathbb{R}_{+})$, is isomorphic to $\mathfrak{C}$, where the isomorphism is defined on the generators by

3.2.   Toeplitz operators with continuous symbols $a({\rm{Im}} \zeta_{1})$

Our next stage is the study of the $C^*$-algebra generated by all Toeplitz operators $T_a$ acting on the poly-Bergman space $\mathcal{A}^{2}_{(1, n)}(D_{2})$, with symbols of the form $\tilde{a}(\zeta) = a({\rm{Im}} \zeta_{1})$. Recall that $\gamma^a$ is given by

for all $(x_1, x_2)\in\Pi = \mathbb{R}\times \mathbb{R}_+$. It is easy to see that $\gamma^a$ is continuous on $\Pi$.

Based on the results obtained in ^[7], we have the following theorem.

Theorem 3.4. The $C^{*}$-algebra generated by all Toeplitz operators of the form $T_{a}$, where $\tilde{a}(\zeta) = a({\rm{Im}} \zeta_1)$ with $a \in C(\overline{ \mathbb{R}})$, is isomorphic and isometric to the $C^{*}$-algebra $C(\triangle)$, where the quotient space $\triangle = \overline{\Pi} / (\overline{ \mathbb{R}} \times \{+\infty\})$ is defined by the identification of $\overline{ \mathbb{R}} \times \{\infty\}$ to a point. Furthermore, the $C^{*}$-algebra generated by all Toeplitz operators $T_{a}$ with $a \in PC(\overline{ \mathbb{R}}, \{0\})$ is isomorphic to the $C^{*}$-algebra $C(\overline{\Pi})$, where $PC(\overline{ \mathbb{R}}, \{0\})$ consists of all functions continuous on $\overline{ \mathbb{R}}\setminus \{0\}$ and have one-sided limits at $0$.

Proof. Note that $\gamma^a$ can be identified with the scalar function

This function was obtained in ^[7] as the spectral function of the Toeplitz operator acting on the Bergman space $\mathcal{A}^{2}(D_{2})$ with symbol $\tilde{a}(\zeta) = a({\rm{Im}} \zeta_{1})$. Theorems 10 and 14 of ^[7] complete the proof. □

4.   Toeplitz operators on the poly-Bergman space $\mathcal{A}^{2}_{(n, 1)}(D_{2})$

In this section, we study certain $C^*$-algebras generated by Toeplitz operators with nilpotent symbols acting on the poly-Bergman-type space $\mathcal{A}^{2}_{(n, 1)}(D_{2})$. We apply techniques as in Section 3. A Toeplitz operator acting on $\mathcal{A}^{2}_{(n, 1)}(D_{2})$ with nilpotent symbol $\tilde{c}(\zeta_{1}, \zeta_{2}) = c({\rm{Im}} \zeta_{1}, {\rm{Im}} \zeta_{2}- |\zeta_{1}|^{2})$ is defined by

where $B_{(n, 1)}$ is the orthogonal projection from $L^{2}(D_{2})$ onto $\mathcal{A}^{2}_{(n, 1)}(D_{2})$. The Bargmann-type operator $R_{(n, 1)}$ identifies the space $\mathcal{A}^{2}_{(n, 1)}(D_{2})$ with $(L^{2}(\mathbb{R}\times \mathbb{R}_{+}))^n$.

Theorem 4.1. Let $\tilde{c}$ be a nilpotent symbol, then the Toeplitz operator $T_{c}$ is unitary equivalent to the multiplication operator $\gamma^{c}I = R_{(n, 1)}T_{c}R_{(n, 1)}^*$, where the matrix-valued function $\gamma^{c} : \mathbb{R} \times \mathbb{R}_{+} \rightarrow M_n(\mathbb{C})$ is given by

Proof. We have

where

$z = (x_1 + iy_1, x_2+iy_2)\in \mathcal{D}$, and $C$ is given in (3.2).

Finally, let $g = (g_1, ..., g_n)^T \in(L^{2}(\mathbb{R} \times \mathbb{R}_{+}))^{n}$ and $B = (R_{(n, 1)}T_{c}R_{(n, 1)}^* g)(x_1, x_2)$, then

Thus, $R_{(n, 1)}T_{c}R_{(n, 1)}^* = \gamma^{c} I,$ where $\gamma^{c}(x_1, x_2)$ is given in (4.1). □

As in Section 3, we consider two specific cases of nilpotent symbols. First, we will take Toeplitz operators with symbols of the form $\tilde{b}(\zeta) = b({\rm{Im}} \zeta_{2}- |\zeta_{1}|^{2})$, for which

This spectral function can be identified with the scalar function

which was studied in ^[16]. Thus, the algebra generated by Toeplitz operators of the form $T_{b}$, where $b\in L^{\infty}_{\{0, +\infty\}}(\mathbb{R}_{+})$, has been completely described.

Second, we analyze Toeplitz operators with symbols of the form $\tilde{a}(\zeta) = a({\rm{Im}} \zeta_{1})$; in this case, we have

From this point on, we focus on describing the $C^*$-algebra generated by matrix-valued functions of this type.

4.1.   Continuity of the spectral function $\gamma^a$

In order to describe the $C^*$-algebra generated by Toeplitz operators acting on $\mathcal{A}^{2}_{(n, 1)}(D_{2})$ with nilpotent symbols of the form $\tilde{a}(\zeta) = a({\rm{Im}} \zeta_{1})$, first we will analyze the continuous extension of $\gamma^{a} = (\gamma_{jk}^a)$ to the compactification $\overline{\Pi}: = \overline{ \mathbb{R}} \times \overline{ \mathbb{R}}_+$. Make the change of variable $y_1 \mapsto 2\sqrt{x_2} y_1+x_1$ in the integral representation of $\gamma_{jk}^{a}$, then

The function $\gamma_{jk}^a$ is continuous at each point $(x_{1}, x_{2}) \in \Pi$ by the continuity of $h_{j-1}h_{k-1}$ and the Lebesgue dominated convergence theorem. Next, we will prove that $\gamma_{jk}^a$ has a one-sided limit at each point of $\mathbb{R} \times \{0\}$. For $a \in L^{\infty}(\mathbb{R})$, we introduce the notation

if such limits exist.

Lemma 4.2. Let $a\in L^{\infty}(\mathbb{R})$ and suppose that $a$ has limits at $\pm\infty$, then for each $x_{0} \in \mathbb{R}$, the spectral matrix-valued function $\gamma^{a}: \Pi \rightarrow M_n(C)$ satisfies

Proof. Let $A$ denote the $(j, k)$-entry of the righthand side of (4.5). Take $\epsilon > 0$. We will prove that there exists $\delta > 0$ such that $|\gamma_{}^{a}(x_{1}, x_{2})-A| < \epsilon$ whenever $|x_{1}-x_{0}| < \delta$ and $0 < x_{2} < \delta$. Note that $|a_-|, |a_+| \leq \|a\|_{\infty}$. Since $C_{jk} = \int_{-\infty}^{\infty} |h_{j-1}(y_{1}) h_{k-1}(y_{1})| dy_1 > 0$, there exists $\delta_{1} > 0$ such that

Then

We have assumed that $a$ converges at $\pm \infty$. Thus, there exists $N > 0$ such that $|a(y)-a_{-}| < \epsilon/(5C_{jk})$ and $|a(y)-a_{+}| < \epsilon/(5C_{jk})$ for $|y| > N$. Let $\delta = \min \{\delta_{1}/2, \delta_1^2/(16N^2) \}$, then we have $\frac{1}{2\sqrt{x_{2}}}|-x_{1}+y_{1}| > N$ if $|x_1-x_0| < \delta$, $0 < x_2 < \delta$ and $|y_1-x_0|\geq \delta_{1}$. Thus,

and

Finally, we conclude that $|\gamma_{jk}^{a}(x_{1}, x_{2})-A| < \epsilon$ whenever $|x_{1}-x_{0}| < \delta$ and $0 < x_{2} < \delta$. □

In general, the matrix-valued function $\gamma^a$ does not converge at the points $(\pm \infty, +\infty)\in \overline{\Pi}$; however, $\gamma^{a}$ has limit values along the parabolas $x_{2} = \alpha (x_{1}^{2}+1)$, with $\alpha > 0$. For this reason, we introduce the mapping $\Phi: \Pi \longrightarrow \Pi$ given by

We will prove that $\phi^{a} = \gamma^a \circ \Phi^{-1}:\Pi\rightarrow M_n(\mathbb{C})$ has a continuous extension to $\overline{\Pi} = \overline{ \mathbb{R}}\times\overline{ \mathbb{R}}_+$ with the usual topology. It is easy to see that $\Phi^{-1}(t_1, t_2) = (t_1, (t_1^2+1)t_2)$. Concerning the spectral properties of $T_a$, the matrix-valued function $\phi^{a}$ contains the same information as $\gamma^a$, but $\phi^a$ behaves much better than $\gamma^a$, at least for $a$ continuous on $\overline{ \mathbb{R}}$. From now on, we take $\phi^a$ as the spectral matrix-valued function for the Toeplitz operator $T_a$. A direct computation shows that

Note that both $\Phi$ and $\Phi^{-1}$ are continuous on $\mathbb{R} \times [0, +\infty)$. In addition, the spectral function $\phi^{a} = \gamma^{a} \circ \Phi^{-1}$ is continuous on $\mathbb{R} \times [0, +\infty)$ because $\gamma^{a}$ is. Since $\Phi^{-1}(t_1, 0) = (t_1, 0)$, we have that $\phi^{a}(t_{1}, 0) = \gamma^{a}(t_1, 0)$ for all $t_1\in \mathbb{C}$.

Theorem 4.3. For $a \in C(\overline{ \mathbb{R}})$, the spectral matrix-valued function $\phi^a: \Pi \rightarrow M_n (\mathbb{C})$ can be extended continuously to $\overline{\Pi} = \overline{ \mathbb{R}} \times \overline{ \mathbb{R}}_+$. Furthermore, $\phi^a$ is constant along $\overline{\mathbb{R}} \times \{+\infty\}$.

Proof. The result follows from Lemmas 4.2 and 4.4–4.6 below. □

Lemma 4.4. Let $a\in L^{\infty}(\mathbb{R})$ and suppose that $a$ converges at $\pm \infty$, then $\phi^{a} = (\phi_{jk}^a)$ satisfies

That is, for $\epsilon > 0$, there exists $\delta > 0$ and $N > 0$ such that $|\phi_{jk}^{a}(t_{1}, t_{2})- \delta_{jk}a(-\infty)| < \epsilon$ whenever $0 < t_{2} < \delta$ and $t_{1} > N$. Analogously,

Proof. Suppose that $a(-\infty) = 0$. Let $\epsilon > 0$. Since $h_{j-1}(s_{1}) h_{k-1}(s_{1})\in L^{2}(\mathbb{R})$, there exists $s_{0} > 0$ such that

Let $C_{jk} = \int_{-\infty}^{\infty} |h_{j-1}(s_{1}) h_{k-1}(s_{1})| dy_1 > 0$, then we have

Since $a$ converges to zero at $-\infty$, there exists $N_{1} > 0$ such that $C_{jk}|a(s)| < \epsilon/2$ for $-s > N_{1}$. Take $\delta = 1/(16N_{1}^{2})$, then we have $\frac{1}{2\sqrt{t_2}} > 2N_1$ for $0 < t_2 < \delta$. On the other hand, assume $t_{1} > s_0$ and $-\infty < s_1 < s_{0}$, then

The righthand side of this inequality converges to $1$ when $t_1$ tends to $+\infty$, thus there exists $N_2 > s_0$ such that $(t_{1}- s_{0})/\sqrt{t_{1}^{2} +1} > 1/2$ for $t_1 > N_2$. Consequently,

For $0 < t_2 < \delta$ and $t_1 > N: = \max\{ s_0, N_2 \}$ we have

We define $\hat{a}(s) = a(s)-a_2$ in the case $a(-\infty) \neq 0$, where $a_2: = a(-\infty)$. Note that $\hat{a}$ converges to zero at $-\infty$ and $\phi^{a_1+a_2} = \phi^{a_1}+\phi^{a_2}$ for any nilpotent symbols $a_1$ and $a_2$, then

Finally, the limit of $\phi^a$ at $(-\infty, 0)$ can be proved analogously. □

Lemma 4.5. Let $t_{0} \in \mathbb{R}_+$. If $a\in L^{\infty}(\mathbb{R})$ is continuous at $-1/(2\sqrt{t_0})$, then the spectral matrix-valued function $\phi^{a}$ satisfies

Analogously, if $a$ is continuous at $1/(2\sqrt{t_0})$, then

Proof. Suppose that $a$ converges to zero at $-1/(2\sqrt{t_{0}})$. Let $\epsilon > 0$. Since $h_{j-1}(s_{1}), h_{k-1}(s_{1}) \in L^{2}(\mathbb{R}, ds_{1})$, there exists $s_{0} > 0$ such that

Take into account $C_{jk} = \int_{-\infty}^{\infty}|h_{j-1}(s_{1}) h_{k-1}(s_{1})|ds_{1} > 0$ in the following computation

Because of the continuity of $a$ at $-1/(2\sqrt{t_0})$, there exists $\delta_{1} > 0$ such that $C_{jk}|a(s)| < \epsilon/3$ for $|s-\frac{-1}{2\sqrt{t_{0}}}| < \delta_{1}$. Let us estimate the value of the argument of $a$:

Choose $\delta > 0$ such that $\left|- \frac{1}{2\sqrt{t_{2}}}+ \frac{1}{2\sqrt{t_{0}}} \right| < \delta_{1} /3$ for $|t_{2}- t_{0}| < \delta$. Pick $N_{1} > 0$ such that $\left|1-\frac{t_{1}}{\sqrt{t_{1}^{2} + 1}}\right| < (2\sqrt{t_{0}}\delta_{1}) /3$ whenever $t_{1} > N_{1}$. Now, assume that $|t_2-t_0| < \delta$ and $|s_{1}| < s_{0}$, then $|\frac{1}{2\sqrt{t_{2}}}| < \frac{1}{2\sqrt{t_{0}}} + \frac{\delta_{1}}{3}$. Thus, $\frac{|s_{1}|}{2\sqrt{t_{2}(t_{1}^{2} + 1)}}$ converges to $0$ when $t_{1}$ tends to $+\infty$. Therefore, there exists $N > N_1$ such that $\frac{|s_{1}|}{2\sqrt{t_{2}(t_{1}^{2} + 1)}} < \delta_{1} /3$ for $t_{1} > N$. The additional condition $t_1 > N$ implies

Hence, $|\phi_{jk}^{a}(t_{1}, t_{2})| < \epsilon$ if $|t_2-t_0| < \delta$ and $t_1 > N$.

If $a$ does not converge to zero at $-\frac{1}{2\sqrt{t_{0}}}$, then take the function $\hat{a}(s) = a(s)-a_2$ and proceed as in the proof of Lemma 4.4, where $a_2 = a\left(-\frac{1}{2\sqrt{t_{0}}}\right)$.

Finally, the justification of the limit of $\phi^a$ at $(-\infty, t_0)$ can be done analogously. □

Lemma 4.6. Let $a\in L^{\infty}(\mathbb{R})$ be continuous at $0\in \mathbb{R}$. For $t_{0} \in \overline{ \mathbb{R}}$, the spectral matrix-valued function $\phi^{a}$ satisfies

Actually, we have uniform convergence of $\phi^a(t_1, t_2)$ at $(t_0, +\infty)$; that is, for $\epsilon > 0$, there exists $N > 0$ such that $|\phi_{jk}^{a}(t_{1}, t_{2})-a(0)| < \epsilon$ for all $t_2 > N$ and for all $t_1\in \overline{ \mathbb{R}}$.

Proof. Suppose that $a(0) = 0$. Let $\epsilon > 0$, and choose $s_{0} > 0$ such that Eq (4.6) holds. Let $C_{jk} = \int_{-\infty}^{\infty} |h_{j-1}(s_{1}) h_{k-1}(s_{1}) | ds_{1} > 0$, then

By the continuity of $a$ at $0$, there exists $\delta_{1} > 0$ such that $|a(s)| < \epsilon/(3C_{jk})$ for $|s| < \delta_{1}$. For $-s_0 < s_1 < s_0$, we have

Take $N = (1+s_{0})^{2}/(4 \delta_{1}^{2})$. The inequality $t_2 > N$ implies $\frac{1}{2\sqrt{t_2}} < \frac{\delta_{1}}{(1+s_0)}$. Thus, if $t_{2} > N$, $t_{1} \in \overline{ \mathbb{R}}$ and $-s_{0} < s_1 < s_{0}$, then

Consequently, $|\phi_{jk}^{a}(t_{1}, t_{2})| < \epsilon$ for all $t_2 > N$ and $t_1 \in \overline{ \mathbb{R}}$.

Finally, in the case $a(0) \neq 0$, the proof can be carried out by considering the symbol $\hat{a}(s) = a(s)-a(0)$. □

For each nilpotent symbol $a\in C(\overline{ \mathbb{R}})$, the spectral function $\phi^{a}$ is continuous on $\overline{\Pi}$ and is constant along $\overline{ \mathbb{R}}\times \{+\infty\}$. In order to obtain a larger algebra, we now consider symbols $a\in PC(\overline {\mathbb{R}}, \{0\})$, where $PC(\overline{ \mathbb{R}}, \{0\})$ is the set of continuous functions on $\overline{ \mathbb{R}}$ with one-sided limits at $0$.

Consider the indicator function $\chi_+ = \chi_{[0, +\infty]}$, for which

Theorem 4.7. Let $a \in PC(\overline{ \mathbb{R}}, \{0\})$, then the spectral matrix-valued function $\phi^a$ can be extended continuously to $\overline{\Pi}$.

Proof. For $a \in PC(\overline{ \mathbb{R}}, \{0 \})$, we have

where $a(0_-)$ and $a(0_+)$ are the one-side limits of $a$ at $0$, and $\hat{a}(s) = a(s)+[a(0_-)-a(0_+)]\chi_+(s)$. Since $\hat{a} \in C(\overline{ \mathbb{R}})$, the spectral function $\phi^{\hat{a}}$ is continuous on $\overline{\Pi}$. According to (4.7), $\phi^{\chi_+}$ is obviously continuous on $\overline{\Pi}$. □

The spectral matrix-valued function $\phi^{\chi_+}$ depends only the real variable $t_1$; thus, it can be identified with the one-variable function

Lemma 4.8. The matrix-valued function $\phi^{+} = (\phi^{+}_{jk})$ satisfies:

(1) $\phi^{+}(-\infty) = I$ and $\phi^{+}(+\infty) = 0$.

(2) For each $t \in \mathbb{R}$, $\phi^{+}(t)$ is symmetric positive definite and $\|\phi^{+}(t)\| \leq 1$, where $\| \cdot\|$ is the uniform norm.

(3) There exists $E \in M_n(\mathbb{C})$ such that for all $t\in \overline{ \mathbb{R}}$, one has that $\phi^{+}(t) = E M_{t} E^T$, where $E \in M_n(\mathbb{C})$ and

(4) For each $t \in \mathbb{R}$ and $\lambda \in \mathbb{C}$, $\det (\lambda I - \phi^{+}(t)) = 0$ if, and only if, $\det (\lambda M_{-\infty}- M_{t}) = 0$.

Proof. Part (1) follows since $\{h_j\}_{j = 0}^{\infty}$ is an orthonormal basis for $L_2(\mathbb{R})$. The matrix $\phi^{+}(t)$ is symmetric for all $t$ because $H(s) H(s)^T$ is symmetric for all $s$. Let $v \in \mathbb{C}^n$ be a unit vector, then

where $e^{s^2}|\langle H(s), v \rangle|^2$ is a nonzero polynomial, thus $\langle\phi^{+}(t) v, v \rangle > 0$. Now, we note that

hence, $\|\phi^{+}(t)\| \leq 1$. This proves $(2)$.

The Hermite function is given by

Define

then $H(s) = (h_0(s), ..., h_{n-1}(s))^T = e^{-s^2 / 2} ES$ and

Therefore, $\phi^{+}(t) = E M_{t} E^T.$ Also, $\det E \neq 0$ since $E$ is a lower triangular matrix and the scalars $c_{jj}$ are nonzero. This proves $(3)$.

Finally, let $\lambda \in \mathbb{C}$. We have $I = \phi^{+}(-\infty) = EM_{-\infty}E^T$, then

Thus, $\det (\lambda I - E M_{t} E^T) = 0$ if, and only if, $\det (\lambda M_{-\infty}- M_{t}) = 0$. □

4.2.   The algebra generated by the Toeplitz operator $T_{\chi^+}$

The $C^*$-algebra generated by $T_{\chi^+}$ is isomorphic to the $C^*$-algebra generated by $\phi^+$. Let $\mathcal{D}_n$ be $C^*$-algebra generated by $I$ and $\phi^{+}$, which is a subalgebra of $M_n(\mathbb{C})\otimes C(\overline{\mathbb{R}})$, where the metric is given by $\|M\| = \max_{t\in \overline{\mathbb{R}}} \|M(t)\|$.

According to Lemma 4.8, the matrix $\phi^{+}(t)$ is diagonalizable for each $t \in \mathbb{R}$ and its spectrum $\sigma (\phi^{+}(t))$ lies in $[0, 1]$. The eigenvalues are given by the equation $\det (\lambda M_{-\infty}- M_{t}) = 0$. There exists an orthogonal matrix $B(t)$ such that

that is, if $B(t) = [v_1(t)\cdots v_n(t)]$, then $\phi^{+}(t) v_j(t) = \lambda_j(t) v_j(t)$ for $j = 1, ..., n$. We may assume that $B$ and $\lambda_j$ are continuous on $\overline{\mathbb{R}}$, and $\lambda_1(t)\leq \lambda_2(t) \leq \cdots \leq \lambda_n(t)$. We have $\lambda_j(-\infty) = 1$ and $\lambda_j(+\infty) = 0$.

Up to isomorphism, the $C^*$-algebra $\mathcal{D}_n$ is equal to the $C^*$-algebra generated by $D$, that is, each element $\varphi \in \mathcal{D}_n$ is a uniform limit of polynomials on $D$:

Therefore, we conclude

Theorem 4.9. Let $\mathcal{C}_n(\overline{\mathbb{R}})$ be the $C^*$-subalgebra of $(C(\overline{\mathbb{R}}))^n$ that consists of all $n$-tuples $f = (f_1, ..., f_n)$ such that

when $\lambda_j(t) = \lambda_k(x)$. We can identify $f$ with $\mathit{\text{diag}}\, \{f_1, ..., f_n\}$, then the $C^*$-algebra $\mathcal{D}_n$ generated by $\phi^+$ is isomorphic to $\mathcal{C}_n(\overline{\mathbb{R}})$, where the isomorphism is given by

4.3.   The algebra generated by the Toeplitz operators with symbols $a\in PC(\overline{ \mathbb{R}}, \{0\})$

In this section, we describe the $C^*$-algebra generated by all the Toeplitz operators $T_a$ or, equivalently, the $C^*$-algebra generated by the matrix-valued functions $\phi^a:\overline{\Pi}\rightarrow \mathbb{C}$ with $a \in PC(\overline{ \mathbb{R}}, \{0\})$. Let $\mathfrak{B}$ be the $C^*$-algebra generated by all the matrix-valued functions $\phi^a$ with $a\in PC(\overline{ \mathbb{R}}, \{0\})$, and let $\mathcal{T}$ be the $C^*$-subalgebra of $M_{n}(C(\overline{\Pi})) = M_{n}(\mathbb{C}) \otimes C(\overline{\Pi})$ consisting of all $M$ such that $M(\pm \infty, t_2) \in \mathbb{C} I$ for each $t_2\in\overline{ \mathbb{R}}_+$ and

We will prove that $\mathfrak{B} = \mathcal{T}$ by using a Stone-Weierstrass theorem for $C^*$-algebras. Recall that a $C^*$-algebra $\mathcal{A}$ is said to be a CCR algebra if for every non-cero irreducible representation $(H, \pi)$ of $\mathcal{A}$ we have $\pi(\mathcal{A}) \subset K(H)$, where $K(H)$ is the ideal of all compact operators acting on the Hilbert space $H$.

Theorem 4.10. ^[26] Let $\mathcal{A }$ and $\mathcal{B}$ be $C^{*}$-algebras such that $\mathcal{B} \subset \mathcal{A}$. If $\mathcal{A}$ is a CCR algebra and $\mathcal{B}$ separates the pure state space of $\mathcal{A}$, then $\mathcal{B} = \mathcal{A}$.

Our main result of this section is the following:

Theorem 4.11. The $C^*$-algebra generated by all matrix-valued functions $\phi^a$, with $a \in PC(\overline{ \mathbb{R}}, \{0\})$, equals $\mathcal{T}$. That is, the $C^*$-algebra generated by all Toeplitz operators $T_a$ is isomorphic and isometric to the algebra $\mathcal{T}$, where the isomorphism is defined on the generators by the rule

Proof. $\mathfrak{B} = \mathcal{T}$ follows from Theorem 4.10. That is, $\mathfrak{B}$ separates the pure state space of $\mathcal{T}$ according to Lemmas 4.12–4.15, 4.17, and 4.19. □

It is easy to see that $\mathfrak{B}$ is contained in $\mathcal{T}$. Let $\langle \cdot, \cdot \rangle$ denote the usual inner product on $\mathbb{C}^n$. Now, the pure state space of the $C^*$-algebra $\mathcal{T}$ consists of all functionals having the form:

$1)$ $f_{(x_1, x_2), v}(M) = \langle M(x_1, x_2) v, v \rangle$ for $(x_1, x_2) \in \Pi$, $v \in \mathbb{C}^n$ a unit vector,

$2)$ $f_{(\pm \infty, t_2)}(M) = \lambda_{\pm t_2}$ for $0\leq t_2 \leq +\infty$, where $\lambda_{\pm t_2}I = M(\pm \infty, t_2)$,

$3)$ $f_{(t_1, \pm \infty), j}(M) = \langle M(t_1, \pm \infty) v_j(t_1), v_j(t_1) \rangle$ for $t_1\in \overline{ \mathbb{R}}$ and $j = , ..., n$,

where $M\in \mathcal{T}$ is arbitrary. The remainder of this section is devoted to separate all the pure states of $\mathcal{T}$.

Lemma 4.12. Let $t_2, \tau_2 \in \overline{ \mathbb{R}}_+$. We have $f_{(-\infty, t_2)}(\phi ^{\chi_+})\neq f_{(+ \infty, \tau_2)}(\phi^{\chi_+}).$ If $t_2 \neq \tau_2$, then there exists $a \in C(\overline{ \mathbb{R}})$ such that $f_{(\pm\infty, t_2)}(\phi ^a)\neq f_{(\pm \infty, \tau_2)}(\phi^a).$

Proof. The pure states $f_{(-\infty, t_2)}$ and $f_{(+ \infty, \tau_2)}$ are separated by $\phi^{\chi_+}$ since $\phi^{\chi_+}(-\infty, +\infty) = I$ and $\phi^{\chi_+}(+\infty, +\infty) = 0$. If $a \in C(\overline{ \mathbb{R}})$, then $\phi^{a}(\pm \infty, t_2) = a(\mp 1/(2\sqrt{t_2}))I$ for $t_2\in \mathbb{R}+$, $\phi^{a}(\pm \infty, 0) = a(\mp \infty)I$, and $\phi^{a}(\pm \infty, +\infty) = a(0)I$. Thus, taking $a(s) = s/\sqrt{s^2 +1}$, we have

Therefore, the pure states $f_{(\pm \infty, t_2)}$ and $f_{(\pm \infty, \tau_2)}$ are separated by $\phi^a$ when $t_2\neq \tau_2$. □

We shall continue separating the rest of pure states using continuous functions on $\overline{ \mathbb{R}}$ and the indicator function $\chi_+$.

Let $v \in \mathbb{C}^n$ be a unit vector. Consider the function $h_{v}(s) = |\langle H(s), v \rangle|^2$. This can be written as $h_{v}(s) = q_v (s)e^{-s^2}$, where

is a polynomial of degree at most $2n-2$ taking nonnegative values.

Lemma 4.13. Let $v \in \mathbb{C}^n$ be a unit vector and $(\pm\infty, t_2)$, $(x_1, x_2)$ be points with $x_1 \in \mathbb{R}$ and $x_2, t_2 \in \overline{ \mathbb{R}}_+$, then there exists a symbol $a \in PC(\overline{ \mathbb{R}}, \{0\})$ such that

Proof. For $x_1 \in \mathbb{R}$ and $x_2 \in \overline{ \mathbb{R}}_+$, we have

Since $q_v$ is not zero and nonnegative, $f_{(x_1, x_2), v}(\phi^{\chi_+}) > 0$. On the other hand, $f_{(+\infty, t_2)}(\phi ^{\chi_+}) = \chi_+ (-1/(2\sqrt{t_2})) = 0$ for $t_2\in [0, +\infty]$. Hence,

We now take $\chi_- = 1-\chi_+$, then

Also, $f_{(-\infty, t_2)}(\phi ^{\chi_-}) = \chi_- (1/(2\sqrt{t_2})) = 0$, then

□

Lemma 4.14. Let $v, w \in \mathbb{C}^n$ be unit vectors. Take $(t_1, 0), (x_1, x_2) \in \overline{\Pi}$, with $t_1 \in \mathbb{R}$, $x_1 \in \mathbb{R}$ and $0 < x_2\leq +\infty$, then there exists a symbol $a \in C(\overline{ \mathbb{R}})$ such that

Proof. Consider $a(s) = 1/(s^2 +1)$ for which $a(-\infty) = 0 = a(+\infty)$, then

Since $a(s) > 0$ for all $s \in \mathbb{R}$, and $q_v(s) = 0$ at most at finite number of values of $s$, we have that $a(s)q_v(s)e^{-s^2} > 0$ almost everywhere, then

Moreover, $\phi^a(x_1, +\infty) = a(0)I = 1\cdot I$. Thus, $f_{(x_1, +\infty), v}(\phi^{a}) = 1.$ We have proved that $f_{(t_1, 0), w}(\phi^{a}) \neq f_{(x_1, x_2), v}(\phi^{a})$. □

Lemma 4.15. Let $v, w \in \mathbb{C}^n$ be unit vectors, $(x_1, x_2) \in \Pi$, and $(t_1, +\infty)\in\overline{\Pi}$ with $t_1 \in \mathbb{R}$, then there exists $a \in C(\overline{ \mathbb{R}})$ such that

Proof. Let $a(s) = |s|/(s^2 +1)$ so that $a(0) = 0$ and $f_{(t_1, +\infty), w}(\phi^{a}) = a(0) = 0$. Since $q_v(s) = 0$ at most at finite number of points,

□

Next, we will separate the pure states associated to the points $(t_1, t_2) \in\Pi$ using continuous symbols indexed by $\alpha > 0$ and $r \in \mathbb{R}$. We introduce

where

Note that the family of functions $a_{\alpha}: = a_{\alpha}^{0}$ is an approximate identity in $L^1(\mathbb{R}, d\mu)$. Since $h_j h_k \in L^1(\mathbb{R})$, we have pointwise convergence in

because $h_j h_k$ is continuous.

Since $\Phi: \Pi \rightarrow \Pi$ is a homeomorphism and $\phi^{a} = \gamma^{a} \circ \Phi^{-1}$, we consider the matrix-valued function $\gamma^a$ in order to carry out the separation of pure states associated to the points in $\Pi$.

Lemma 4.16. Let $(x_1, x_2) \in \Pi$ and $a_{r, \alpha} = a_{\frac{\alpha}{2\sqrt{x_2}}}^r$, with $\alpha > 0$ and $r \in \mathbb{R}$, then the matrix-valued function $\gamma^{a_{r, \alpha}}$ satisfies

Further,

where $\beta_i = x_1+ 2\sqrt{x_2}r _i$ for $i = 1, 2$.

Proof. Take into account that $\{a_{\alpha}\}$ is an approximate identity and $a_{\alpha}(y-x) = a_{\alpha}(x-y)$ for any $x, y \in \mathbb{R}$. Calculate the entries of $\gamma^{a_{r, \alpha}}$:

Since $a_{\alpha}$ is an approximate identity, we have that

This completes the proof of (4.10). Finally,

□

For $v$ and $w$ unit vectors in $\mathbb{C}^n$ and $(t_1, t_2)\neq (x_1, x_2)$ $\in \Pi$, the following result says that the pure states $f_{(t_1, t_2), w}$ and $f_{(x_1, x_2), v}$ can be separated by $\mathfrak{B}$.

Lemma 4.17. Let $v$, $w$ $\in \mathbb{C}^n$ be unit vectors, $(t_1, t_2)$, $(x_1, x_2)$ $\in \Pi$, and $a_{r, \alpha, t_2} = a_{\frac{\alpha}{4\sqrt{x_2 t_2}}}^r$, with $\alpha > 0$ and $r\in \mathbb{R}$. If

then $(t_1, t_2) = (x_1, x_2)$.

Proof. It is easy to see that

and

Since Eq (4.12) holds for all $\alpha > 0$, we can take the limit in both sides of it when $\alpha \rightarrow 0$; then for all $r \in \mathbb{R}$, we have

where $q_v$ and $q_w$ are polynomials of degree at most $2n-2$. Thus, there is a constant $C\in \mathbb{R}$ such that

Therefore, (4.13) holds if, and only if, $x_1 = t_1$ and $x_2 = t_2$. □

For the separation of pure states attached to the same fiber, we will use the following lemma.

Lemma 4.18. ^[19] Let $y_1, ..., y_n$ be real numbers different from each other, then $\{H(y_1), ..., H(y_n)\}$ is a basis for $\mathbb{C}^n$.

Lemma 4.19. Let $w, v \in \mathbb{C}^n$ be unit vectors and $(x_1, x_2) \in \Pi$. Take the matrix-valued functions $\gamma^{a_{r_1, \alpha}}$ and $\gamma^{a_{r_2, \alpha}}$ with symbols as defined in Lemma 4.16, where $r_1, r_2 \in \mathbb{R}$ and $\alpha > 0$. Suppose that

Then $w = \lambda v$, where $\lambda$ is a uni-modular complex number; that is, $f_{(x_1, x_2), w} = f_{(x_1, x_2), v}$.

Proof. Define $\beta_i = 2\sqrt{x_2}r _i + x_1$ for $i = 1, 2$. The real number $[H(\beta_1)]^T H(\beta_2)$ could be zero only for a finite number of values of $\beta_1$ and $\beta_2$. We also have $x_2 > 0$. By continuity and (4.11), the following equality

is reduced to

Without loss of generality, we can assume that $x_1 = 0$ and $x_2 = 1/4$, then

This equality holds for all $r_1$ and $r_2$. In particular, take $r = r_2 = r_1$; thus, $|\langle w, H(r) \rangle| = |\langle v, H(r) \rangle|$ for all $r$. We can write $\langle w, H(r) \rangle = \langle v, H(r) \rangle e^{i\theta(r)}$ with $\theta(r)\in \mathbb{R}$, then

Thus, $e^{i\theta(r_2)-i\theta(r_1)} = 1$ for all $r_1, r_2$, which means that $\langle w, H(y) \rangle = e^{i\theta_0} \langle v, H(y) \rangle$ for all $y \in \mathbb{R}$ and some constant $\theta_0$. Take $u = w- e^{i\theta_0}v$, then $\langle u, H(y) \rangle = 0$. According to Lemma 4.18, the set $\{H(y_k)\}_{k = 1}^n$ is a basis for $\mathbb{C}^n$ and

Therefore, $u$ must be the zero vector. □

5.   Conclusions

Recall that a nilpotent symbol for the Siegel domain $D_2$ has the form $c(\text{Im}\, \zeta_{1}, \text{Im}\, \zeta_{2} - |\zeta_1|^{2})$. Certainly each Toeplitz operator $T_c:\mathcal{A}^2_{L}(D_2) \rightarrow \mathcal{A}^2_{L}(D_2)$ can be unitarily identified with a multiplication operator $\gamma^c I$, but the $C^*$-algebra generated by all of them is large enough to fully describe its space of irreducible representations. The problem arises because $\gamma^c$ admits a continuous extension to the spectrum of the algebra and such spectrum is uknown in general. For this reason, we confine ourselves to two subclasses of nilpotent symbols in two particular cases of poly-Bergman-type spaces.

In the case of the poly-Bergman-type space $\mathcal{A}^{2}_{(1, n)}(D_{2})$, in Theorem 3.3 we described the $C^*$-algebra generated by all Toeplitz operators with symbols of the form $\tilde{b}(\zeta) = b({\rm{Im}} \zeta_2 -|\zeta_1|^{2})$, whereas in Theorem 3.4 we used symbols of the form $a({\rm{Im}} \zeta_1)$. Concerning the poly-Bergman-type space $\mathcal{A}^{2}_{(n, 1)}(D_{2})$, Theorem 4.11 is our main result, where we described the $C^*$-algebra generated by all Toeplitz operators with symbols of the form $a({\rm{Im}} \zeta_1)$ using the Stone-Weierstrass theorem for noncommutative $C^*$-algebras ^[26]. The $C^*$-algebra generated by all Toeplitz operators with symbols of the form $\tilde{b}(\zeta) = b({\rm{Im}} \zeta_2 -|\zeta_1|^{2})$ was studied in ^[16]. The description of the $C^*$-algebra generated by all Toeplitz operators with nilpotent symbols without restrictions is still an open problem.

Use of AI tools declaration

The authors declare they have not used Artificial Intelligence (AI) tools in the creation of this article.

Acknowledgments

This work was supported by Universidad Veracruzana, and the first author is grateful to CONAHCyT for its role in supporting this work through graduate and postdoctoral fellowships. The authors also thank Matthew G. Dawson for his useful comments and suggestions that improved this work.

Some of the results in this paper were obtained in the research of the first author for her doctoral dissertation ^[27].

Conflict of interest

All authors declare no conflicts of interest in this paper.