Toeplitz Operators Acting on True-Poly-Bergman Type Spaces of the Two-Dimensional Siegel Domain: Nilpotent Symbols

We describe certain $C^*$-algebras generated by Toeplitz operators with nilpotent symbols and acting on a poly-Bergman type space of the Siegel domain $D_{2} \subset \mathbb{C}^{2}$. Bounded measurable functions of the form $c(\text{Im}\, \zeta_{1}, \text{Im}\, \zeta_{2} - |\zeta_1|^{2})$ are called nilpotent symbols. In this work we consider symbols of the form $a(\text{Im}\, \zeta_1) b(\text{Im}\, \zeta_2 -|\zeta_1|^{2})$, where both limits $\lim\limits_{s\rightarrow 0^+} b(s)$ and $\lim\limits_{s\rightarrow +\infty} b(s)$ exist, and $a(s)$ belongs to the set of piece-wise continuous functions on $\overline{\mathbb{R}}=[-\infty,+\infty]$ and having one-side limit values at each point of a finite set $D\subset \mathbb{R}$. We prove that the $C^*$-algebra generated by all Toeplitz operators $T_{ab}$ is isomorphic to $C(\overline{\Pi})$, where $\overline{\Pi}=\overline{\mathbb{R}} \times \overline{\mathbb{R}}_+$ and $\overline{\mathbb{R}}_+=[0,+\infty]$.

In the study of Toeplitz operators one of the common strategies consists in selecting a set of symbols E ⊂ L ∞ in such a way that the algebra generated by Toeplitz operators with symbols in E can be described up to isomorphism, say, with an algebra of continuous functions or finding its spectrum. In this paper, we study Toeplitz operators with nilpotent symbols and acting on a poly-Bergman type space of the Siegel domain D 2 ⊂ C 2 . In [12,13,21] the authors have fully described all commutative C * -algebras generated by Toeplitz operators with symbols invariant under the action of a maximal Abelian subgroup of biholomorphisms and acting on the Bergman spaces of both the unit disk D and the Siegel domain D n ⊂ C n . For the unit disk, they discovered three families of symbols associated to commutative C * -algebras of Toeplitz operators, while for the Siegel domain they found n + 2 classes of symbols. Each class of symbols is invariant under the action of a maximal abelian group of biholomorphism. Certainly, one can use these classes of symbols to study Toeplitz operators acting on poly-Bergman type spaces of the unit disk or the Siegel domain.
Let Π = {z = x + iy ∈ C : y > 0} be the upper half-plane. Toeplitz operators with vertical symbols, which depend on y = Im z, and acting on Bergman type spaces have been studied. In [7,8] the authors proved that the algebra generated by Toeplitz operators with vertical symbols and acting on the weighted Bergman space A 2 λ (Π) is isometrically isomorphic to the algebra V SO(R + ) of all bounded functions that are very slowly oscillating on R + . Taking vertical symbols having limit values at y = 0 and y = ∞, in [14] the authors found that R + = [0, +∞] is the spectrum of the algebra generated by all Toeplitz operators on the true-poly Bergman space A 2 (n) (Π). Similar research was made for Toeplitz operators on poly-Bergman spaces with homogeneous symbols ( [11,17]). Other works about it were made in [9,10,16], where the authors studied Toeplitz operators acting on A 2 (n) (Π) from the point of view of wavelet spaces. On the other hand, in [4,5] the authors studied Toeplitz operators on the Fock space F 2 1 (C) with radial and bounded horizontal symbols, they found that spectral functions are uniformly continuous with respect to an adequate metric. Taking horizontal symbols having one-side limits at x = ±∞, in [18] the authors studied Toeplitz operators acting on poly-Fock spaces F 2 k (C), they found the spectrum of the C * -algebra generated by such Toeplitz operators. Even though the authors described the C * -algebras generated by all spectral functions, the spectrum of the algebras is not fully understood in some cases, for this reason additional conditions on the symbols are imposed.
In [12,13] the authors made remarkable research on the study of Toeplitz operators acting on the Bergman space of the Siegel domain D n ⊂ C n . In particular, they studied the C * -algebra T Nn generated by all Toeplitz operators with bounded nilpotent symbols, which are functions of the form a(ζ) = a(Im ζ 1 , ..., Im ζ n−1 , Im ζ n − |ζ ′ | 2 ), where ζ ′ = (ζ 1 , ..., ζ n−1 ). Let us denote this kind of symbols by N n . Although T Nn is commutative, it is too large, so it is impossible to figure out what its spectrum is. In particular, in [20] the authors described the algebra generated by Toeplitz operators acting on the weighted Bergman space A 2 λ (D 3 ) over three-dimensional Siegel domain D 3 using nilpotent symbols of the form c(y 2 )g(Im The main purpose of the paper is to find the spectrum of the algebra generated by Toeplitz operators acting on the true-poly-Bergman type space A 2 (L) (D 2 ) over two-dimensional Siegel domain D 2 by selecting a particular set of nilpotent symbols. In this sense, we just consider nilpotent symbols of the form a(Im ζ 1 ) and b(Im ζ 2 − |ζ 1 | 2 ). This paper is organized as follows. In Section 2 we recall how poly-Bergman type spaces are defined for the Siegel domain, and how they can be identified with a L 2 -space through a Bargmann type transform. In Section 3 we introduce Toeplitz operators acting on A 2 (L) (D 2 ) with nilpotent symbols, we show that such Toeplitz operators are unitary equivalent to multiplication operators. In Section 4 we take symbols of the form b(Im ζ 2 − |ζ 1 | 2 ) for which both limits lim exist, it is proved that the C * -algebra generated by all Toeplitz operators T b is isomorphic to C(R + ), where R + = [0, +∞] is the one-point compactification of [0, +∞). In Section 5 we take nilpotent symbols of the form a(Im ζ 1 ), where a ∈ C(R) and R = [−∞, +∞] is the two-point compactification of R, we prove that the C * -algebra generated by all Toeplitz operators T a is isomorphic to C(△), where △ = Π/(R × {+∞}) and Π = R × R + . In Section 6 we describe the C * -algebra generated by all Toeplitz operators T d , where d(Im ζ 1 ) ∈ P C(R, D) and P C(R, D) is the set of all piece-wise continuous functions on R having one-side limit values at each point of a finite set D ⊂ R. Finally, in Section 7 we describe the C * -algebra generated by all Toeplitz operators T ab = T a T b = T b T a .
2 Poly-Bergman type spaces of the Siegel domain.
For each multi-index L = (l 1 , ..., l n ) ∈ N n , the poly-Bergman type space In particular, for L = (1, ..., 1), A 2 λL (D n ) is just the Bergman space. Likewise, the anti-poly-Bergman type space A 2 λL (D n ) is defined to be the complex conjugate of A 2 λL (D n ). Thus, we introduce true-poly-Bergman type spaces as follows: where e m = (0, ..., 1, ..., 0) and the 1 is placed at the m-entry. We assume that A 2 λ(L) (D n ) = {0} whenever L ∈ Z n \ N n . In [15] the authors proved that L 2 (D n , dµ λ ) equals to the direct sum of all the true-poly-Bergman type spaces: The authors also proved that A 2 λ(L) (D n ) is isomorphic and isometric to the tensor product  where c m = m!/Γ(m + λ + 1) and Γ is the usual Gamma function. It is well known that {h m } ∞ m=0 and {ℓ λ m } ∞ m=0 are orthonormal bases for L 2 (R) and L 2 (R + , y λ dy), respectively. Finally, H m = span{h m } and L m = span{ℓ λ m }. In this work we restrict ourselves to the study of Toeplitz operators acting on the true-poly-Bergman type spaces over two-dimensional Siegel domain D 2 with the Lebesgue measure dµ (λ = 0). Henceforth, the space A 2 0(L) (D 2 ) will be simply denoted by A 2 (L) (D 2 ); similarly, ℓ m (y) and L m (y) stand for ℓ 0 m (y) and L 0 m (y), respectively. The true-poly-Bergman type space A 2 (L) (D 2 ) can be identified with L 2 (R×R + ) through a Bargmann type transform ( [15]), such identification fits to the study of Toeplitz operators with nilpotent symbols. Several operators are needed to define such identification. To begin with, we introduce the flat domain Then D can be identified with D 2 using the mapping Thus we have the unitary operator U 0 : L 2 (D 2 , dµ) −→ L 2 (D, dη) given by where dη(w) = dµ(w). Take w = (w 1 , w 2 ) ∈ C × Π, with w m = u m + iv 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 where F is the Fourier transform acting on L 2 (R) by the rule Consider now the following two mappings acting on D: where ξ = (ξ 1 , ξ 2 ), z = (z 1 , z 2 ) ∈ D, ξ m = t m + is m and z m = x m + iy m . Both functions ψ 1 and ψ 2 lead to the following unitary operators acting on L 2 (D, dη): Henceforth L = (l 1 , l 2 ) = (j, k).
Introduce the isometric linear embedding R 0(L) : Of course H + (L) is the range of R 0(L) , and it is also the image of A 2 (L) (D 2 ) under U . Thus, the operator isometrically maps the true-poly-Bergman type space . In addition, the operator R * (L) = U * R 0(L) plays the role of the Segal-Bargmann transform for the true-poly-Bergman type space 3 Toeplitz operators with nilpotent symbols.
In this section, we study Toeplitz operators with nilpotent symbols and acting on the true-poly-Bergman type space A 2 (L) (D 2 ). In [21] the author has widely developed the theory of Toeplitz operators on the Bergman spaces, and the author's techniques can be applied to the study of Toeplitz operators acting on A 2 (L) (D 2 ). To begin with, a function c ∈ L ∞ (D 2 , dµ) is said to be a nilpotent symbol if it has the form c(ζ 1 , ζ 2 ) = c(Im ζ 1 , Im ζ 2 − |ζ 1 | 2 ). Then the Toeplitz operator acting on , and it fits properly in the study of the Toeplitz operator T c .
Proof. We have Thus By Theorem 3.1, the C * -algebra generated by all Toeplitz operators T c is commutative (See [12,21]), but its spectrum is difficult to figure out what it is. For this reason, we assume certain continuity conditions on the nilpotent symbols in order to describe the spectrum of the subalgebra generated by the Toeplitz operators. We will split our research into two cases concerning the symbols. Firstly, we study Toeplitz operators with symbols of the form b(Im ζ 2 − |ζ 1 | 2 ), for which Secondly, we analyze Toeplitz operators with symbols of the form a(Im ζ 1 ), for which As mentioned above, the C * -algebra generated by all Toeplitz operators T a is still complicated to be fully described despite its commutative property.
Fortunately, the C * -algebra generated by all operators T a can be described when the symbols a are taken to be continuous on R = [−∞, +∞], where R is the two-point compactification of R. Even more, the C * -algebra of Toeplitz operators T a can be still described for symbols having finitely many jump discontinuities, as shown in Section 6. Finally, we analyze Toeplitz operators with symbols of the form c(ζ 1 , In this section, we study the C * -algebra generated by all Toeplitz operators T b with symbols of the form b(Im ζ 2 − |ζ 1 | 2 ), where b(y) has limit values at y = 0, +∞. Under this continuity condition, we will see that γ b is continuous on Π := R × R + , where R + = [0, +∞] is the two-point compactification of R + = (0, +∞). Apply the change of variable y 2 → 2x 2 y 2 in the integral representation of γ b , then Actually γ b depends only on the variable x 2 , and is continuous on R + because of the continuity of ℓ k−1 (y) and the Lebesgue dominated convergence theorem.
consisting of all functions having limit values at 0 and +∞.
It is worth mentioning that γ b was obtained in [14] as the spectral function of a Toeplitz operator acting on a true-poly-Bergman space of the upper half-plane. Thus, we have at least two scenarios in which γ b appears as a spectral function.
According to Lemma 4.1 and Theorem 4.8 in [14] we have the following , is isomorphic and isometric to the algebra C[0, ∞]. That is, the C * -algebra generated by all Toeplitz operators where the isomorphism is defined on the generators by Obviously the spectral function γ b (x 2 ) is defined and continuous on Π but it is constant along each horizontal straight line. Thus, γ b is identified with a continuous function on the quotient space Π/R, which is homeomorphic to R + . 5 Toeplitz operators with continuous symbols a(Im ζ 1 ).
In this section, we study the C * -algebra generated by all Toeplitz operators T a , where symbols a(Im ζ 1 ) are taken to be continuous on R. Once again, such a C * -algebra can be identified with the algebra of all continuous functions on a quotient space of Π. Henceforth, (x 1 , x 2 ) will denote points in Π instead of intervals.
It is fairly simple to see that γ a is continuous on Π. Take the change of variable y 1 → 2 √ x 2 y 1 + x 1 in the integral representation of γ a , then The function γ a is continuous at each point (x 1 , x 2 ) ∈ Π because of the continuity of h j−1 and the Lebesgue dominated convergence theorem. Next, we will prove that γ a has one-side limit value at each point of R × {0}. For a ∈ L ∞ (R) we introduce the notation if such limits exist.

Modified spectral function for T a .
Let Φ : Π −→ Π be the mapping It is easy to see that Φ −1 (t 1 , t 2 ) = (t 1 , (t 2 1 + 1)t 2 ). Concerning the spectral properties of T a , the function φ a := γ a • Φ −1 is as important as γ a , but φ a behaves much better than γ a , at least for a continuous on R. From now on we take φ a as the spectral function for the Toeplitz operator T a . A direct computation shows that Both Φ and Φ −1 are continuous on Π. Besides, the spectral function For any domain X ⊂ R m and a function ϕ : X → C, we write ϕ(x 0 ) to mean the limit value of ϕ at x 0 , even if x 0 does not belong to X. For example, a(+∞) means lim x→+∞ a(x). Lemma 5.3. Let a ∈ L ∞ (R), and suppose that a(y) converges at the points y = ±∞. Then φ a satisfies That is, for ǫ > 0 there exists δ > 0 and N > 0 such that |φ a (t 1 , t 2 )| < ǫ whenever 0 < t 2 < δ and t 1 > N . Analogously, Take into account ∞ −∞ (h j−1 (s 1 )) 2 ds 1 = 1 in the following computation Since a(s) converges to zero at −∞, there exists N 1 > 0 such that |a(s)| < ǫ/2 for −s > N 1 . Take δ = 1/(16N 2 1 ). Then we have 1 2 √ t 2 > 2N 1 for 0 < t 2 < δ. On the other hand, assume t 1 > s 0 and −∞ < s 1 < s 0 . Then The right-hand side of this inequality converges to 1 when t 1 tends to +∞, thus there exists N 2 > s 0 such that (t 1 − s 0 )/ t 2 1 + 1 > 1/2 for t 1 > N 2 . Consequently, .
Choose δ > 0 in such a way − 1 converges to 0 when t 1 tends to +∞. Therefore, there exists N > N 1 such that . Finally, the justification of the limit of φ a (t 1 , t 2 ) at (−∞, t 0 ) can be done analogously.
For each nilpotent symbol a(Im ζ 1 ) ∈ C(R), the spectral function φ a is continuous on Π and is constant along R × {+∞}. For this reason, the C *algebra generated by all spectral functions φ a is not C(Π), but it coincides with the algebra of continuous functions on a quotient space of Π.

Toeplitz operators T a with continuous symbols a(Im ζ 1 ).
Introduce the quotient space △ := Π/(R × {+∞}). By Lemma 5.5, the spectral function φ a can be identified with a continuous function on △, which will be also denoted by φ a : △ → C. We establish now one of our main results in this work.
Theorem 5.6. The C * -algebra generated by all spectral functions φ a , with a(Im ζ 1 ) ∈ C(R), is isomorphic and isometric to the algebra C(△). That is, the C * -algebra generated by all Toeplitz operators T a is isomorphic to C(△), where the isomorphism is defined on the generators by the rule Proof. The functions φ a separate the points of △ according to Lemmas 5.7, 5.8, 5.9 below. The Stone-Weierstrass theorem completes the proof.
This formula also says that φ a 1 (±∞, 0) = ∓1. On the other hand, the Hermite function h j−1 is continuous and it has just a finitely many roots. Hence, φ a 1 (t 1 , 0) is monotonically decreasing with respect to t 1 . Thus, points in R × {0} are separated by φ a 1 .
This proves that all points of Π can be separated from points of R × {0}.
Take a finite subset S = {β 0 , ..., β m } ⊂ R and let P C(R, S) be the set of functions continuous on R\S and having one-side limit values at each point of S. In this section we study the C * -algebra generated by all Toeplitz operators T a , where a(Im ζ 1 ) ∈ P C(R, S). Obviously, we have to study the algebra generated by the spectral functions φ a . To begin with, take the indicator function χ + = χ [0,+∞] . Then the spectral function φ χ + is continuous on Π \ (R × {+∞}) according to Lemmas 5.1, 5.3 and 5.4. Actually, where ϕ is defined in (8). Hence φ χ + is continuous on Π because ϕ is. Of course we have now a spectral function which is not constant along R×{+∞} anymore.
with β j < β j+1 , we can construct mappings Φ 3 , ..., Φ m in such a way that φ χ βm • Φ −1 1 • · · · • Φ −1 m is continuous on Π. Let T Sm denote the C * -algebra generated by all Toeplitz operators T a with nilpotent symbols a(Im ζ 1 ) ∈ P C(R, S m ), where S m = {β 0 , ..., β m }. For simplicity in our explanation, we assume that β 0 = 0 and S m−1 ⊂ S m ⊂ R + ∪ {0}. We will explain how the Toeplitz algebra T Sm increases as S m does. By Theorem 6.1, T S 0 is isomorphic to C(Π), where the isomorphism is given on the generators by the rule Consider now the algebra T S 1 , where S 1 = {0, β 1 }. By Lemma 6.3, φ a • Φ −1 1 is continuous on Π for every a ∈ P C(R, S 1 ). Then the algebra T S 1 is also isomorphic and isometric to C(Π), where the isomorphism is given by At first sight, both algebras T S 0 and T S 1 seem to have the same spectrum Π but they do not, they are identified with C(Π) through different isomorphisms. Of course, T S 0 is a subalgebra of T S 1 . If T a ∈ T S 0 , then φ a • Φ −1 1 ∈ C(Π) and has constant value along the line segment I 1 = {−∞} × J β 1 . According to the isomorphism Ψ 1 , we can say that the spectrum of T S 1 equals Π meanwhile the spectrum of T S 1 is the quotient space Π/I 1 . This phenomenon persists as long as the set S m grows. Theorem 6.4. Let S m = {0, β 1 , ..., β m } ⊂ R + . Then there exist bijective continuous functions Φ j : Π → Π, j = 1, ..., m, such that φ a • Φ −1 1 • · · · • Φ −1 m admits a continuous extension to Π for each a(Im ζ 1 ) ∈ P C(R, S m ). The C * algebra generated by all Toeplitz operators T a is isomorphic and isometric to C(Π). The isomorphism is defined on the generators by the rule Note that for each piece-wise continuous symbol a(Im ζ 1 ) ∈ P C(R, S m ), in general the spectral function γ a : Π → C does not admit a continuous extension to Π but γ a • Φ −1 • Φ −1 1 • · · · • Φ −1 m does, which means that γ a is uniformly continuous with respect to a new metric on Π, this metric is the pushforward of the usual metric using the mapping Φ −1 • Φ −1 1 • · · · • Φ −1 m .
Proof. Similar to the proof of Lemma 6.2.
Implicitly, we have considered several compactifications of Π associated to the C * -algebras studied here in, each compactification depends on the kind of symbols. Take Q − = (−∞, +∞) and Q + = (+∞, +∞), let us explain the situation in the case of the algebra generated by the Toeplitz with symbols a(Im ζ 1 ) ∈ C(R). Essentially the corresponding compactification of Π is obtained from Π \ ({Q − , Q + }) by gluing a line segment at each corner Q − and Q + . Each spectral function γ a (x 1 , x 2 ) is continuous on Π \ ({Q − , Q + }), and has limit values when (x 1 , x 2 ) moves along the parabolas x 2 = µ + µx 2 1 and tends to Q ± . For a net {X λ } tending to Q + , {γ a (X λ )} converges if {X λ } is eventually in gaps between two parabolas close enough from each other with respect to the parameter µ.