Toeplitz operators on two poly-Bergman-type spaces of the Siegel domain D 2 ⊂ C 2 with continuous nilpotent symbols

: We studied Toeplitz operators acting on certain poly-Bergman-type spaces of the Siegel domain D 2 ⊂ C 2 . Using continuous nilpotent symbols, we described the C ∗ -algebras generated by such Toeplitz operators. Bounded measurable functions of the form ˜ c ( ζ ) = c (Im ζ 1 , Im ζ 2 − | ζ 1 | 2 ) are called nilpotent symbols. In this work, we considered symbols of the form ˜ a ( ζ ) = a (Im ζ 1 ) and ˜ b ( ζ ) = b (Im ζ 2 − | ζ 1 | 2 ), where both limits lim s → 0 + b ( s ) and lim s → + ∞ b ( s ) exist, and a belongs to the set of piece-wise continuous functions on R = [ −∞ , + ∞ ] and with one-sided limits at 0. We described certain C ∗ -algebras generated by such Toeplitz operators that turned out to be isomorphic to subalgebras of M n ( C ) ⊗ C ( Π ), where Π = R × R + and R + = [0 , + ∞ ]

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 ⊂ L ∞ in such a way that the C * -algebra and von Neumann algebra generated by Toeplitz operators with symbols in E can be explicitly described up to isomorphism, say, as an algebra of matrix-valued functions.
In this paper, we study Toeplitz operators with nilpotent symbols and acting on a poly-Bergman-type space on the Siegel domain D 2 ⊂ C 2 .In [4,5,6], the authors fully described all commutative C * -algebras generated by Toeplitz operators acting on the Bergman spaces of both the unit disk D and the Siegel domain D n ⊂ C n with symbols invariant under the action of a maximal Abelian subgroup of biholomorphisms.
Let Π = {z = x + iy ∈ C | y > 0} be the upper half-plane.Toeplitz operators with vertical symbols (which depend on y = Im z) acting on poly-Bergman-type spaces are well studied.Taking vertical symbols with limits at y = 0 and y = ∞, in [7,8] the authors described the C * -algebra generated by all Toeplitz operators on the poly-Bergman space A 2 n (Π); this algebra is isomorphic to a subalgebra of M n (C) ⊗ C[0, +∞].Similar research has studied Toeplitz operators on poly-Bergman spaces with homogeneous symbols ( [9,10]).In [11,12,8], the authors studied Toeplitz operators acting on the true-poly-Bergman space A 2  (n) (Π) from the point of view of wavelet spaces.Taking horizontal symbols having one-sided limits at x = ±∞, in [13,14] the authors studied Toeplitz operators acting on poly-Fock spaces F 2 k (C), they proved that the C * -algebra generated by such Toeplitz operators is isomorphic to a subalgebra of M n (C) ⊗ C[−∞, +∞].In [15,16,17,18], the authors studied the decomposition of the von Neumann algebra of radial operators acting on the poly-Fock spaces F 2 k (C) and weighted poly-Bergman spaces A 2 n (D).
In [4,5] the authors made remarkable progress in 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 Furthermore, in [19] the authors studied Toeplitz operators on the truepoly-Bergman-type space A 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 an L 2 -space through a Bargmann-type transform.In Section 3 we introduce the Toeplitz operators acting on A 2  (1,n) (D 2 ) with nilpotent symbols, and we show that such Toeplitz operators are unitarily equivalent to multiplication operators. In ) denote the set of all functions continuous on R \ {0} and having one-side limit values at 0, where R is the two-point compactification of R. In Section 3.2 we take nilpotent symbols of the form ã(ζ) = a(Im ζ 1 ), where a ∈ P C(R, {0}); the C * -algebra generated by all Toeplitz operators T a is isomorphic to C(Π), where Π = R × R + .
In Section 4 we introduce Toeplitz operators acting on A 2 (n,1) (D 2 ) with nilpotent symbols c, we show that such Toeplitz operators are unitarily equivalent to multiplication operators γ c I acting on L 2 (R × R + ), where γ c is a continuous matrix-valued function on Π.In this work we take symbols of the form ã(ζ) = a(Im ζ 1 ), where a ∈ P C(R, {0}).In Section 4.1 we prove that the matrix-valued function γ a can be continuously extended to Π under a change of variable, this 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 (C) ⊗ C(Π), 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 [20] that are needed in this paper.Each ζ ∈ C n will be represented as an ordered pair The Euclidean norm will be denoted by | • |.The n-dimensional Siegel domain is defined by We will study Toeplitz operators acting on certain poly-Bergman-type subspaces of L 2 (D n , dµ λ ), where dµ λ and dµ(ζ) is the usual Lebesgue measure.We clarify that if X is any positive-measure subset of a Euclidean space, then L 2 (X) refers to L 2 (X, dµ), where dµ is Lebesgue measure restricted to X.
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 λL (D n ) = {0} whenever L ∈ Z n \ N n .In [20] the authors proved that L 2 (D n , dµ λ ) is the direct sum of all the true-poly-Bergman-type spaces: The authors also constructed a unitary map from for m = 0, 1, 2, ... Recall also the Hermite and Laguerre functions 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 poly-Bergman-type spaces over the two-dimensional Siegel domain D 2 with Lebesgue measure dµ (that is, λ = 0).Henceforth, the space A 2 0L (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.We will study Toeplitz operators with nilpotent symbols acting on A 2 L (D 2 ) in both cases L = (1, n) and L = (n, 1).The poly-Bergman-type spaces A 2 (1,n) (D 2 ) and A 2 (n,1) (D 2 ) can be identified with (L 2 (R × R + )) n through a Bargmann type transform ( [20]).Several operators are needed to define this unitary map.To begin, we introduce the flat domain D = C × Π, where Π = R × R + ⊂ C. Then D can be identified with D 2 using the mapping whose real Jacobian determinant is everywhere equal to one.Thus we have the unitary operator U 0 : L 2 (D 2 ) −→ L 2 (D) given by We introduce the map where F is the Fourier transform acting on L 2 (R) by the rule Consider now the following two mappings acting on D: where we write 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): ) is an isometric isomorphism onto the space For each L = (n, 1) ∈ N 2 , the operator U restricted to A 2 (n,1) (D 2 ) is an isometric isomorphism onto the space Moreover, for each L = (j, k) ∈ N 2 , the operator U restricted to the true-Bergman-type space ) is an isometric isomorphism onto the space We introduce the isometric linear embedding R 0(L) : Of course H + (L) is the image of R 0(L) , and it is also the image of A 2 (L) (D 2 ) under U .Thus, the operator unitarily 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 A 2 (L) (D 2 ), where the adjoint operator Similarly, we introduce the following isometric linear embeddings defined by Clearly, H + (1,n) and H + (n,1) are the ranges of R 0(1,n) and R 0(n,1) respectively; they are also the images of , where B (1,n) and B (n,1) are the orthogonal projections from ) play the role of the Segal-Bargmann transform for the poly-Bergman-type spaces , where the adjoint operators where 3 Toeplitz operators on the poly-Bergman space In this section we study Toeplitz operators with nilpotent symbols acting on the poly-Bergman-type space A 2 (1,n) (D 2 ).In [6] the author thoroughly developed the theory of Toeplitz operators on Bergman spaces, and the author's techniques can be applied to the study of Toeplitz operators acting on ) n , and it fits properly in the study of the Toeplitz operators T c .Theorem 3.1.Let c be a nilpotent symbol.Then the Toeplitz operator T c is unitary equivalent to the multiplication operator M γ c , and in fact, , where the matrix-valued function Proof.We have Thus 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 b Secondly, we analyze Toeplitz operators with symbols of the form ã(ζ) = a(Im ζ 1 ), for which Apply the change of variable y 2 → 2x 2 y 2 in the integral representation of γ b jk .Then We sometimes think of γ b jk as a function from R + to C, as it depends only on the variable x 2 , and is continuous on R + because of the continuity of ℓ j−1 (y)ℓ k−1 (y) and the Lebesgue dominated convergence theorem.
The form of the matrix-valued function γ b was obtained in [7] 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 γ b appears as a spectral matrix-valued function.Obviously, in this context, the spectral matrix-valued function γ b 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 R + .

Toeplitz operators with continuous symbols a(Im ζ 1 ).
Our next stage is the study of the C * -algebra generated by all Toeplitz operators T a acting on the poly-Bergman space A 2 (1,n) (D 2 ), with symbols pf the form ã(ζ) = a(Im ζ 1 ).Recall that γ a is given by Note that γ a can be identified with the scalar function This function was obtained in [19] as the spectral function of the Toeplitz operator acting on the Bergman space A 2 (D 2 ) with symbol ã(ζ) = a(Im ζ 1 ).
Based on the results obtained in [19], we state the following theorem in the context of Toeplitz operators acting on A 2 (1,n) (D 2 ).
Theorem 3.4.The C * -algebra generated by all Toeplitz operators of the form T a , where ã(ζ) = a(Im ζ 1 ) for some a ∈ C(R), is isomorphic and isometric to the C * -algebra C(△), where the quotient space △ = Π/(R×{+∞}) is defined by the identification of R × {∞} to a point.Futhermore, the C *algebra generated by all Toeplitz operators T a with a ∈ P C(R, {0}), is isomorphic to the C * -algebra C(Π), where P C(R, {0}) consists of all functions continuous on R \ {0} and having one-sided limits at 0.
In this section we study certain C * -algebras generated by Toeplitz operators with nilpotent symbols acting on the poly-Bergman-type space A 2 (n,1) (D 2 ).We apply techniques as in Section 3. A Toeplitz operator acting on Theorem 4.1.Let c be a nilpotent symbol.Then the Toeplitz operator T c is unitary equivalent to the multiplication operator , where the matrix-valued function γ c : R × R + → M n (C) is given by Proof.We have where Thus R (n,1) T c R * (n,1) = γ c I, where γ c (x 1 , x 2 ) is given in (5).
As in Section 3, we consider two specific cases of nilpotent symblos.Firstly, we will take Toeplitz operators with symbols of the form b This spectral function can be identified with the scalar function which was studied in [7].Thus, the algebra generated by Toeplitz operators of the form T b , where b ∈ L ∞ {0,+∞} (R + ), has been completely described.Secondly, we analyze Toeplitz operators with symbols of the form ã(ζ) = a(Im ζ 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 γ a .
In order to describe the C * -algebra generated by Toeplitz operators acting on A 2 (n,1) (D 2 ) with nilpotent symbols of the form ã(ζ) = a(Im ζ 1 ), first we will analyze the continuous extension of γ a = (γ a jk ) to the compactification Π := R × R + .Make the change of variable y 1 → 2 √ x 2 y 1 + x 1 in the integral representation of γ a jk , then The function γ a jk is continuous at each point (x 1 , x 2 ) ∈ Π by the continuity of h j−1 h k−1 and the Lebesgue dominated convergence theorem.Next, we will prove that γ a jk has a one-sided limit at each point of R × {0}.For a ∈ L ∞ (R) we introduce the notation if such limits exist.
In general, the matrix-valued function γ a does not converge at the points (±∞, +∞) ∈ Π; however, γ a has limit values along the parabolas x 2 = α(x 2 1 +1), with α > 0. For this reason we introduce the mapping Φ : Π −→ Π given by We will prove that φ a = γ a • Φ −1 : Π → M n (C) has a continuous extension to Π = R × R + with the usual topology.It is easy to see that Concerning the spectral properties of T a , the matrix-valued function φ a contains the same information 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 matrix-valued function for the Toeplitz operator T a .A direct computation shows that Note that both Φ and Φ −1 are continuous on R × [0, +∞).In addition, the spectral function That is, for ǫ > 0 there exists δ > 0 and Since a converges to zero at −∞, there exists ).Then we have > 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, .
Because of the continuity of a at −1/(2 √ t 0 ), there exists Let us estimate the value of the argument of a: .
converges to 0 when t 1 tends to +∞.Therefore, there exists N > N 1 such that , then take the function â(s) = a(s) − a 2 and proceed as in the proof of Lemma 4.4, where Finally, the justification of the limit of φ a at (−∞, t 0 ) can be done analogously.
For each nilpotent symbol a ∈ C(R), the spectral function φ a is continuous on Π and is constant along R × {+∞}.In order to obtain a larger algebra, we now consider symbols a ∈ P C(R, {0}), where P C(R, {0}) is the set of continuous functions on R with one-sided limits at 0.
(3) There exists C ∈ M n (C) such that for all t ∈ R, one has that φ + (t) = CM t C T , where C ∈ M n (C) and e −s 2 SS T ds, S = (1, s, ..., s n−1 ) T .
Proof.Part (1) follows since {h j } ∞ j=0 is an orthonormal basis for L 2 (R).The matrix φ + (t) is symmetric for all t because of H(s)H(s) T is symmetric for all s.Let v ∈ C n be an unit vector, then where e s 2 | H(s), v | 2 is a nonzero polynomial, thus φ + (t)v, v > 0. Now, we note that The Hermite function is given by and define

4.3
The algebra generated by the Toeplitz Operators with symbols a ∈ P C(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 φ a : Π → C, with a ∈ P C(R, {0}).Let B be the C * -algebra generated by all the matrix-valued functions φ a , with a ∈ P C(R, {0}), and let T be the We will prove that B = T by using a Stone-Weierstrass theorem for C *algebras.Our main result of this section is the following: Theorem 4.11.The C * -algebra generated by all matrix-valued functions φ a , with a ∈ P C(R, {0}), equals T .That is, the C * -algebra generated by all Toeplitz operators T a is isomorphic and isometric to the algebra T , where the isomorphism is defined on the generators by the rule T a → φ a .
Proof.B = T follows from Theorem 4.10.That is, B separates the pure state space of T according to Lemmas 4.12, 4.13, 4.14, 4.15, 4.17 and 4.19.
It is easy to see that B is contained in T .Let •, • denote the usual inner product on C n .Now the pure state space of the C * -algebra T consists of all functionals having the form: where M ∈ T is arbitrary.
We shall continue separating the rest of pure states using continuous functions on R and the indicator function χ + .
Next we will separate the pure states associated to the points (t 1 , t 2 ) ∈ Π using continuous symbols indexed by α > 0 and r ∈ R. We introduce Note that the family of functions a α := a 0 α is an approximate identity in L 1 (R, dµ).Since h j h k ∈ L 1 (R), we have pointwise convergence in lim α→0 (a α * h j h k )(y) = (h j h k )(y) because h j h k is continuous.
Since Φ : Π → Π is a homeomorphism and φ a = γ a • Φ −1 , we consider the matrix-valued function γ a in order to carry out the separation of pure states associated to the points in Π. Proof.Take into account that {a α } is an approximate identity and a α (y − x) = a α (x − y) for any x, y ∈ R. Calculate the entries of γ a r α 2 Since a α is an approximate identity, we have that Take into account that For the separation of pure states attached to the same fiber we will use the following lemma.Thus e iθ(r 2 )−iθ(r 1 ) = 1 for all r 1 , r 2 , which means that w, H(y) = e iθ 0 v, H(y) for all y ∈ R and some constant θ 0 .Take u = w − e iθ 0 v, then u, H(y) = 0.According to Lemma 4.18, the set {H(y k )} n k=1 is a basis for C n and u, H(y k ) = 0, k = 1, ..., n.
Therefore u must be the zero vector.
The C * -algebra generated by all Toeplitz operators T c (with nilpotent symbol) is large enough to fully describe its space of irreducible representations, for this reason we confine ourselves to consider a subclass of nilpotent symbols.
Section 3.1 we take symbols of the form b(ζ) = b(Im ζ 2 − |ζ 1 | 2 ) for which both limits lim s→0 + b(s) and lim s→+∞ b(s) exist; the C * -algebra generated by all Toeplitz operators T b is isomorphic to Both the H m and L m are one-dimensional spaces defined below.Recall the Hermite and Laguerre polynomials: H m (y) := (−1) m e y 2 d m dy m (e −y 2 ), L λ m (y) := e y y −λ m! d m dy m (e −y y m+λ )

By Lemma 3 . 2 and
Theorem 4.8 in [7], we have the following Theorem 3.3.For all b ∈ L ∞ {0,+∞} (R + ), the spectral matrix-valued function γ b belongs to the C * -algebra C.Moreover, the C * -algebra generated by all the matrix-valued functions γ b : Π → M n (C), with b ∈ L ∞ {0,+∞} (R + ), is equal to C. That is, the C * -algebra generated by all the Toeplitz operators T b , with b ∈ L ∞ {0,+∞} (R + ), is isomorphic to C, where the isomorphism is defined on the generators by T b −→ γ b .

Theorem 4 .
10 ([21]).Let A and B be C * -algebras such that B ⊂ A. If A is a CCR algebra and B separates the pure state space of A, then B = A.