Toeplitz Operators on Higher Cauchy–Riemann Spaces Over the Unit Ball

In this paper, we investigate some algebraic properties of Toeplitz operators over higher Cauchy–Riemann spaces $$C_{\alpha ,m}$$Cα,m on the unit ball $$\mathbb {B}^d$$Bd with $$d\ge 2$$d≥2. We first discuss the Berezin transform on higher Cauchy–Riemann spaces. By making use of Berezin transform, we completely characterize (semi-)commuting Toeplitz operators with bounded pluriharmonic symbols over higher Cauchy–Riemann space $$C_{\alpha ,m}$$Cα,m. Moreover, we show that compact products of finite Toeplitz operators with a class of bounded pluriharmonic symbols only happen in the trivial case.


Introduction
This paper is an attempt to study Toeplitz operators theory over higher Cauchy-Riemann spaces on bounded symmetric domain of complex spaces. While there exist abundant works on function theory for higher Cauchy-Riemann spaces, investigation of the operator aspect was started only a few years.
Let us start by recalling a few basic facts and some terminologies. Let Ω be an irreducible bounded symmetry domain, and G = Aut(D) the biholomorphic automorphism group. It is well known [9,11] that via change of variables, the group G unitarily acts on a series of weighted Bergman spaces L 2 a (Ω, dμ α ) with dμ α = h(z, z) α dm(z), α > −1, here h(z, w) is a sesquiholomorphic polynomial called the quasi-determinant. Indeed it may quickly follow by that the fact L 2 a (Ω, dμ α ) = kerD ∩ L 2 (Ω, dμ α ), whereD is the invariant Cauchy-Riemann operator on Ω. We now denote by C α,m (Ω) = kerD m+1 ∩ L 2 (Ω, dμ α ) the mth Cauchy-Riemann space. Those G-invariant spaces, called as nearly holomorphic function spaces, was first studied by Shimura. In the paper [16], he found some important connections between nearly holomorphic functions and nonholomorphic automorphic forms in Kähler manifold. Lately Engliš, Peetre, Peng and Zhang made a thorough study on them via the invariant Cauchy-Riemann operatorD. We refer the readers to [6,7,12,13,18] for more details. These results reveal a beautiful relationship among Kähler geometry, Lie group representation theory and classical holomorphic function theory.
We now turn to operator aspect of higher Cauchy-Riemann spaces. It is natural to expect the structure of Toeplitz operators over higher Cauchy-Riemann spaces reveal a rich function theory picture directly dependent on the symbol of the Toeplitz operator and geometry of symmetric domains. One such instance has been made by Engliš and Zhang [8]. In their breakthrough, they discuss Toeplitz operators on the higher Cauchy-Riemann space C α,m and established an interesting Dixmier trace formula involving pseudodifferential symbols and geometry of the boundary.
Initiated by a seminal work of Brown and Halmos [2], the study of algebraic properties for Toeplitz operators has inspired much deep researches and prompted many interesting problems on the classical Hardy space H 2 (D), the classical Bergman space L 2 a (D) or others; see, for example, [19,20] and the references therein. We will attempt to start an systematic study in higher Cauchy-Riemann spaces setting. In the present paper we will begin our study with the case of the unit ball B d , the bounded symmetric domain of rank 1. It will be shown that some results in classical Bergman space L 2 a (D) of the unit disc D have free analogues in higher Cauchy-Riemann spaces C α,m (B d ) setting.
To ease the notation, we write C α,m = C α,m (B d ) = kerD m+1 ∩ L 2 (B d , dμ α ) for some nonnegative integer. Some careful arguments of their function theory have appeared in the paper of Zhang and his colleague [13,17,18]. In particular, one sees that C α,m = C α,D(α) whenever m ≥ D(α); see the definition D(α) in Eq. 2.1. Therefore, from now on we will always assume that m is a fixed integer such that 0 ≤ m ≤ D(α). We first introduced Berezin transform in the C α,m setting. It is proved that Berezin transform B α,m on C α,m can be written a finite linear combination of some usual weighted Berezin transforms. We also show a fixed point theorem for Berezin transform. With its help we completely characterize when Toeplitz operators with bounded pluriharmonic symbols commute. More precisely, T f T g = T g T f for bounded pluriharmonic functions f, g if and only if one of the following holds: (1) Both f and g are holomorphic.
(2) Bothf andḡ are holomorphic. ( We also characterize semi-commuting Toeplitz operators. We final turn to the compact properties. It is proved that there aren't nontrivial compact Toeplitz operators with bounded pluriharmonic symbols on C α,m . We also consider the problems of compact product of finite Toeplitz operators with bounded pluriharmonic symbols. Suppose that all bounded functions u i have continuous extensions to some common nonempty relatively open subset of the boundary. We proved that in this case, the product of Toeplitz operators T ui is compact if and only if at least one of u i is zero. For general symbols, the problem still remains a mystery. The paper is organized as follows. In Sect. 2, we recall some necessary background and give an explicit description of higher Cauchy-Riemann spaces. In Sect. 3, we introduce the Berezin transform and show a fixed point theorem. We next characterize (semi-)commuting Toeplitz operators with bounded pluriharmonic symbol functions on C α,m in Sect. 4. Finally, in Sect. 5 we will discuss compact products of finite Toeplitz operators with bounded pluriharmonic symbol functions.

Preliminaries
In this section we introduce some notations of the higher Cauchy-Riemann spaces on bounded symmetric domains; especially in the case of the unit ball, the bounded symmetric domain of rank 1. We first recall the definition of the invariant Cauchy-Riemann operator.
Let Ω be a bounded symmetric domain with the Bergman kernel K(z, w). It is well known that Ω has a Kähler metric given by the matrix Then for a Hermitian vector bundle E over Ω, there exists an invariant Cauchy-Riemann operator involving this metric, where T (1,0) (Ω) is the holomorphic tangent bundle of Ω. Indeed, let e β be a collection of local trivializing sections of E. For a smooth section f = f β e β ∈ C ∞ (Ω, E), the operatorD E is given locally bȳ where {hj ,i } is the inverse of the matrix {h i,j }, and {∂ i } is the standard base of T (1,0) (Ω). It follows the important invariance property ofD E ; namely, for a biholomorphism mapping ϕ of Ω, where U ϕ is the action on sections of E induced by ϕ. Moreover, by iteration one may get the invariant operatorD m E for any positive integer m, which is an operator such that ). We refer the reader to [6,8] for more details.
We now return to the case of unit ball B d in complex space C d . Take E the trivial bundle E = C, then the smooth sections of E are just smooth functions C ∞ (B d , C). For any f ∈ C ∞ (B d , C), the invariant Cauchy-Riemann operatorD : (1,0) ) is given bȳ  (1,0) ) is well defined for every positive integer m. The function in the space ∪ ∞ m=1 kerD m is called nearly holomorphic function in the sense of Shimura, see [16].
For α > −1, let L 2 (B d , dμ α ) be the L 2 -integrable function space with respect to the weighted measure be the mth Cauchy-Riemann spaces, which are closed in L 2 (B d , dμ α ) by the fact that differential operatorsD m+1 are closed. By [16,18], It also proved in [13,18] that when 0 ≤ m ≤ D(α), Here for any mult-index I, Then we immediately have the following relationship By the definition, it is easy to check C α,m is invariant under the action of any biholomorphism. More precisely, for every biholomorphism ϕ ∈ Aut(B d ), define a unitary action U ϕ on L 2 (B d , dμ α ) by where J ϕ (z) is the complex Jecobian of ϕ at point z. By the invariant property ofD, we have that U ϕ (C α,m ) = C α,m . Clearly it yields a natural unitary representation of Aut(B d ) on C α,m .
Moreover, denote A α,2 j by the subspace of L 2 (B d , dμ α ) generated by D(α), are exactly all irreducible representations of Aut(B d ), and each of them is called discrete series of L 2 (B d , dμ α ) as in [15,18]. In particular, in the case j = 0, we have that the subspace A α,2 is the standard weighted Bergman space. With some arguments of representation theory, Zhang [17] showed the that for 0 ≤ m ≤ D(α), C α,m has an irreducible orthogonal decomposition . This implies that the following holds. Proposition 2.1. The space C α,m is a reproducing kernel space with the reproducing kernel Proof. It is [17] proved that each A α,2 j is a reproducing kernel space with the kernel with the constat denoted by the Pochhammer for any complex number ν. This implies that C α,m is also a reproducing kernel space with the kernel which leads to the desired result by a direct computation.
For each bounded function f ∈ L ∞ (B d ), we define the Toeplitz operator T where P is the orthogonal projection from L 2 (B d , dμ α ) onto C α,m . In the following context, we write them by T f whenever no confusion arises. Obviously one has that

Berezin Transform on C α,m
In this section we will define Berezin transform on C α,m . We first recall Berezin transform on weighted Bergman space L 2 a (B d , dμ α ). For a function f ∈ L 1 (B d , dμ α ), the Berezin transform of B α f is given by and ϕ a is the Möbius transform at the point a ∈ B d , see Sect. 2.2.1 of [14]. We note that B α 1 = 1, and B α f is smooth inside the unit ball for any f ∈ L 1 (B d , dμ α ). Now we turn to define Berezin transform on C α,m . From (2.4) and (2.6), we know that It follows that the normalized reproducing kernel at z, w ∈ B d is given by For convenience, we introduce some notations. Denote by B(X, Y ) the set of all bounded operators from Banach space X to Banach space Y , and we also use B(X) instead of B(X, X) as usual. Moreover, we write the function space L p (B d , dμ α ) in simplified form L p (dμ α ). Then we give the definition of Berezin transform on C α,m .
(2) For T ∈ B(C α,m ), the Berezin transform B α,m for T is given by By definition, one sees that B α,m T f = B α,m f for any f ∈ L ∞ (B d ) as usual. However, the property for B α,m becomes much compliant. For example, B α,m is even not injective in general cases.
We next show that B α,m has still a close relation with the Berezin transform B α over the weighted Bergman space L 2 a (B d , dμ α ). We first notice that B α−j belongs to the set B(L 1 (dμ α−2m ), L 1 (dμ α−2m+1 )) for 0 ≤ j ≤ 2m, by Proposition 1.4.10 of [14]. For our purpose, we give the definition as following.
This type operaor has been consider by Lee [10] in the case of nonnegative coefficients. We first remark the following obersevation. Proof. It suffices to show that if Indeed, by the definition of B α in (3.1) and the unitary invariance of dm, we have that by the Riesz's representation theorem. Thus The following shows a direct connection between the Berezin transform B α,m on C α,m and the Berezin transform B α on the standard weighted Bergman spaces.  (α, j) such that for f ∈ L 1 (dμ α−2m ) and z ∈ B d , On the other hand, note that B α,m 1 = 1 and B α−j 1 = 1, j = 0, . . . , 2m.
We remark some examples as follows.
(2) When m = 1 and d = 1, a direct computation shows Note that the second coefficient is not nonnegative even in the simplest case.
In the following, we will show some properties for the Berezin transform sum operator for the further research.

Lemma 3.5. Suppose A is a Berezin transform sum operator. Then, for any
Proof. It follows from Proposition 2.1 of [1] and Definition 3.3. Proof. By dominated convergence theorem, it can be shown that for each β > −1, B β f ∈ C(B d ) and B β f = f on the boundary ∂B d . See Proposition 6.14 in [21] for case d = 1. Therefore, for a Berezin transform sum operator For f ∈ C 2 (B d ), the invariant Laplacian Δ is defined by We say f is M-harmonic function if Δf = 0. Proof. Since f is M-harmonic function, by Eq. (6) in [4] we know that

Proposition 3.7. Let A be a Berezin transform sum operator. For
The converse of the above result is one of the center problem in the study of the Berezin transforms. In the case B α,m over higher Cauchy-Riemann spaces, we establish the following result. Put g = f − h, then g ∈ C 0 (B d ) and B α,m g = g. We need to prove that g is zero. Without loss of generality, we may assume that g is a real function. Then it can get some extreme value inside B d . So, we may assume that there exists a point a ∈ B d such that g(z) ≤ g(a) on B d . Then Lemma 3.5 and the definition of B α,m implies that (3.7) Moreover, from (2.4), (2.6) and (3.2), we know that By fundamental theorem of algebra, we know that the zero set of k (α,m),0 (w) on B d is the union of at most finite spheres in the ball B d . This implies that Let w = ϕ a (z), z ∈ B d in (3.7), we get that Since g ∈ C 0 (B d ) one sees that g = 0, completing the proof. In the paper [10], Lee proved a fixed theorem in the case f ∈ L ∞ (B d ) when all the coefficients of the Berezin transform sum operator are nonnegative. In our situation, it would be hard to establish the fixed point theorem in the case f ∈ L ∞ (B d ).

(Semi-)Commuting Toeplitz Operators on C α,m
This section is devoted to the study for (semi-)commuting Toeplitz operators with bounded pluriharmonic symbols on C α,m . A function f from B d to C is said to be pluriharmonic, if f is twice differentiable on B d such that f is harmonic on every complex line inside B d . It is equivalent to that there are two holomorphic functions f 1 and f 2 on B d , such that f = f 1 +f 2 ; see [14,19]. We first prove the following theorem for semi-commuting Toeplitz operators. The proof of the theorem are divided to several steps. Let R denote the radialization operator for continuous functions on B d , i.e. for any f ∈ C(B d ), One important property for R is that it commutes with usual weighted Berezin transform and also commutes with the invariant Laplacian Δ. (2) If f is twice differentiable on B d , then Δ(Rf ) = R( Δf ).
We next show a fixed point theorem for the general functions.
Proof. First we assume that f is M-harmonic function. Then by Proposition 3.7, we have that B α,m f = f. By M-mean value property of M-harmonic functions [14], one sees is a constant function.
(2) From (1), we know that f, g ∈ H 2 (B d ). By using the integration formula in polar coordinates, we get the desired result.
We also need the following two results from [20].  Now we turn to the proof of Theorem 4.1. The Proof of Theorem 4.1. Note that C α,m is invariant under the multiplication by bounded holomorphic functions. It is trivial that T f T g = T fg whenf or g is holomorphic.
In the rest of this section, we prove the following theorem for commuting Toeplitz operators. (1) Both f and g are holomorphic.
(2) Bothf andḡ are holomorphic. ( Proof. It is easy to check that if f and g satisfy one of conditions (1)-(4).
In the next, we will prove another direction. Suppose f = f 1 +f 2 , g = g 1 +ḡ 2 are two bounded pluriharmonic function on B d with f 1 , f 2 , g 1 , g 2 holomorphic functions on B d and T f T g = T g T f on C α,m . According to (4.1)-(4.4), we know that and From (4.4) and the proof of (3) of Lemma 4.4, we know that Then g 1f2 − f 1ḡ2 is M-harmonic by Proposition 4.2. According to (1) of Lemma 4.4 and Theorem 4.6, we have that f 1 , f 2 , g 1 , g 2 satisfy one of the following conditions: (e) There is a nonzero constant b such that bf 2 − g 2 and g 1 −bf 1 are constants.

Compact Toeplitz Operators on C α,m
In this section we discuss compact products of finite Toeplitz operators with bounded pluriharmonic symbol functions on C α,m . We first show the following theorem for compact single Toeplitz operator. To complete the proof, we need two lemmas.
for any holomorphic polynomial p and multi-index L such that 0 ≤ |L| ≤ m ≤ D(α). Indeed, for any z ∈ B d , we have Then, by Proposition 1.4.10 of [14], we get that Thus, by (5.1) we get that This completes the proof.
(2) The proof is similar to (1). We omit it.
Proof. Since T f is compact Toeplitz operator on C α,m with bounded symbol f, it follows that by (1) When m = 0, i.e. in the case weighted Bergman space L 2 a (dμ α ), we know that the converse of Lemma 5.3 is also true, and it is even true in the weighted Bergman spaces over bounded symmetric domains [5]. However, it is unknown how to characterize the compact Toeplitz operators with bounded symbols on C α,m for m > 0. Now we prove the Theorem 5.1. The Proof of Theorem 5.1. Suppose f = 0, then T f = 0 is compact on C α,m .
Conversely, suppose f is bounded pluriharmonic function. Note that pluriharmonic functions are M-harmonic functions. Then by Proposition 3.7, it follows that Hence, together with Lemma 5.3, we get that Note that f is pluriharmonic on B d , then f = 0 on B d , by Theorem 4.3.3 of [14].
In the rest of the section, we consider the problem of compact finite products of Toeplitz operators. Inspired by [3] we show the following theorem. As a consequence, we immediately obtain the following result about zero product problems of the Toeplitz operators on C α,m .
Proof. Since U ϕ maps C α,m onto C α,m and U ϕ is unitary, it follows that where P is the orthogonal projection from L 2 (dμ α ) onto C α,m . Then, ∀g ∈ C α,m , we have that To prove Theorem 5.4, we give the α-Berezin transform on C α,m . The α-Berezin transform of T ∈ B(C α,m ) is defined by B α (T )(z) = T k α,z , k α,z , z ∈ B d , (5.2) where k α,z (w) is the normalized reproducing kernel of Bergman space L 2 a (dμ α ). For operator T that can be written the finite sum of products of finite Toeplitz operators, we have the following result.  In the next, we will prove (5.4) by induction. For n = 1. We know that function ϕ z → ξ pointwisely as z → ξ. Since by the dominated convergence theorem. Then 5) where P is the orthogonal projection from L 2 (dμ α ) onto C α,m . This proves (5.4) for n = 1. We now proceed by induction on n. Assume that (5.4) holds for some n ≥ 1 and consider the case n + 1. By induction hypothesis, we have that Conversely, suppose n j=1 T uj is compact on C α,m . By (2) of Lemma 5.2, we know that k α,z → 0 weakly on C α,m , z → ∂B d . Since n j=1 T uj is compact on C α,m , it follows that n j=1 T uj k α,z converges to 0 on norm topology as z → ∂B d . Then, we have that T uj k α,z , k α,z ≤ n j=1 T uj k α,z → 0, z → ∂B d .
Since u 1 , . . . , u n ∈ C(B d ∪ W ) for some nonempty relatively open set W ⊆ ∂B d , together with Proposition 5.8, we get that