Properties of Commutativity of Dual Toeplitz Operators on the Orthogonal Complement of Pluriharmonic Dirichlet Space over the Ball

We completely characterize the pluriharmonic symbols for (semi)commuting dual Toeplitz operators on the orthogonal complement of the pluriharmonic Dirichlet space in Sobolev space of the unit ball. We show that, for f and g pluriharmonic functions, SfSg = SgSf on (Dh) ⊥ if and only if f and g satisfy one of the following conditions: (1) both f and g are holomorphic; (2) both f and g are holomorphic; (3) there are constants α and β, both not being zero, such that αf + βg is constant.


Introduction
For any integer  > 1, let   denote the open unit ball in   .The boundary of   is the sphere   and the closure of   with the Euclidean metric on   is denoted by   .Let ] denote the Lebesgue volume measure on the unit ball   of   , normalized so that the measure of   equals 1.The Sobolev space  1,2 =  1,2  } ] ()] where /  , /  is the weak partial derivative.The  1,2 where ⟨⋅, ⋅⟩ 2 denotes the inner product in the Lebesgue space  2 (  , ]).The Dirichlet space D = D(  , ]) is the closed subspace of  1,2 (  , ]) consisting of all holomorphic functions, and let  denote the orthogonal projection from  1,2 (  , ]) onto D(  , ]).Then  is an integral operator represented by where   () = (, ) is the reproducing kernel of D. By computation, we know that  (, ) = 1 + ∑ where {0} = (0, . . ., 0),  = ( 1 , . . .,   ) ∈ N  , ! =  1 !⋅ ⋅ ⋅   !,   = Given a function  ∈ They are all bounded linear operators.Under the decomposition  1,2 = D ℎ ⊕ (D ℎ ) ⊥ , the multiplication operator   is represented as This shows close relationships among the above four types of operators.Many studies for dual Toeplitz operators offer some insights into the study for Toeplitz operators.So it is reasonable to focus on the dual Toeplitz operators.Although dual Toeplitz operators differ in many ways from Toeplitz operators, they do have some of the same properties.The general problem that we are interested in is the following: what is the relationship between their symbols when two dual Toeplitz operators commute?For Toeplitz operators, this problem has been studied for a long time.In the case of the classical Hardy space, Brown and Halmos [1] showed that two Toeplitz operators with general bounded symbols commute if and only if either both symbols are analytic, both symbols are conjugate analytic, or a nontrivial linear combination of the symbols is constant.
Initiated by Brown and Halmos's pioneering work, the problem of characterizing when two Toeplitz operators commute has been one of the topics of constant interest in the study of Toeplitz operators on classical function spaces over various domains.On the Bergman space of the unit disk, Axler and Čučković [2] studied commuting Toeplitz operators with harmonic symbols and obtained a similar result to that of Brown and Halmos.Stroethoff [3] later extended that result to essentially commuting Toeplitz operators.Axler et al. [4] showed that if two Toeplitz operators commute and the symbol of one of them is nonconstant analytic, then the other one must be analytic.Čučković and Rao [5] studied Toeplitz operators that commute with Toeplitz operators with monomial symbols.On the Bergman space of several complex variables, by making use of M-harmonic function theory, Zheng [6] characterized commuting Toeplitz operators with pluriharmonic symbols on the Bergman space of the unit ball.Choe and Lee [7][8][9] studied commuting and essentially commuting Toeplitz operators with pluriharmonic symbols on the unit ball.Lu [10] characterized commuting Toeplitz operators on the bidisk with pluriharmonic symbols.Choe et al. [11] obtained characterizations of (essentially) commuting Toeplitz operators with pluriharmonic symbols on the Bergman space of the polydisk.
The fact that the product of two harmonic functions is no longer harmonic adds some mystery to the study of operators on harmonic Bergman space.Many methods which work for the operators on analytic Bergman space lose their effectiveness on harmonic Bergman space.On the harmonic Bergman space of the unit disk, Ohno [12] first characterized the commutativity of   and   , where  is an analytic function.Choe and Lee [13] studied commuting Toeplitz operator with harmonic symbols and one of the symbols is a polynomial.In [14], Choe and Lee proved that if ,  ∈  ∞ and supposedly one of them is noncyclic, then     =     if and only if either  or  is constant.On the pluriharmonic Bergman space of the unit ball, commuting Toeplitz operators were studied in [15,16].
However, the study on the problem for dual Toeplitz operators started recently.Stroethoff and Zheng [17] characterized the commutativity of dual Toeplitz operators with bounded symbols on the orthogonal complement of the Bergman space of the unit disk and studied algebraic and spectral properties of dual Toeplitz operators.On the Bergman space of the unit ball and the polydisk, commuting dual Toeplitz operators were studied in [18][19][20].Yang and Lu [21] gave complete characterization for the (semi)commuting dual Toeplitz operators with harmonic symbols on harmonic Bergman space.
In recent years the Dirichlet space has received a lot of attention from mathematicians in the areas of modern analysis, probability, and statistical analysis.Many mathematicians are interested in function theory and operator theory on the Dirichlet space.Yu and Wu [22,23] investigated commuting dual Toeplitz operators with harmonic symbols on the Dirichlet space.Yu [24] obtained the commutativity of dual Toeplitz operators with general symbols on Dirichlet space.
In this paper, we want to characterize commuting dual Toeplitz operators with pluriharmonic symbols on the orthogonal complement of the pluriharmonic Dirichlet space in Sobolev space of the unit ball.
We state our main result now.We postpone the proofs of these theorems until Section 3.
Theorem 1. Suppose that ,  ∈  1,∞ (  ) are pluriharmonic functions; then   =     if and only if one of the following statements holds: (1) Both  and  are holomorphic.
Theorem 2. Suppose that ,  ∈  1,∞ (  ) are pluriharmonic functions, then     =     if and only if one of the following statement holds: (1) Both  and  are holomorphic.
(3) There are constants  and , both not being zero, such that  +  is constant.
For  = 1, two dual Toeplitz operators with harmonic symbols always commute on the orthogonal complement of harmonic Dirichlet space; that is,     =     holds for all harmonic functions  and .
A pluriharmonic function in the unit ball is the sum of a holomorphic function and the conjugate of a holomorphic function.It is clear that all pluriharmonic functions on   are M-harmonic.A good reference for the function theory of the unit ball is Rudin's book [25].
The difficult part of the proof of Theorem 2 is to answer the following question about pluriharmonic functions.
This question is very subtle.If  = 2, this question is a special case of Theorem 5.6 in [6].In [26], Choe et al. gave a necessary and sufficient condition for this question in Lemma 4.7, which is useless to the proof of Theorem 2. In this paper, we give another characterization to the question and induce the proof of Theorem 2.

Some Lemmas
The following Lemma has been known to be true for  = 1 in [24].For  > 1, the following lemma may be known, but we cannot find its proof; for completeness, we give its proof.Lemma 3. The set of all polynomials in  and  is dense in  1,2 (  ).
Proof.We will discuss it in the case of real variables.For  ∈  1,2 where for all ( 1 ,  1 , . . .,   ,   ) in , and let Similarly, we also can define   for  = 1, . . ., 2.It is obtained that The standard orhonormal basis for C  consists of the vectors  1 ,  2 , . . .,   , where   is the ordered -tuple that has 1 in the th spot and 0 everywhere else.A direct computation gives that Let N = span{    − (    ) : ,  ≥ 0} and we have the following Lemma.
Proof.Since polynomials are dense in  1,2 by Lemma 3 and − is a bounded operator, we get that  is dense in D ⊥ ℎ .
The following lemma will be useful for the proof of the main theorem.

Lemma 5. Suppose that 𝑓
is holomorphic, we have  = ∑ ≥0     .For  = , it follows that where The last case is similar; we omit the proof.Hence we get that if  ∈  1,∞ and  is holomorphic, we have   ((D ℎ ) ⊥ ) ⊂ D. As the same discussion, we can deduce that   ((D ℎ ) ⊥ ) ⊂ D.
In the following proposition, we give an answer to the question that when a dual Toeplitz operator equals zero.Proposition 6. Suppose that  ∈  1,∞ is a pluriharmonic function.Then   = 0 if and only if  ≡ 0.
Proof.Assume that   = 0. Let A direct computation gives that Since || < 1, it follows that  ≡ 0. The converse part is easy to see.
Then we get the following equations: ) satisfying the lemma.

Proofs of Main Theorems
In this section, we will present the proofs of the main results.
Proof of Theorem 1.If (1) holds, we have the fact that   (D ⊥ ℎ ) is contained in (  ).It follows that     = 0.The desired result follows from the equation   =     +     .Case (2) is similar.Case (3) is easy to get the desired result.
Then it suffices to prove that  1 = 0 in condition (4) when both  1 and  2 are not constants.For fixed 1 ≤  ≤ , let In the same way, we get the following: Applying Theorem 5.6 in [6] again, there exist two constants  2 ,  3 such that Therefore we have for all  ∈ N  − {0} and 1 ≤  ≤ .