Toeplitz Operators on Dirichlet-Type Space of Unit Ball

and Applied Analysis 3 By Hölder’s and Poincaré’s inequality, 󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 ∫


Introduction
Let B  represent the open unit ball in several complex spaces C  .The Sobolev space  ) ) where  denotes the normalized Lebesgue volume measure on B  . 2,1 (B  ) is a Hilbert space with inner product where ⟨⋅, ⋅⟩ denotes the inner product in  2 (B  , ).The Dirichlet-type space D is the subspace of all analytic functions  in  2,1 (B  ) with (0) = 0.Then, D is a closed subspace of the  2,1 (B  ).Let  be the orthogonal projection from  2,1 (B  ) onto D.  is an integral operator represented by where   () = (, ) is the reproducing kernel of D. By computation, we know

Schatten 𝑝-Class Toeplitz Operators with Unbounded Symbols
In general, if  ∉  ∞ (B  ), the space of essentially bounded functions on B  and then   is densely defined only.In the case of Hardy space, it is well known that   is bounded if and only if  is essentially bounded, and   is compact if and only if  = 0 (see Douglas [6] and Davie and Jewell [7]).However, there are indeed bounded and compact Toeplitz operators with unbounded symbols on Bergman spaces of one complex variable; in fact, Miao and Zheng [8] have introduced a class of functions, called BT, which contains  ∞ , for  ∈ BT;   is compact on  2  if and only if the Berezin transform of   vanishes on the unit circle.Zorboska [9] has proved that if  belongs to the hyperbolic BMO space, the   is compact if and only if the Berezin transform of  vanishes on the unit circle.Cima and Cuckovic [10] construct a class of unbounded functions over a Cantor set; the Toeplitz operators with these functions are compact.Essentially, if the values of the function   vanish rapidly near the unit circle in the sense of measure , then   will be compact.Cao [11] also construct compact Toeplitz operators and trace class Toeplitz operators with unbounded symbols on Bergman space of several complex variables.
In this section, we construct the compact Toeplitz operators and the Schatten -class Toeplitz operators for 0 <  < ∞ with unbounded symbols on Dirichlet-type space of several complex variables.
For preparation, we introduce a special set in B  , which is important for building our main results in this section.
For  > 0 and  ∈ B  , the boundary of B  , let then it is obvious that (, ) is a domain in B  , which is called circular-like cone with vertex , because it looks like a circular cone.For 0 <  < 1, set B  () = { ∈ C  : || < } to be the ball with radius .We use   to denote the area measure on B  (), for  = 1;  1 =  is the normalized area measure on B  ; then there holds   (B  ()) = ( 2−1 ) and  1 (B  ) = 1.For  > 0, because of the property of (, ), there exits proper () > 0 such that, for any 0 <  < 1, where  is a constant number that is independent of  and .
Using Lemma 3, we can construct a compact Toeplitz operator with a symbol that is unbounded on any neighborhood of every point in unit surface.
Following the above theorem, we construct Schatten class Toeplitz operators for all 0 <  < ∞ whose symbols are also unbounded on any neighborhood of every point in unit surface.
where  is a constant independent of .Integrating by parts, we find where () is a constant that depends on .Let then  is compact operator by Theorem 4. Note that  is positive operator; thus, for 0 <  ≤ 1, where  = ∑ ∞ =1 (1/2  ).Changing the order of summation, we have Note that, for any continuous function  on B  , we have where   = ( 1 , . . .,  −1 ),   =   ,  = √ −1.Then, by induction, we see obviously that Further, there are constants  1 ,  2 > 0 such that Namely,   is a   -class operator for 0 <  ≤ 1.
For 1 <  < ∞, let  =  + 2 + 2; let  be the operator defined in (27).By the above proof, we have By the convergence in (33), we know that there exists a  0 such that, for That is,   is a   -class operator for 1 <  < ∞.

Toeplitz Operators with Radial Symbols
Is the product of two Toeplitz operators equal to a Toeplitz operator?In general, the answer is negative, but Brown and Halmos [14] showed that two bounded Toeplitz operators   and   commute on the Hardy space if and only if (I) both  and  are analytic, (II) both  and  are analytic, or (III) one is a linear function of the other.For more details on the same question for Toeplitz operators on the Bergman spaces of one variable, see Cuckovic et al. [15,16] and Louhichi et al. [17,18].For the case of the Bergman spaces of several variable, Zheng [19] studied commuting the Toeplitz operators with pluriharmonic symbols on the unit ball in C  .Recently, Quiroga-Barranco and Vasilevski [20,21] gave the description of many (geometrically defined) classes of commuting Toeplitz operators on the unit ball.Zhou and Dong [22] discussed commuting Toeplitz operators with radial symbols on the unit ball.
In this section, we discuss the same questions for the Dirichlet-type space on the unit ball.The rest of this section is organized as follows.First, we introduce some basic properties of the Mellin transform and Mellin convolution which will be needed later.Second, we discuss when the product of two Toeplitz operators with radial symbols is a Toeplitz operator.Then, the zero-product problem for several Toeplitz operators with radial symbols on the Dirichlet-type space is investigated.Finally, the corresponding commuting problem of Toeplitz operators with quasihomogeneous symbols is studied.

Mellin Transform and Mellin Convolution.
Mellin transform, the most useful tool we use later, is defined as follows.
It is known that û is a bounded analytic function in the half plane { : Re() > 2}.It is important and helpful to know that the Mellin transform is uniquely determined by its values on an arithmetic sequence of integers.In fact, we have the following classical theorem (see [23]).
When considering the product of two Toeplitz operators, we need a known fact about the Mellin convolution of their symbols.If  and  are defined on [0, 1), then their Mellin convolution is defined by The Mellin convolution theorem states that and that if  and  are in  1 ([0, 1], ), then so is (  *  )().

Products of Toeplitz Operators with Radial Symbols.
For convenience, we use  −  to denote The notations  ⪰  and  ⊥  mean, respectively, if () = () for any unitary transform  of C  .Then, for each radial function , we define the function ũ on [0, 1) by ũ() = (e) where e is a unit vector in C  .It is trivial that ũ is well defined.In the following, we will often identify an integrable radial function  on the unit ball with the corresponding function ũ defined on the interval [0, 1).
In the following, some basic results concerning Toeplitz operators with radial symbols on the Dirichlet-type space of the unit ball are obtained.
Theorem 10 (see Lu and Sun [5]).Let  be a radial function and Theorem 11.Let  ∈  1 (B  ) be a radial function in which   is bounded on D; then, for any  ∈ N  − {(0, . . ., 0)}, Proof.By the definition of Toeplitz operator, we have From [13], we know the unique nonzero item in (43) is A direct computation shows that Theorem 11 shows that the Toeplitz operator with a radial symbol on the Dirichlet-type space of the unit ball acts in a very simple way.On the other hand, the Toeplitz operators of Theorem 11 must be Toeplitz operators with radial symbols.Theorem 12. Let  ∈  1 (B  ).For each multi-index , if there exists  || ∈ C which depends only on || such that     =  ||   , then  is a radial function.
Proof.Suppose     =  ||   ; then, for any unitary transform  of C  with there holds where  1 , . . .,   are multi-indexes.By the definition of Toeplitz operator, note that ⟨, ⟩ = ∑  =1     ; we have From Theorem 12, V is a radial function.Moreover,  V is obviously a bounded operator.
Grudsky et al. [24] gave a particular answer to this question considering the Toeplitz operators with radial symbols.In [22], Zhou and Dong discussed the same question about the Toeplitz operators with radial symbols; they gave a different way to characterize when the product of two Toeplitz operators is equal to a Toeplitz operator.Let  1 and  2 be two radial functions on B  which induce bounded Toeplitz operators, and let further The formal construction (inverse Fourier-Laplace transform) defines a holomorphic function in the upper half plane Π ⊂ C which coincides on the real axis with the inverse Fourier transform F −1  1 () of the function  1 ().Theorem 3.7 of [24] shows that if the function belongs to Wiener ring  0 of the inverse Fourier transforms of summable functions, then there exists a Toeplitz operator with the radial symbol V such that   1   2 =  V .The following theorem will give the condition for the product of two Toeplitz operators with radial symbols to be a Toeplitz operator on Dirichlet-type space.
Theorem 14.Let  1 and  2 be two radial functions on B  which induce bounded Toeplitz operators.Then,   1   2 is equal to the Toeplitz operator  V if and only if Proof.For any , it follows from (42) and (49) that if and only if A direct computation gives where 4 −2 () is well defined if Re() > 2.By Remark 9, (56) is equivalent to (53).
For some products of Toeplitz operators, the following fun result is obtained.
Corollary 15.Let  and  be two real numbers greater than or equal to −2.Then, Proof.By Theorem 14,  ||   ||  is equal to the Toeplitz operator  V if and only if A direct calculation shows that The desired result follows from (59).
Axler and Cuckovic [16] and Choe and Koo [25] study, respectively, the zero-product problem for two Toeplitz operators with harmonic symbols on unit disk and on the Bergman spaces of the unit ball.In [22], Zhou and Dong discuss the same question about the Toeplitz operators with radial symbols.In the following theorem, we will solve the zero-product problem for several Toeplitz operators with radial symbols acting on the Dirichlet-type space of the unit ball.
By Theorem 16, we can show that the only idempotent Toeplitz operators with radial symbols are 0 and .

Commuting Toeplitz Operators with Quasihomogeneous
Symbols on Dirichlet-Type Space.In this subsection, commuting Toeplitz operators with bounded quasihomogeneous symbols on the Dirichlet-type space of the unit ball are discussed.The definition of the quasihomogeneous function on the unit disk has been given in [17,26], and the definition on the unit ball has been given in [22].Definition 18.Let  ∈ Z  and  ∈  1 (B  , ). is called a quasihomogeneous function of quasihomogeneous degree  if  is of the form   , where  is a radial function; that is, for any  in the unit sphere  and  ∈ [0, 1).

Remark 19.
It is obvious that any  ∈ Z  can be uniquely written as  − , where  and  are two multi-indexes such that  ⊥ .Thus, in this paper, we always define the function for any  ∈ Z  .
In the following lemma, a result which we will use often is given.(66)

Lemma 20. Let 𝛽, 𝛾 be two multi-indexes and let
By calculation, we get the desired results.Now, we can discuss the commuting problem of Toeplitz operators with quasihomogeneous symbols on Dirichlet-type space.
Theorem 21.Let ,  be two multi-indexes and let  and V be two bounded radial functions on B  .If  is not equal to 0, then Assume that || ̸ = ||; without loss of generality, we can also assume that || > ||, for otherwise we could take the adjoins.Set It will show later that ∑ ∈ (1/||) = ∞, which implies that V = 0 by Remark 9.
If ∑ ∈ (1/||) < ∞, we will induce a contradiction.Let where   is the complement of  in N; then On the other hand, for any || ∈   , (70) gives Denote then  is analytic and bounded on { : Re() > 0} since V is bounded.Moreover, (74) implies that According to Theorem 8,  must be zero; thus, For any integer  0 greater than 2, the above equation gives that If we denote (1/( 0 − 2))û( 0 ) by the constant , we obtain Multiply two sides of (79) by 1/( 0 + 2(|| − ||)), which leads to where I denotes the constant function with value one, for any  ∈ N, by Remark 9 again, and, then, clearly, (I * || −2 * ) is constant.By calculation, we know that then, derive (81) with respect to variable ; we get and, then, clearly,  is 0, which is a contradiction.Thus, we conclude that either || = || or V = 0. Conversely, if || = || or V = 0, then we can easily show that (70) holds, and, consequently, for each multi-index , which implies      V and   commute.
It was shown in [27] that a Toeplitz operator with a radial symbol on the Bergman spaces of the unit disk  may only commute with another such operator with a radial symbol, but it is not true in higher dimensions by the theorem above.It is known that every function  ∈  2 (, ) has the decomposition where   () are square integrables in [0, 1] with respect to the measure .More details can be found in [27].Similarly, let  = { : B  → C radial : ∫ Therefore,   commutes with   if and only if (  ) = () for almost all  ∈ R and  ∈ B  .
Remark 24.In the one-dimensional case, a function satisfying (  ) = () is exactly a radial function, so this corollary coincides with Theorem 6 of [27].
There are lots of examples of functions of the form   , which are the symbols of commuting Toeplitz operators (see [27]), but the following theorem will show that two Toeplitz operators with quasihomogeneous symbols of degrees  and −, respectively, commute only in the trivial case.
Theorem 25.Suppose that  and  are two nonzero multiindexes, and let  and V be two bounded radial functions on B  .If then,  = 0 or V = 0. Proof.

Theorem 4 .
There exists a function  ∈