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Abstract and Applied Analysis
Volume 2012 (2012), Article ID 958201, 21 pages
Toeplitz Operators on the Dirichlet Space of
1School of Mathematical Sciences, Dalian University of Technology, Liaoning 116024, China
2College of Computer and Information, Fujian Agriculture and Forestry University, Fujian 350002, China
Received 13 August 2012; Accepted 10 October 2012
Academic Editor: Alberto Fiorenza
Copyright © 2012 HongZhao Lin and YuFeng Lu. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
We study the algebraic properties of Toeplitz operators on the Dirichlet space of the unit ball . We characterize pluriharmonic symbol for which the corresponding Toeplitz operator is normal or isometric. We also obtain descriptions of conjugate holomorphic symbols of commuting Toeplitz operators. Finally, the commuting problem of Toeplitz operators whose symbols are of the form is studied.
For any integer , let denote the open unit ball of and denote the normalized Lebesgue measure on . The Sobolev space is defined to be the completion of smooth functions on which satisfy The inner product on is defined by The Dirichlet space of is the closed subspace consisting of all holomorphic functions in . It is easily verified that each point evaluation is a bounded linear functional on . Hence, for each , there exists a unique reproducing kernel such that Actually, it can be calculated that , where is a multi-index, , and . For multi-indexes and , the notation means that and means that and .
Let be the orthogonal projection from onto . By the explicit formula for , we have Let . Given , the Toeplitz operator with symbol is the linear operator on defined by It is easy to verify that the Toeplitz operator is always bounded, whenever .
On the Hardy space of the unit circle, a well-known theorem of Brown and Halmos  has shown that two Toeplitz operators with bounded symbols commute if and only if one of the followings holds: (i) both symbols are holomorphic; (ii) both symbols are antiholomorphic; (iii) a nontrivial linear combination of the symbols is constant.
On the Bergman space, the commuting problem is more complicated. Axler and uković  proved that Brown-Halmos Theorem also holds for Toeplitze operators with bounded harmonic symbols. However, the corresponding problem for Toeplitz operator with general symbol remains open.
In recent years, more and more attention has been paid to the Toeplitz operators on Dirichlet spaces. The algebric properties of the Toeplitz operators on the classical Dirichlet spaces of the unit disc have been investigated intensively in [6–13]. Cao considered Fredholm properties of Toeplitz operators with symbols in . Lee showed in  that Brown-Halmos's result with harmonic symbols remains vaild on the Dirichlet space of the unit disc. In , Duistermaat and Lee gave the following characterizations of the harmonic symbols for which the associated Toeplitz operators are commuting, self-adjoint, or isometric: for a harmonic symbol , is self-adjoint if and only if is a real constant function; for a harmonic symbol , is an isometry if and only if is a constant function of modulus 1; for two harmonic symbols , , and commute if and only if either and are holomorphic or a nontrivial linear combination of and is constant on . In , the corresponding problems have been investigated on the polydisc Dirichlet spaces and similar results have been obtained.
Motivated by the work of [12, 13], we study the corresponding problems on the Dirichlet spaces of . In Section 2, we give the characterizations of the pluriharmonic symbol for which the associated Toeplitz operator is self-adjoint or an isometry. In Section 3, we discuss when two Toeplitz operators with conjugate holomorphic symbols commute. At last, we concern with the commuting Toeplitz operators with symbols .
2. Characterization of Normality and Isometry
In this section, we will give the condition under which Toeplitz operators with pluriharmonic symbols are self-adjoint or isometric on . Before doing this, we first exhibit some properties of Toeplitz operators on .
Lemma 2.1. Let be holomorphic. Then the following statements hold:(1); (2); (3).
Proof. By the definition of the Toeplitz operators and the properties of the reproducing kernel, we obtain that
Lemma 2.2. Let , where and are holomorphic. If is normal, then where .
The next lemma shows there is only trivial normal Toeplitz operator with holomorphic (or antiholomorphic) symbols.
Lemma 2.3. Let be holomorphic. Then the following statements are equivalent:(1) is normal;(2) is normal;(3) is normal;(4) is a constant function on .
Proof. By Lemma 2.2 (with ), we have that
This along with the fact that proves that , for . This shows that is a constant.
Since is normal, it follows from Lemma 2.2 that which implies that is a constant function for is holomorphic.
On the other hand, Lemma 2.1 ensures that It follows that , for all . Hence is a constant, as desired.
Suppose is normal. Using Lemma 2.2, where . We conclude that , for , since by direct computation and Lemma 2.1 Hence is a constant.
The converse implications are clear. The proof is complete.
On the Hardy space and the Bergman space, we always have . So it is easy to see that is self-adjoint (i.e., ) if and only if is a real-valued function. However, on the Dirichlet space of disc and polydisc, the situations are different because is not equal to in both cases. In the following, we will study the adjoint of Toeplitz operators with pluriharmonic symbols on the Dirichlet space of .
Theorem 2.4. Let , where and are holomorphic. Then if and only if u is a constant function.
Proof. First, assume that is holomorphic. Since , for each multi-index , we have
In fact, for ,
On the other hand,
we conclude that, , for .
Second, assume that is the general pluriharmonic symbol and . In particular, we have By Lemma 2.1, we get that Since where , it follows that Equivalently, This implies that , for . So is holomorphic. The desired result follows immediately from the previous holomorphic case.
The converse implication is clear. The proof is complete.
We now characterize pluriharmonic symbols inducing self-adjoint Toeplitz operators.
Theorem 2.5. Let , where , are holomorphic. Then if and only if is a real constant function.
Proof. The “if” part is clear. Suppose and . It follows from Lemma 2.1 that
Since , by comparing the coefficients of the above two equations, we have that
Let , where . Then we have This shows that On the other hand, if is holomorphic, then Therefore, A direct computation shows that Hence,
Comparing the expressions of and , we obtain
It follows from (2.26) and (2.34) that which, according to (2.27), implies that is a real constant function. This complete the proof of the theorem.
Corollary 2.6. Let , where and are holomorphic. Then is a projection operator if and only if or .
Proof. The “if” part is clear. If is a projection, then . Theorem 2.5 implies that where is a real. Since , we see that . This proves or .
Next, we will characterize pluriharmonic symbols for which the corresponding Toeplitz operator is an isometry.
Theorem 2.7. Let , where and are holomorphic. Then the following statements are equivalent:(1) is unitary;(2) is isometric;(3) is a constant function of modulus 1.
Proof. That implies follows from the fact that unitary operator is isometric.
To prove that implies , we denote and . Recalling the proof of Theorem 2.5, we have that Calculating the norms of the above items, it follows that By the assumption, (2.37), (2.38), (2.39), and (2.40) are all equal to 1 since as well as is an isometry.
Note that , for and . Comparing (2.39) and (2.40), we obtain that and , for and . Then (2.37) implies that Therefore, and .
Finally, if is a constant function, by Theorem 2.4, we have . The desired implication follows from the fact that and .
3. Commuting Toeplitz Operators with Conjugate Holomorphic Symbols
In this section, we will study the commuting problems of Toeplitz operators with conjugate holomorphic symbols. By the definition of , if is holomorphic, then . Therefore, for two holomorphic symbols , , , and commute. It is natural to ask when and commute, The following theorem shows that commutes with only in the trival case. In this section, we may always assume and .
Theorem 3.1. Let and be holomorphic. Then if and only if for , , ,
Proof. Suppose reproducing kernel = . Without loss of generality, we may assume .
Note that for , where . It follows that Therefore, we have Similarly, we have that Observe that (3.4) and (3.5) have the same coefficients of , it follows that if and only if Since , the desired result is obtained.
Corollary 3.2. If for , where is a positive integer, then .
For example, since . On the Dirichlet space of the unit disc or polydisc, Dusitermaat, Lee, Geng, and Zhou prove that for holomorphic functions and , if and only if , , and 1 are dependent, see [12, 13]. However, this is not true on the unit ball Dirichlet space by Corollary 3.2. Indeed, the condition that , , and 1 are linearly dependent is sufficient but not necessary for the commuting of and .
Next, we will discuss when Toeplitz operator with holomorphic symbol and Toeplitz operator with conjugate holomorphic symbol will commute.
Theorem 3.3. Let and be holomorphic. Then if and only if or is a constant function.
Proof. The “if” part implication is obvious. Now suppose . For each multi-index , we have
Assume that is not a constant function. Hence there exists such that .
For (3.8), let and , the coefficient of is On the other hand, if we let and in (3.9), then the coefficient of is Since and , we deduce that which implies that is a constant function. The proof is complete.
4. Commuting Toeplitz Operators with Symbols
Zhou and Dong  discussed the commuting and zero product problems of Toeplitz operators on the Bergman space of the unit ball in whose symbols are of the form where is a radial function. In , they generalized the case of the radial symbols to that of the separately quasi-homogeneous symbols. In , Grudsky et al. considered weighted Bergman spaces on the unit ball in . In terms of the Wick symbol of a Toeplitz operator, the complete information about the operator with radial symbols was given. Vasilevski  studied the Toeplitz operators with the quasi-radial quasi-homogeneous symbol. For the case of Dirichlet spaces, Chen et al. [18, 19] studied the quasi-radial Toeplitz operaors on the disk. However, little work has been done in the unit ball case. The commuting problem on it is subtle and no general answer is known. Dong and Zhou  have shown that any function in has the decomposition , where is separately radial. In this section, the commuting and zero product problems of Toeplitz operators will be concerned, which may be helpful to the further study of the commuting Toeplitz operators with general symbols. We denote and is absolutely continuous on . In the remaining part of this paper, we will always assume . A direct calculation gives the following lemma.
Lemma 4.1. Let and be radial functions. Then where .
Proof. To simiplify the statement, we denote reproducing kernel by . Notice that . For , with integration in polar coordinates we have
Since with integration by part, the desired result is obvious.
For , it is easy to see that The proof is complete.
We now characterize the commuting Toeplitz operators whose symbols are of the form , where .
Theorem 4.2. Let , , , . if and only if + holds for any multi-index .
Proof. For any multi-index , by Lemma 4.1, it follows that Since , the result is followed.
A particular case of the above theorem is . In this case if and only if Thus we immediately have the following result.
Corollary 4.3. Let , , . If , then .
If is absolutely continuous on [0,1), integrating by parts, one has , for any positive integer . Thus, using Lemma 4.1 one can get the following lemma which will be often used in the sequel.
Lemma 4.4. Let , , one has
By Lemma 4.4, a regular argument shows the results below.
Theorem 4.5. Let and . Then the followings hold.(1). (2) if and only if .(3)Let for , if and only if each .
Before discussing the commutivity of Toeplitz operator with symbols , one needs the following lemma which can be obtained by direct computation.
Lemma 4.6. Let multi-index and . Then where and means that there exists such that .
Proof. For , We get that
For , we have
If there exists such that , then Thus the proof is complete.
Note that if , then . It follows that
The following theorem gives some properties of the Toeplitz operator with symbols .
Theorem 4.7. Let , , , for and , , . Then the following assertions hold.(1) if and only if = .(2) if and only if one of the following holds:(i) and ;(ii)There exists where such that .(3) if and only if and for each , .
Proof. Assertions and are the direct consequence of Lemma 4.6. We only need to prove assertion . By Lemma 4.6, for , since , we have
It is obvious that
holds for .
For , we obtain Since , then the desired result is obvious.
In the assertion of Theorem 4.7, if , then we get
Therefore, it is easy to get the following corollary.
Corollary 4.8. Let , . Then if and only if .
It is well known that on the Hardy space if and only if either or is holomorphic. However, Lemma 4.6 and Theorem 4.7 implies that a similar result does not hold on the Dirichlet space of the unit ball. Indeed, by the computation in Lemma 4.6 and Theorem 4.7, it is easy to verify that for any , , .
Theorem 4.9. Let , and , . Then if and only if or and .
Proof. For each multi-index , by Lemmas 4.4 and 4.6, we have that
Consequently, for , we conclude
Since , , it is clear that can not always equal to for all . Thus, we get
On the other hand, for , we have Combining (4.19) and (4.20), we obtain the desired result.
Finally, we will discuss when Toepitze operaor commute with .
Theorem 4.10. Let , and , . Then the following assertions hold.(1)If , if and only if or