1. Introduction

Let be a Hilbert space with an inner product, and let be a bounded linear operator on . The numerical range of is the subset of the complex plane defined by It is well known that is a convex set whose closure contains the spectrum of , which denoted by . If is a normal operator, then the closure of is the convex hull of . Moreover, it is also well known that each extreme point of is an eigenvalue of . See [1, 2] for more information of the numerical range of a operator.

Brown and Halmos in [3] and Klein in [4] studied the numerical range of arbitrary Toeplitz operator on the Hardy space of the unit disk. Thukral studied the numerical range of Toeplitz operator with harmonic symbol on the Bergman space of the unit disk in [2]. On the Bergman space and pluriharmonic Bergman space of the unit ball, the numerical range and normality of Toeplitz operator was described in [5].

In this paper, we consider the same problem on the Bergman space and pluriharmonic Bergman space of the polydisk. We first study some relations between the numerical range and normality of the Toeplitz operator with -harmonic function symbols acting on the Bergman space on the polydisk. Next, we consider the same problem on the pluriharmonic Bergman space of the polydisk. Our results show that as case of the ball hold on the polydisk.

2. Toeplitz Operators on the Bergman Space of the Polydisk

Let be the unit disk in the complex plane. For a fixed integer , the unit polydisk is the cartesian product of copies of . Let denote the usual Lebesgue space with respect to the volume measure on normalized to have total mass . The Bergman space is the closed subspace of consisting of all holomorphic functions on . Some information on Bergman-type spaces on the polydisk (including Bergman projections) can be found, for example, in [6–10] (see also the reference therein).

Let be the orthogonal projection from onto . The Toeplitz operator with symbol is the linear operator defined by for functions .

A function is called -harmonic if is harmonic in each variable separately. More explicitly, is -harmonic if Here denotes the complex partial differentiation with respect to the th variable.

Recall that a complex-valued function is said to be pluriharmonic if Note that each pluriharmonic function is -harmonic function.

In this section, we give characterizations of Toeplitz operator with symbols -harmonic acting on the Bergman space on the polydisk. For this purpose, we need the following result (see [11]).

Lemma 2.1. Let . Then is an -harmonic function on if and only if for every .

Theorem 2.2. Let be a bounded -harmonic function on . Then .

Proof. For each , let denote the normalized kernel, namely, It is obvious that and for every .
For each , we let , where each is the usual Mbius map on given by
then is an automorphism of and is the identity on . Since the real Jacobian of is given by , we have whenever the integrals make sense. In particular, by Lemma 2.1, we obtain for function integrable and holomorphic on . Hence for every . Therefore .

Recall that means for every . Using Theorem 2.2, we obtain the following result.

Theorem 2.3. Let be a bounded -harmonic function on . Then if and only if .

Proof. First we assume that . By the definition of , we have . By Theorem 2.2, we see that that is, .
Conversely, suppose that . For every , we have
Hence by the arbitrary of . The proof of the theorem is completed.

Theorem 2.4. Let be a bounded -harmonic function on . If lies in the upper half-plane and contains , then must be self-adjoint.

Proof. We modify the proof of Theorem in [5]. From the assumption, we have for every . In addition, for every . Hence by the arbitrary of . By Theorem 2.3, we have .
On the other hand, since contains , there exist some with such that . Therefore,
We obtain by the fact that . Because , we see that . It follows that is real and is self-adjoint.

Theorem 2.5. Let be a bounded -harmonic function on . If is not open in , then is normal on .

Proof. The proof is similar to the proof of Theorem of [5]. We omit the details.

Since an open convex set is the interior of its closure, we obtain the following corollary.

Corollary 2.6. Let be a bounded -harmonic function on . If is not normal on , then is the interior of its closure.

Lemma 2.7 (See [1]). If is a line segment, then is normal.

We will consider the problem of when the converse of this fact is also true. First, we prove the following three results.

Proposition 2.8. Let be bounded real n-harmonic on . If is nonconstant, then , where and .

Proof. If , then is an extreme point of and hence is an eigenvalue of . Therefore, there exists a nonzero such that , that is, . We obtain Since on , we get . Because is nonzero, we obtain . Therefore is a constant, which is a contradiction. So .
Similarly to the above proof we get .

Theorem 2.9. Let be a bounded real function on . Then , where and .

Proof. Suppose that , then either or on . First we assume that and choose such that We obtain It follows from the last inequality that and are invertible. Because , we get .
Now we assume that on . From the above proof and the facts that and
we get the desired result.

Theorem 2.10. Let be a bounded nonconstant real -harmonic function on . Then

Proof. Using Proposition 2.8, Theorems 2.2 and 2.9, similarly to the proof of Theorem of [5], we get the desired result. We omit the details.

Lemma 2.11. Let be a bounded pluriharmonic function on . Then is normal on if and only if is a part of a line in .

Proof. The proof is similar to [12, Proposition ]. We omit the details.

Theorem 2.12. Let be a bounded nonconstant pluriharmonic function on . If is normal on , then is an open line segment.

Proof. By Lemma 2.11, is a part of a line in when is normal. Therefore, there exist constants and a nonconstant bounded real pluriharmonic function such that on . Since each pluriharmonic function is -harmonic function, by Theorem 2.10, we have , where and . For a given bounded linear operator T on a Hilbert space, we note that It follows from that Therefore is an open line segment.

3. Toeplitz Operators on the Pluriharmonic Bergman Space

In this section, we consider the same problem for Toeplitz operators acting on the pluriharmnoic Bergman space in the polydisk. The pluriharmonic Bergman space is the space of all pluriharmonic functions in . It is well known that is a closed subspace of and hence is a Hilbert space. Hence, for each , there exists a unique function called the pluriharmonic Bergman kernel, which has the following reproducing property: for every . From this reproducing formula, it follows that the orthogonal projection from onto is realized as an integral operator for .

It is well known that a function is pluriharmonic if and only if it admits a decomposition , where and are holomorphic. Furthermore, if , then it is not hard to see . Hence Therefore where is the well-known holomorphic Bergman kernel. By (3.2) and (3.4), we see that admits the following integral representation: for .

Let . The Toeplitz operator with symbol is defined by for . The operator is densely defined. In fact, we have for any . Using the the same arguments as the Section 2, we have the following results.

Theorem 3.1. Let be a bounded n-harmonic function on . Then .

Theorem 3.2. Let be bounded n-harmonic function on . Then if and only if .

Theorem 3.3. Let be bounded n-harmonic function on . If lies in the upper half-plane and contains 0, then must be self-adjoint.

Theorem 3.4. Let be bounded n-harmonic function in . If is not open in , then is normal on .

Theorem 3.5. Let be nonconstant real n-harmonic function on . Then one has where and .

We also need a corresponding result of Lemma 2.11 (see [13, Theorem ]).

Lemma 3.6. Let be a bounded pluriharmonic function on . Then is normal on if and only if is a part of a line in .

Theorem 3.7. Let be bounded nonconstant pluriharmonic function on . If is normal on , then is an open line segment.