The Numerical Range of Toeplitz Operator on the Polydisk

and Applied Analysis 3 It is obvious that ka ∈ A2 D and 〈ka, ka〉 ∫


Introduction
Let H be a Hilbert space with an inner product, and let T be a bounded linear operator on H.The numerical range W T of T is the subset of the complex plane C defined by It is well known that W T is a convex set whose closure contains the spectrum of T , which denoted by σ T .If T is a normal operator, then the closure of W T is the convex hull of σ T .Moreover, it is also well known that each extreme point of W T is an eigenvalue of T .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 n-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.
Let D be the unit disk in the complex plane.For a fixed integer n, the unit polydisk D n is the cartesian product of n copies of D. Let L 2 D n denote the usual Lebesgue space with respect to the volume measure V V n on D n normalized to have total mass 1.The Bergman space A 2 D n is the closed subspace of L 2 D n consisting of all holomorphic functions on D n .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 P be the orthogonal projection from L 2 D n onto A 2 D n .The Toeplitz operator is the linear operator defined by Here ∂ j denotes the complex partial differentiation with respect to the jth variable.
Recall that a complex-valued function f ∈ C 2 D n is said to be pluriharmonic if

2.3
Note that each pluriharmonic function is n-harmonic function.
In this section, we give characterizations of Toeplitz operator with symbols n-harmonic acting on the Bergman space on the polydisk.For this purpose, we need the following result see 11 .
Proof.For each a ∈ D n , let k a denote the normalized kernel, namely,

2.5
Abstract and Applied Analysis then ϕ a is an automorphism of D n , and ϕ a • ϕ a is the identity on D n .Since the real Jacobian of ϕ a is given by |k a z | 2 , we have whenever the integrals make sense.In particular, by Lemma 2.1, we obtain for function h integrable and holomorphic on D n .Hence Recall that T u ≥ 0 means T u f, f ≥ 0 for every f ∈ A 2 D n .Using Theorem 2.2, we obtain the following result.Proof.First we assume that T u ≥ 0. By the definition of W T u , we have W T u ⊂ 0, ∞ .By Theorem 2.2, we see that that is, u ≥ 0.

Abstract and Applied Analysis
Conversely, suppose that u ≥ 0. For every f ∈ A 2 D n , we have Hence T u ≥ 0 by the arbitrary of f.The proof of the theorem is completed.
Theorem 2.4.Let u be a bounded n-harmonic function on D n .If W T u lies in the upper half-plane and contains 0, then T u must be self-adjoint.
Proof.We modify the proof of Theorem 3 in 5 .From the assumption, we have Im T u f, f ≥ 0 for every f ∈ A 2 D n .In addition, for every f ∈ A 2 D n .Hence T Imu ≥ 0 by the arbitrary of f.By Theorem 2.3, we have Imu ≥ 0. On the other hand, since W T u contains 0, there exist some g ∈ A 2 with g, g 1 such that T u g, g 0. Therefore, 0 Im T u g, g Im Imu g 2 dV.

2.14
We obtain Imu |g| 2 0 by the fact that Imu ≥ 0. Because g / 0, we see that Imu 0. It follows that u is real and T u is self-adjoint.
Theorem 2.5.Let u be a bounded n-harmonic function on Proof.The proof is similar to the proof of Theorem 4 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 u be a bounded n-harmonic function on D n .If T u is not normal on A 2 D n , then W T u is the interior of its closure.
Lemma 2.7 See 1 .If W T is a line segment, then T is normal.
We will consider the problem of when the converse of this fact is also true.First, we prove the following three results.Proof.If m ∈ W T u , then m is an extreme point of W T u and hence is an eigenvalue of T u .Therefore, there exists a nonzero f ∈ A 2 D n such that T u f mf, that is, P uf − mf 0. We obtain Since u−m ≥ 0 on D n , we get u−m |f| 2 0. Because f is nonzero, we obtain u m.Therefore u is a constant, which is a contradiction.So m / ∈ W T u .Similarly to the above proof we get M / ∈ W T u .
Theorem 2.9.Let u be a bounded real function on D n .Then σ T u ⊂ m, M , where m inf u and M sup u.

2.16
We obtain

2.17
It follows from the last inequality that T u z −λ and T u−λ are invertible.Because T u−λ T u − λ, we get λ / ∈ σ T u .Now we assume that u−λ < 0 on D n .From the above proof and the facts that −u λ > 0 and

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 b 2 D n is the space of all pluriharmonic functions in L

Theorem 2 . 3 .
Let u be a bounded n-harmonic function on D n .Then T u ≥ 0 if and only if u ≥ 0.

Proposition 2 . 8 .
Let u be bounded real n-harmonic on D n .If u is nonconstant, then m, M / ∈ W T u , where m inf u and M sup u.

3
It is obvious that k a ∈ A 2 D n and , a 2 , . . ., a n ∈ D n , we let ϕ a ϕ a 1 , ϕ a 2 , . . ., ϕ a n , where each ϕ a i is the usual M öbius map on D given by Let u be a bounded pluriharmonic function on D n .Then T u is normal onA 2 D n if and only if u D n is a part of a line in C.Proof.The proof is similar to 12, Proposition 13 .We omit the details.Let u be a bounded nonconstant pluriharmonic function onD n .If T u is normal on A 2 D n , then W T u is an open line segment.Proof.By Lemma 2.11, u D n is a part of a line in C when T u is normal.Therefore, there exist constants s, t ∈ C and a nonconstant bounded real pluriharmonic function v such that u sv t on D n .Since each pluriharmonic function is n-harmonic function, by Theorem 2.10, we have W T v m, M , where m inf v and M sup v.For a given bounded linear operator T on a Hilbert space, we note that u is an open line segment.
2D n .It is well known that b 2 D n is a closed subspace of L 2 and hence is a Hilbert space.Hence, for each z ∈ D n , there exists a unique function R z ∈ b 2 D n called the pluriharmonic Bergman kernel, which has the following reproducing property: ∈ b 2 .From this reproducing formula, it follows that the orthogonal projection Q from L 2 D n onto b 2 D n is realized as an integral operator L 2 .It is well known that a function f ∈ C 2 D n is pluriharmonic if and only if it admits a decomposition f g h, where g and h are holomorphic.Furthermore, if f ∈ b 2 D n , then it is not hard to see g, h ∈ A 2 D n .Hence