# The Numerical Range of Toeplitz Operator on the Polydisk

Authors: Dinggui Gu
Journal: Abstract and Applied Analysis
 (2009)

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## Abstract

The numerical range and normality of Toeplitz operator acting on the Bergman space and pluriharmonic Bergman space on the polydisk is investigated in this paper.


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## Body

## 1. Introduction

LetH 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
(1.1)W(T)={〈Tx,x〉:x∈H,〈x,x〉=1}.
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 withn-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

LetD be the unit disk in the complex plane. For a fixed integer n, the unit polydisk Dn is the cartesian product of n copies of D. Let L2(Dn) denote the usual Lebesgue space with respect to the volume measure V=Vn on Dn normalized to have total mass 1. The Bergman space A2(Dn) is the closed subspace of L2(Dn) consisting of all holomorphic functions on Dn. 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).LetP be the orthogonal projection from L2(Dn) onto A2(Dn). The Toeplitz operator Tu:A2(Dn)→A2(Dn) with symbol u∈L∞(Dn) is the linear operator defined by
(2.1)Tuf=P(uf)
for functions f∈A2(Dn).A functionu∈C2(Dn) is called n-harmonic if u is harmonic in each variable separately. More explicitly, u is n-harmonic if
(2.2)∂j∂̅ju=0,j=1,2,…,n.
Here ∂j denotes the complex partial differentiation with respect to the jth variable.Recall that a complex-valued functionf∈C2(Dn) is said to be pluriharmonic if
(2.3)∂j∂¯kf=0,j,k=1,2,…,n.
Note that each pluriharmonic function is n-harmonic function.In this section, we give characterizations of Toeplitz operator with symbolsn-harmonic acting on the Bergman space on the polydisk. For this purpose, we need the following result (see [11]).Lemma 2.1.
Letu∈L∞(Dn). Then  u is an  n-harmonic function on  Dn if and only if
(2.4)∫Dnu∘φadV=u(a)
for every a∈Dn.Theorem 2.2.
Letu be a bounded  n-harmonic function on  Dn. Then  u(Dn)⊂W(Tu).Proof.
For eacha∈Dn, let ka denote the normalized kernel, namely,
(2.5)ka(z)=Ka(z)Ka(a)=∏j=1n(1-|aj|2)(1-aj¯zj)2.
It is obvious that ka∈A2(Dn) and
(2.6)〈ka,ka〉=∫Dn|ka|2dV=1
for every a∈Dn.
For eacha=(a1,a2,…,an)∈Dn, we let φa=(φa1,φa2,…,φan), where each φai is the usual Möbius map on D given by
(2.7)φai(zi)=ai-zi1-ai¯zi,
then φa is an automorphism of Dn, and φa∘φa is the identity on Dn. Since the real Jacobian of φa is given by |ka(z)|2, we have
(2.8)∫Dnh∘φadV=∫Dnh|ka(z)|2dV
whenever the integrals make sense. In particular, by Lemma 2.1, we obtain
(2.9)∫Dnh∘φadV=h(a)
for function h integrable and holomorphic on Dn. Hence
(2.10)〈Tuka,ka〉=〈P(uka),ka〉=〈uka,ka〉=∫Dnu|ka|2dV=∫Dnu∘φadV=u(a)
for every a∈Dn. Therefore u(Dn)⊂W(Tu).Recall thatTu≥0 means 〈Tuf,f〉≥0 for every f∈A2(Dn). Using Theorem 2.2, we obtain the following result.Theorem 2.3.
Letu be a bounded  n-harmonic function on  Dn. Then  Tu≥0 if and only if  u≥0.Proof.
First we assume thatTu≥0. By the definition of W(Tu), we have W(Tu)⊂[0,∞). By Theorem 2.2, we see that
(2.11)u(Dn)⊂W(Tu)⊂[0,∞),
that is, u≥0.
Conversely, suppose thatu≥0. For every f∈A2(Dn), we have
(2.12)〈Tuf,f〉=〈P(uf),f〉=〈uf,f〉=∫Dnu|f|2dV≥0.
Hence Tu≥0 by the arbitrary of f. The proof of the theorem is completed.Theorem 2.4.
Letu be a bounded  n-harmonic function on  Dn. If  W(Tu) lies in the upper half-plane and contains  0, then  Tu must be self-adjoint.Proof.
We modify the proof of Theorem3 in [5]. From the assumption, we have Im〈Tuf,f〉≥0 for every f∈A2(Dn). In addition,
(2.13)Im〈Tuf,f〉=Im〈P(uf),f〉=Im〈uf,f〉=∫Dn(Imu)|f|2dV=〈TImuf,f〉
for every f∈A2(Dn). Hence TImu≥0 by the arbitrary of f. By Theorem 2.3, we have Imu≥0.
On the other hand, sinceW(Tu) contains 0, there exist some g∈A2 with 〈g,g〉=1 such that 〈Tug,g〉=0. Therefore,
(2.14)0=Im〈Tug,g〉=Im∫Dnu|g|2dV=∫Dn(Imu)|g|2dV.
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 Tu is self-adjoint.Theorem 2.5.
Letu be a bounded  n-harmonic function on  Dn. If  W(Tu) is not open in  ℂ, then  Tu is normal on  A2(Dn).Proof.
The proof is similar to the proof of Theorem4 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.
Letu be a bounded  n-harmonic function on  Dn. If  Tu is not normal on  A2(Dn), then  W(Tu) is the interior of its closure.Lemma 2.7 (See [1]).
IfW(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.Proposition 2.8.
Letu be bounded real n-harmonic on  Dn. If  u is nonconstant, then  m,M∉W(Tu), where  m=infu and  M=supu.Proof.
Ifm∈W(Tu), then m is an extreme point of W(Tu) and hence is an eigenvalue of Tu. Therefore, there exists a nonzero f∈A2(Dn) such that Tuf=mf, that is, P(uf-mf)=0. We obtain
(2.15)0=〈P(uf-mf),f〉=〈(u-m)f,f〉=∫Dn(u-m)|f|2dV.
Since u-m≥0 on Dn, 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(Tu).
Similarly to the above proof we getM∉W(Tu).Theorem 2.9.
Letu be a bounded real function on  Dn. Then  σ(Tu)⊂[m,M], where  m=infu and  M=supu.Proof.
Suppose thatλ∉[m,M], then either u-λ>0 or u-λ<0 on Dn. First we assume that u-λ>0 and choose ϵ>0 such that
(2.16)supz∈Dn|ϵ(u(z)-λ)-1|<1.
We obtain
(2.17)∥Tϵ(u-λ)-I∥=∥Tϵ(u-λ)-1∥≤∥ϵ(u-λ)-1∥∞<1.
It follows from the last inequality that Tϵ(u(z)-λ) and Tu-λ are invertible. Because Tu-λ=Tu-λ, we get λ∉σ(Tu).
Now we assume thatu-λ<0 on Dn. From the above proof and the facts that -u+λ>0 and
(2.18)Tu-λ=Tu-λ=-T-u+λ,
we get the desired result.Theorem 2.10.
Letu be a bounded nonconstant real  n-harmonic function on  Dn. Then
(2.19)W(Tu)=(m,M),wherem=infu,M=supu.Proof.
Using Proposition2.8, Theorems 2.2 and 2.9, similarly to the proof of Theorem 8 of [5], we get the desired result. We omit the details.Lemma 2.11.
Letu be a bounded pluriharmonic function on  Dn. Then  Tu is normal on  A2(Dn) if and only if  u(Dn) is a part of a line in  ℂ.Proof.
The proof is similar to [12, Proposition 13]. We omit the details.Theorem 2.12.
Letu be a bounded nonconstant pluriharmonic function on  Dn. If  Tu is normal on  A2(Dn), then  W(Tu) is an open line segment.Proof.
By Lemma2.11, u(Dn) is a part of a line in ℂ when Tu is normal. Therefore, there exist constants s,t∈ℂ and a nonconstant bounded real pluriharmonic function v such that u=sv+t on Dn. Since each pluriharmonic function is n-harmonic function, by Theorem 2.10, we have W(Tv)=(m,M), where m=infv and M=supv. For a given bounded linear operator T on a Hilbert space, we note that
(2.20)W(αT+β)=αW(T)+β,α,β∈ℂ.
It follows from Tu=sTv+t that
(2.21)W(Tu)=sW(Tv)+t=(sm+t,sM+t).
Therefore W(Tu) 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 spaceb2(Dn) is the space of all pluriharmonic functions in L2(Dn). It is well known that b2(Dn) is a closed subspace of L2 and hence is a Hilbert space. Hence, for each z∈Dn, there exists a unique function Rz∈b2(Dn) called the pluriharmonic Bergman kernel, which has the following reproducing property:
(3.1)f(z)=∫Dnf(w)Rz(w)¯dV(w)
for every f∈b2. From this reproducing formula, it follows that the orthogonal projection Q from L2(Dn) onto b2(Dn) is realized as an integral operator
(3.2)Q(φ)(z)=∫Dnφ(w)Rz(w)¯dVn(w),z∈Dn
for φ∈L2.It is well known that a functionf∈C2(Dn) is pluriharmonic if and only if it admits a decomposition f=g+h̅, where g and h are holomorphic. Furthermore, if f∈b2(Dn), then it is not hard to see g,h∈A2(Dn). Hence
(3.3)b2(Dn)=A2(Dn)+A2(Dn)¯.
Therefore
(3.4)Rz=Kz+Kz¯-1,
where Kz is the well-known holomorphic Bergman kernel. By (3.2) and (3.4), we see that Q admits the following integral representation:
(3.5)Q(φ)(z)=∫Dnφ(w)(Kz(w)¯+Kz(w)-1)dV(w),z∈Dn
for φ∈L2(Dn).Letu∈L2(Dn). The Toeplitz operator tu with symbol u is defined by
(3.6)tuf=Q(uf)
for f∈b2(Dn). The operator tu is densely defined. In fact, we have Q(uf)∈b2(Dn) for any f∈H∞(Dn). Using the the same arguments as the Section 2, we have the following results.Theorem 3.1.
Letu be a bounded n-harmonic function on  Dn. Then  u(Dn)⊂W(tu).Theorem 3.2.
Letu be bounded n-harmonic function on  Dn. Then  tu≥0 if and only if  u≥0.Theorem 3.3.
Letu be bounded n-harmonic function on  Dn. If  W(tu) lies in the upper half-plane and contains 0, then  tu must be self-adjoint.Theorem 3.4.
Letu be bounded n-harmonic function in  b2(Dn). If  W(tu) is not open in  ℂ, then  tu is normal on  b2(Dn).Theorem 3.5.
Letu be nonconstant real n-harmonic function on  Dn. Then one has  W(tu)=(m,M) where  m=infu and  M=supu.We also need a corresponding result of Lemma2.11 (see [13, Theorem 1.2]).Lemma 3.6.
Letu be a bounded pluriharmonic function on  Dn. Then  tu is normal on  b2(Dn) if and only if  u(Dn) is a part of a line in  ℂ.Theorem 3.7.
Letu be bounded nonconstant pluriharmonic function on  Dn. If  Tu is normal on  b2(Dn), then  W(tu) is an open line segment.



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*Source: 757964-2009-08-24.xml*