# Numerical Ranges of Normal Weighted Composition Operators on the Fock Space ofCN

Authors: Lili Lang; Liankuo Zhao
Journal: Journal of Function Spaces
 (2018)

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## Abstract

Numerical ranges of normal weighted composition operators on the Fock space ofCN are completely characterized. The main result shows that numerical ranges of such operators are closely related to their composition symbols.


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## Body

## 1. Introduction

Inspired by results on numerical ranges of (weighted) composition operators on Hardy space [1], in this paper we give a complete characterization of numerical ranges of normal weighted composition operators on the Fock space of CN(N≥1), the N-dimensional complex Euclidean space.Recall that for a bounded operatorT on a complex Hilbert space H, the numerical range W(T) of T is defined as(1)WT=Tf,fH∣f∈H,fH=1.The Fock spaceF2 over CN is the space of analytic functions f on CN with(2)f2=12πN∫CNfz2exp⁡-z22dm2Nz,where |z| is the norm for z∈CN and dm2N denotes usual Lebesgue measure on CN. It is well known that F2 is a reproducing kernel Hilbert space with inner product(3)f,g=12πN∫CNfzgz¯exp⁡-z22dm2Nz,f,g∈F2and reproducing kernel functions(4)Kwz=exp⁡z,w2,w,z∈CN,where 〈z,w〉 denotes the inner product for z,w∈CN and |z|2=〈z,z〉. Usually, kw denotes the normalization of Kw,(5)kwz=KwzKw=exp⁡z,w2-w24.A weighted composition operatorCψ,φ on F2 with ψ an analytic function on CN and φ an analytic self-map of CN is defined as(6)Cψ,φf=ψf∘φ,f∈F2.Weighted composition operators on F2 have been studied extensively. It is noteworthy that weighted composition operators on F2 of C are closely related to some important transforms on L2(R) [2].In [3], normal weighted composition operators on F2 are characterized completely. We have the following result.Theorem 1.
Letψ be a nonzero analytic function on CN and φ be an analytic self-map of CN. Then Cψ,φ is a bounded normal operator on F2 if and only if(7)φz=Az+b,ψz=sKcz,z∈CN,where A is a normal operator on CN with |A|≤1,  b,c∈CN with (I-A)c=(I-A∗)b,  |b|=|c|, and s is a nonzero constant. Furthermore(8)Aξ,b+Ac=0whenever |Aξ|=|ξ| for ξ∈CN.For the presentation of our main result, we make some notations.LetA be an operator on CN. |A| means the norm of A. ker⁡A={z∈CN∣Az=0} and ran⁡A={Az∣z∈CN}. For z∈CN,  A-1z={w∈CN∣Aw=z}. I is the identity operator on CN.Forλ=λ1,λ2,⋯,λN∈CN,  α=(α1,α2,⋯,αN)∈N0N (N0 denotes the set of nonnegative integers), let λα=λ1α1λ2α2⋯λnαn.For a setE⊂C, let E¯ and co⁡E denote the closure of E and the convex set generated by E, respectively. For s∈C,  sE={su∣u∈E}. D is the open unit disk of C and T is the unit circle of C. For r>0,  D(0,r)={u∈C∣|u|<r}.Based on Theorem1, our main result reads as follows.Theorem 2.
Letφ(z)=Az+b and ψ(z)=Kc(z) such that Cψ,φ is a bounded normal operator on F2. Let λ=(λ1,λ2,⋯,λN),  λ1,λ2,⋯,λN be eigenvalues of A.
(1) Ifb∈ran⁡(I-A), then(9)WCψ,φ=exp⁡p,c2co⁡λα∣α∈N0Nwith p∈(I-A)-1b.
(2) Ifb∉ran⁡(I-A) and b=b1+b2 with b1∈ran⁡(I-A)⊥,  b2∈ran⁡(I-A), then(10)WCψ,φ=D0,exp⁡p,c+b1,p2+b124with p∈(I-A)-1b2.In [4], the spectrum of normal weighted composition operators on F2 is considered. Although the spectrum of a normal operator is closely related to its numerical ranges, it seems that a complete characterization of spectrum of normal weighted composition operators on F2 is a little more difficult. For more results on numerical ranges of (weighted) composition operators on Hardy space, see [5–8].


## 2. Proof of Main Result

In this section, we give the proof of our main result, Theorem2. First, we recall some known results on numerical ranges of operators and a class of very important unitary operators on the Fock space.LetT be a bounded operator on a complex Hilbert space H. Denote σ(T) with the spectrum of T and σp(T) the eigenvalues of T. u∈σp(T) if and only if there exists f∈H,f≠0 such that Tf=uf, and f is called an eigenvector of T corresponding to the eigenvalue u. It is well known that σp(T)⊂W(T)⊂D(0,‖T‖H)¯. If T is a normal operator, then eigenvectors that correspond to distinct eigenvalues are orthogonal. Moreover, for a normal operator T,  W(T)¯=co⁡(σ(T)) [9, Theorem 1.4-4].Forp∈CN, denote φp(z)=z-p,Up=Ckp,φp. Then Up is a unitary operator on F2 and Up-1=U-p [10, Proposition 2.3].For convenience, we make some notations.Letψ be an analytic function on CN and φ be an analytic mapping of CN. Denote(11)Φpz=φp∘φ∘φ-pzand(12)Ψpz=k-p·ψ∘φ-p·kp∘φ∘φ-pz.Then(13)U-pCψ,φUp=Ck-pψ∘φ-pkp∘φ∘φ-p,φp∘φ∘φ-p=CΨp,Φpand Cψ,φ is unitarily equivalent to CΦp,Ψp.Letφ(z)=Az+b,ψ(z)=Kc(z). Then(14)Φpz=Az+b-I-Apand(15)Ψpz=exp⁡z,c-I-A∗p2+p,c+b,p-I-Ap,p2.Now we consider normal weighted composition operators onF2 from a different view, which will simplify the characterization of such operators’ numerical ranges.For an operatorA on CN, let CA denote the composition operator defined by CAf(z)=f(Az).Theorem 3.
LetCψ,φ be a bounded normal operator on F2 with φ(z)=Az+b,ψ(z)=Kc(z).
(1) Ifb∈ran⁡(I-A), then Cψ,φ is unitarily equivalent to exp⁡(〈p,c〉/2)CA with p∈(I-A)-1b.
(2) Ifb∉ran⁡(I-A), then for any t∈R,  Cψ,φ is unitarily equivalent to eitCψ,φ.
(3) Ifb∉ran⁡(I-A) and b=b1+b2 with b1∈ran⁡(I-A)⊥,  b2∈ran⁡(I-A), then Cψ,φ is unitarily equivalent to exp⁡(〈p,c〉+〈b1,p〉/2)Cψ1,φ1, where φ1(z)=Az+b1,  ψ1(z)=K-b1(z) and p∈(I-A)-1b2.Proof.
(1) Ifb∈ran⁡(I-A), then there exists p∈CN such that(16)I-Ap=b.
SinceCψ,φ is bounded and normal, |A|≤1,  A is normal and (I-A)c=(I-A∗)b by Theorem 1. So we have(17)I-Ac=I-A∗b=I-A∗I-Ap=I-AI-A∗p.I.e., (I-A)(c-(I-A∗)p)=0. It follows that(18)c-I-A∗p∈ker⁡I-A.
For anyξ∈ker⁡(I-A),  Aξ=ξ and by Proposition 3.1 in [11, Ch I], A∗ξ=ξ. Since ran⁡(I-A)⊥=ker⁡(I-A∗),  ξ⊥b. By Theorem 1, 〈Aξ,b+Ac〉=0. But(19)Aξ,b+Ac=ξ,b+Ac=ξ,b+A∗ξ,c=ξ,c.So 〈c,ξ〉=0. Hence(20)c-I-A∗p,ξ=c,ξ-p,ξ+p,Aξ=c,ξ=0,which implies that(21)c-I-A∗p⊥ker⁡I-A.It follows from (18) and (21) that(22)I-A∗p=c.
By formulas (14) and (15), Φp(z)=Az and Ψp(z)=exp⁡(〈p,c〉/2). So Cψ,φ is unitarily equivalent to CΨp,ϕp=exp⁡(〈p,c〉/2)CA.
(2) Ifb∉ran⁡(I-A), then b≠0 and there exist b1∈ran⁡(I-A)⊥ and b2∈ran⁡(I-A) such that(23)b=b1+b2.Obviously, b1≠0.
For anyt∈R, let p=itb1,  i is the imaginary unit. Notice ker⁡(I-A)=ker⁡(I-A∗)=ran⁡(I-A)⊥ by Proposition 3.1 in [11, Ch I], so we have(24)A∗p=Ap=p.By Theorem 1, 0=〈Ap,Ac+b〉=〈p,Ac+b〉=〈A∗p,c〉+〈p,b〉=〈p,c〉+〈p,b〉. So(25)p,c+b,p=-p,b+b,p=-2itb12.
By formulas (14) and (15),(26)Φpz=Az+b=φz,Ψpz=exp⁡z,c2-itb12=exp⁡-itb12ψz.So Cψ,φ is unitarily equivalent to exp⁡(-it|b1|2)Cψ,φ.
(3) Sinceb2∈ran(I-A), there exists p∈CN such that(27)I-Ap=b2.By Theorem 1, A is normal and (I-A)c=(I-A∗)b. We have(28)I-Ac-I-A∗p=I-Ac-I-AI-A∗p=I-A∗b-I-A∗I-Ap=I-A∗b1+b2-I-A∗b2=I-A∗b1=0,since b1∈ran⁡(I-A)⊥=ker⁡I-A∗=ker⁡(I-A) by Proposition 3.1 in [11, Ch I], which implies that c-(I-A∗)p∈ker⁡(I-A). Hence(29)c-I-A∗p+b1∈ker⁡I-A.
On the other hand, for anyξ∈ker⁡(I-A), by Ch 1, Proposition 3.1 in [11],(30)ξ∈ker⁡I-A∗=ran⁡I-A⊥.So, by Theorem 1,(31)0=Aξ,b+Ac=ξ,b+Ac=ξ,b1+b2+ξ,c=ξ,b1+ξ,c.But(32)c-I-A∗p+b1,ξ=c,ξ-I-A∗p,ξ+b1,ξ=c,ξ-p,I-Aξ+b1,ξ=c,ξ+b1,ξ.So we have 〈c-(I-A∗)p+b1,ξ〉=0, which implies that(33)c-I-A∗p+b1⊥ker⁡I-A.It follows from (29) and (33) that(34)c-I-A∗p+b1=0.
By formulas (14) and (15),(35)Φpz=Az+b1,Ψpz=exp⁡z,-b12+p,c+b1,p2=exp⁡p,c+b1,p2K-b1z.So Cψ,φ is unitarily equivalent to exp⁡(〈p,c〉+〈b1,p〉/2)Cψ1,φ1.For an analytic functionφ on CN, it is well known that the composition operator Cφ on F2 is a normal operator if and only if φ(z)=Az for some normal operator A on CN with |A|≤1. In fact, such operators are diagonalizable. In [8], the numerical ranges of such operators on Hardy space of D are studied.Lemma 4.
LetA be a normal operator on CN with |A|≤1 and σ(A)={λ1,λ2,⋯,λN}. Then on F2, ‖CA‖=1.Proof.
SinceA is a normal operator on CN, there exists a unitary operator U on CN such that(36)UAU∗≔λ1λ2⋱λk+1⋱λn=DThen CU∗CACU=CD and ‖CA‖=‖CD‖.
For anyf∈F2, let f(z)=∑α∈N0Nc(α)zα; then(37)CDfz=fDz=∑α∈N0Ncαλαzα,where λ=(λ1,λ2,…,λN). Since |λα|≤1,  ‖CDf‖≤‖f‖. It follows that ‖CD‖≤1. But CD1=1. So ‖CD‖=1.Theorem 5.
LetCψ,φ be a bounded normal operator on F2 with φ(z)=Az+b,ψ(z)=Kc(z).
(1) Ifb∈ran⁡(I-A), then ‖Cψ,φ‖=exp(〈p,c〉/2) with p∈(I-A)-1b.
(2) Ifb∉ran⁡(I-A) and b=b1+b2 with b1∈ran⁡(I-A)⊥,  b2∈ran⁡(I-A), then ‖Cψ,φ‖=exp(〈p,c〉+〈b1,p〉/2+|b1|2/4), where p∈(I-A)-1b2.Proof.
(1) The conclusion follows from Theorem3 (1) and Lemma 4.
(2) By Theorem3(3), Cψ,φ is unitarily equivalent to exp⁡(〈p,c〉+〈b1,p〉/2)Cψ1,φ1 with(38)φ1z=Az+b1,ψ1z=K-b1zand p∈(I-A)-1b2.
For anyf∈F2,(39)Cψ1,φ1f2=12πN∫CNψ1zfφ1z2exp⁡-z22dm2Nz=12πN∫CNfAz+b12exp⁡-z2+z,b1+b1,z2dm2Nz=exp⁡b12212πN∫CNfAz+b12exp⁡-z+b122dm2Nz=exp⁡b12212πN∫CNfAz-b1+b12exp⁡-z22dm2Nz.Since b1∈ran⁡(I-A)⊥=ker⁡I-A∗=ker⁡(I-A),  b1=Ab1. Therefore,(40)Cψ1,φ1f2=exp⁡b12212πN∫CNfAz2exp⁡-z22dm2Nz=exp⁡b122CAf2.It follows from Lemma 4 that ‖Cψ1,φ1‖=exp⁡(|b1|2/4). Hence(41)Cψ,φ=exp⁡p,c+b1,p2+b124.The following result is well known.Lemma 6.
LetA be a normal operator on CN,  |A|≤1 and σ(A)={λ1,λ2,⋯,λN}. Then on F2,(42)σpCA=λα:α∈N0N,σCA=λα:α∈N0N¯=σpCA¯,where λ=(λ1,λ2,⋯,λN).Theorem 7.
LetA be a normal operator on CN,  |A|≤1, and σ(A)={λ1,λ2,⋯,λN}. Then on F2, W(CA)=coσp(CA).Proof.
Note thatCA is a normal operator on F2; we have(43)co⁡σpCA⊂WCA⊂WCA¯=co⁡σCA.
Ifλ=(λ1,λ2,⋯,λN)=(0,0,⋯,0), then A=0. In this case, σ(CA)=σp(CA)={0,1}. By formula (43),(44)WCA=co⁡σpCA=0,1.
Ifλ=(λ1,λ2,⋯,λN)=(1,1,⋯,1), then A=I. In this case, σ(CA)=σp(CA)={1}. By formula (43),(45)WCA=co⁡σpCA=1.
In the following, we assume(0,0,⋯,0)≠(λ1,λ2,⋯,λN)≠(1,1,⋯,1). We take the proof into several parts.
Case I.There exists λj∈σ(A),1≤j≤N, such that |λj|=1;  λj is not a root of 1.
It is well known that{λjm}m=0∞ is dense in T. So we have(46)D⊂co⁡λjmm=0∞⊂co⁡σpCA⊂WCA⊂D0,CA¯=D¯,the last equality follows from Lemma 4.
For anyu∈T, if u∈W(CA), then by [1, lemma 2.3], u∈σp(CA) since |u|=1=CA. So(47)WCA⊂D∪σpCA⊂co⁡σpCA.Therefore W(CA)=co⁡σp(CA).
Case II. σ ( A ) = { λ 1 , λ 2 , ⋯ , λ N } ⊂ ( 0,1 ].
In this case,CA is a positive operator on F2. Since (λ1,λ2,⋯,λN)≠(1,1,⋯,1), there exists λj,1≤j≤N,  |λj|<1. Obviously, 1∈σp(CA)⊂(0,1]. Since limm→∞⁡λjm=0, we have(48)0,1=co⁡σpCA⊂WCA⊂WCA¯=co⁡σCA=0,1.
If there existsf∈F2 such that 〈CAf,f〉=0, then CAf=0 and hence f=0. Therefore 0∉W(CA). So W(CA)=co⁡σp(CA)=(0,1].
Case III.For any j,1≤j≤N, either |λj|<1 or λj is a root of 1, and there exists λn∈σ(A),1≤n≤N, λn∉(0,1].
Note that, in this case,σ(CA)=σp(CA)¯⊂σp(CA)∪{0}. Since |λn|<1 or λn is a root of 1 and λn∉(0,1],(49)0∈co⁡λnmm=0∞⊂co⁡σpCASo we obtain(50)co⁡σpCA⊂WCA⊂WCA¯=co⁡σCA⊂co⁡σpCA∪0⊂co⁡σpCA.It follows that W(CA)=co⁡σp(CA).Lastly, we give the proof of Theorem2. Some ideas are derived from [1, 8].Proof.
(1) Since unitarily equivalent operators have the same numerical ranges, the conclusion follows from Theorems3(1), and 7 and Lemma 6.
(2) SinceCψ,φ is normal, there exists v∈σ(Cψ,φ) such that |v|=‖Cψ,φ‖. By Theorem 3(2), for any t∈R,  eitv∈σ(Cψ,φ). It follows that(51)u∈C∣u=Cψ,φ⊂σCψ,φ.So we have(52)WCψ,φ¯=co⁡σCψ,φ=co⁡u∈C∣u=Cψ,φ=D0,Cψ,φ¯.
If there existsv∈{u∈C∣u=‖Cψ,φ‖} such that v∈W(Cψ,φ), then by [1, Lemma 2.3] and Theorem 3(2), for any t∈R,  eitv∈σp(Cψ,φ), which implies that there is an uncountable collection of orthogonal vectors in F2 since Cψ,φ is normal and eigenvalues of Cψ,φ that correspond to distinct eigenvalues are orthogonal, a contradiction. So(53)WCψ,φ⊂D0,Cψ,φand W(Cψ,φ) is dense in D(0,‖Cψ,φ‖). Since W(Cψ,φ) is convex, we must have D(0,‖Cψ,φ‖)⊂W(Cψ,φ). Hence(54)WCψ,φ=D0,Cψ,φ=D0,exp⁡p,c+b1,p2+b124by Theorem 5(2).Remark 8.
In Theorem2(2), we can choose p∈ker⁡(I-A)⊥ such that (I-A)p=b2, and then 〈b1,p〉=0 since b1∈ran⁡(I-A)⊥=ker⁡I-A∗=ker⁡(I-A). So(55)WCψ,φ=D0,exp⁡p,c2+b124,where p∈(I-A)-1b2∩ker⁡(I-A)⊥.



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