# Self-Commutators of Composition Operators with Monomial Symbols on the Dirichlet Space

Authors: A. Abdollahi; S. Mehrangiz
Journal: Abstract and Applied Analysis
 (2011)

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## Abstract

Letφ(z)=zn,z∈𝕌, for some positive integer n and Cφ the composition operator on the Dirichlet space 𝒟 induced by φ. In this paper, we completely determine the point spectrum, spectrum, essential spectrum, and essential norm of the operators Cφ∗Cφ,CφCφ∗ and self-commutators of Cφ, which expose that the spectrum and point spectrum coincide. We also find the eigenfunctions of the operators.


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## Body

## 1. Introduction

Letφ be a holomorphic self-map of the unit disk 𝕌∶={z∈ℂ:|z|<1}. The function φ induces the composition operatorCφ, defined on the space of holomorphic functions on 𝕌 by Cφf=f∘φ. The restriction of Cφ to various Banach spaces of holomorphic functions on 𝕌 has been an active subject of research for more than three decades, and it will continue to be for decades to come (see [1–3]). Let 𝒟 denote the Dirichlet space of analytic functions on the unit disk with derivatives that are square integrable with respect to the area measure on the disk. In recent years, the study of composition operators on the the Dirichlet space has received considerable attention (see [4–9] and references cited therein).Letφ(z)=zn,z∈𝕌, for some positive integer n, and Cφ:𝒟→𝒟 the composition operator on the Dirichlet space 𝒟 induced by φ. The main aim here is to find the spectrum, point spectrum, essential spectrum, and essential norm of Cφ*Cφ,CφCφ*, self-commutator [Cφ*,Cφ]=Cφ*Cφ-CφCφ* and anti-self-commutator {Cφ*,Cφ}=Cφ*Cφ+CφCφ*, for composition operators Cφ on the Dirichlet space.In [10], by using Cowen’s formula for the adjoint of Cφ on H2(𝕌), the authors have completely determined the spectrum, essential spectrum, and point spectrum for selfcommutators of automorphic composition operators acting on the Hardy space of unit disk. In [4], the first author, has extended these results from the Hardy space to the Dirichlet space.The other problem which is important to the study of composition operators is finding the relationships between the properties of the symbolφ and essential normality of the composition operator Cφ. Recall that an operator T on a Hilbert space ℋ is called essentially normal if its image in the Calkin algebra is normal or equivalently if the self-commutator[T*,T]=T*T-TT* is compact on ℋ.In [11], the authors have determined which composition operators with automorphism symbol are essentially normal on A2(BN) and H2(BN) for N≥1. They have shown that the only essential normal automorphic composition operators are actually normal. This was first shown in the setting H2(𝕌) by Zorboska in [12]. The related works and some historical remarks can be found in [10–13].In [5], the authors consider composition operators Cφ, where φ is a linear-fractional self-map of the unit disk 𝕌, acting on the Dirichlet space 𝒟. By using the E. Gallardo and A. Montes’ adjoint formula given in [6], they show that the essentially normal linear fractional composition operators on 𝒟 are precisely those whose symbol is not a hyperbolic nonautomorphism with a boundary fixed point. They also obtained conditions for the linear fractional symbols φ and ψ of the unit disk for which Cψ*Cφ or CφCψ* is compact.In the next section, after giving some background material and presenting formula for the adjoint ofCφ on 𝒟, we give useful formula for the operators Cφ*Cφ, CφCφ*, [Cφ*,Cφ], and {Cφ*,Cφ}, when φ is an arbitrary monomial symbol φ(z)=zn. In Section 3, we completely determine the point spectrum, spectrum, and essential spectrum of Cφ*Cφ and CφCφ*. Finally, in Section 4, we determine the same for [Cφ*,Cφ] and {Cφ*,Cφ}.


## 2. Preliminaries

Throughout the paper, for a Hilbert spaceℋ, ℬ(ℋ) denotes the set of bounded operators on ℋ and ℬ0(ℋ) denotes the closed ideal of all compact operators in ℬ(ℋ). The natural homomorphism of ℬ(ℋ) onto the quotient Banach algebra ℬ(ℋ)/ℬ0(ℋ)=ℬ/ℬ0—the Calkin algebra—is denoted by T↦T̃=T+ℬ0(ℋ).For an operatorT∈ℬ(ℋ), the essential norm of T is defined by (2.1)‖T‖e∶=inf⁡⁡{‖T+K‖:K∈B0(H)},
and the essential spectrumσe(T) is defined as the spectrum of the image T̃ of T in the Calkin algebra ℬ(ℋ)/ℬ0(ℋ). It is well known that the essential spectrum of a normal operator consists of all points in the spectrum of the operator except the isolated eigenvalues of finite multiplicity (see [14]).As we mentioned in the Introduction, an operatorT on a Hilbert space ℋ is called essentially normal if its image in the Calkin algebra is normal or equivalently if the self-commutator [T*,T]=T*T-TT* is compact on ℋ.TheDirichlet space, which we denote by 𝒟, is the set of all analytic functions f on the unit disk 𝕌 for which (2.2)∫U|f′(z)|2dA(z)<∞,
where dA denote the normalized area measure, and equivalently an analytic function f is in 𝒟 if ∑n=1∞n|f̂(n)|2<∞, where f̂(n) denotes the n-th Taylor coefficient of f at 0. Background on the Dirichlet space can be found in [15] and the references cited therein.For each holomorphic self-mapφ of 𝕌, we define the composition operatorCφ by Cφf=f∘φ(f∈𝒟).Martín and Vukotić in [9] express and prove formulas for the adjoint of Cφ on the Hardy space, when φ is finite Blaschke product and also is rational self-map of the unit disk 𝕌. By using the same arguments as in [9] for the Hardy space, one can prove the following theorem for the Dirichlet space case.Theorem 2.1.
Letφ(z)=zn. For an arbitrary point  w=reiθ in  𝕌, writing its nth roots as  rk,w=r1/nei(θ+2kπ)/n, k=0,1,…,n-1. The adjoint of  Cφ (viewed as an operator on the Dirichlet space  𝒟) is given by the formula
(2.3)Cφ*f(w)=∑k=0n-1f(rk,w)-(n-1)f(0).Throughout this paper, we denote byℳ the closed subspace of 𝒟 spanned by the monomials {znk:k=0,1,…} and by Pℳ:𝒟→ℳ the corresponding orthogonal projection onto ℳ.Remark 2.2.
Letφ(z)=znandrk,φ(w)=r1/nei(θ+2kπ)/n,k=0,1,…,n-1, be thenth roots ofφ(z)=φ(w). Forf∈𝒟withf(z)=∑m=0∞cmzm, we have(2.4)∑k=0n-1f(rk,φ(w))=∑k=0n-1∑m=0∞cmrk,φ(w)m=∑m=0∞cm∑k=0n-1rk,φ(w)m=n∑j=0∞cnjwnj=n(PMf)(w).Before stating our main results, we also need the next results.Theorem 2.3.
Letφ(z)=zn. Then,
(2.5)Cφ*Cφ=nImod⁡B0(D),(2.6)(CφCφ*f)(w)=n(PMf)(w)-(n-1)f(0),ℬ0(𝒟) is the ideal of compact operators on 𝒟.Proof.
By a simple computation and using formula (2.3), it follows that
(2.7)(Cφ*Cφf)(w)=nf(w)-(n-1)f(0),(2.8)(CφCφ*f)(w)=∑k=0n-1f(rk,φ(w))-(n-1)f(0),
for each w∈𝕌,f∈𝒟. Thus,
(2.9)Cφ*Cφ=nImod⁡B0(D),(CφCφ*f)(w)=n(PMf)(w)-(n-1)f(0).


## 3. Spectrum ofCφ*Cφ and CφCφ*

Letφ(z)=zn. In this section, we are going to find the point spectrum, spectrum, essential spectrum, and the eigenfunctions of the operators Cφ*Cφ and CφCφ*.Theorem 3.1.
Letφ(z)=zn. Then,
(3.1)σe(Cφ*Cφ)={n},σp(Cφ*Cφ)=σ(Cφ*Cφ)={1,n},
and, for n≥2,
(3.2)σe(CφCφ*)={0,n},σp(CφCφ*)=σ(CφCφ*)={1,0,n},
and, in the case that n=1,
(3.3)σp(CφCφ*)=σe(CφCφ*)=σ(CφCφ*)={1}.Proof.
Since the operatorCφ*Cφ is a finite rank perturbation of nI, the essential spectrum of this operator is {n}. Since any points in the spectrum of a normal operator which are not in the essential spectrum are isolated eigenvalues of finite multiplicity, it is enough to find the eigenvalues. We first do this for the operator Cφ*Cφ. Let λ∈ℂ be an eigenvalue of the operator Cφ*Cφ with corresponding eigenvector f∈𝒟. Then, Cφ*Cφf=λf. By using formula (2.7) for Cφ*Cφ, we have
(3.4)nf(w)-(n-1)f(0)=λf(w),w∈U.
By putting w=0, it follows that nf(0)-(n-1)f(0)=λf(0). If f(0)≠0, then λ=1. Thus for the case λ=1, the function f(w)=f(0) is a nonzero function in 𝒟 that satisfies the equation, and, hence, λ=1 is an eigenvalue of the operator Cφ*Cφ. If f(0)=0, then λ=n, and, in this case, the function wk,k≥1 is a nonzero function in 𝒟 that satisfies (3.4). Hence, λ=n is an eigenvalue of the operator Cφ*Cφ with infinite multiplicity. So
(3.5)σp(Cφ*Cφ)=σ(Cφ*Cφ)={1,n}.
Now, let λ∈ℂ be an eigenvalue of the operator CφCφ* with corresponding eigenvector f∈𝒟. Then, CφCφ*f=λf. By using formula (2.6) for CφCφ*, we have
(3.6)n(PMf)(w)-(n-1)f(0)=λf(w).
By putting w=0, it follows that
(3.7)nf(0)-(n-1)f(0)=λf(0).
If f(0)≠0, then λ=1. Thus,
(3.8)n(PMf)(w)-(n-1)f(0)=f(w).
Let f(z)=∑m=0∞cmzm. Then,
(3.9)n∑j=0∞cnjwnj-(n-1)c0=∑m=0∞cmwm,w∈U.
For n≥2, it follows that cm=0, whenever m≥1. So f≡1 is an eigenfunction corresponding to λ=1. Thus, λ=1 is an eigenvalue of the operator CφCφ*.
Now suppose thatf(0)=0. Then, nPℳf=λf and so λ/n is an eigenvalue of Pℳ. Hence, λ=0 or λ=n.
So, for the caseλ=0 and n≥2, the function w is a nonzero function in 𝒟 that satisfies (3.6), and, hence, λ=0 is an eigenvalue of the operator CφCφ*.
So whenn≥2, the eigenvalues of CφCφ* are {1,0,n}. In the case that λ=n, for each natural number k, wnk is a nonzero function in 𝒟 that satisfies (3.6), and, hence, the essential spectrum CφCφ* contains n. If λ=0 and n≥2, then we conclude that for each natural number m which is not a multiple of n, wm is a nonzero function in 𝒟 that satisfies (3.6), and, hence, λ=0 is an eigenvalue of the operator CφCφ* with infinite multiplicity. So the essential spectrum CφCφ* contains 0. Since
(3.10)σ(CφCφ*+B0(D))∪{0}=σ(Cφ*Cφ+B0(D))∪{0},
and σe(Cφ*Cφ)={n}, for n≥2, we conclude that
(3.11)σe(CφCφ*)={0,n}.
So when φ(z)=zn and n≥2,
(3.12)σp(CφCφ*)=σ(CφCφ*)={1,0,n},
and, for n=1,
(3.13)σp(CφCφ*)=σe(CφCφ*)=σ(CφCφ*)={1}.


## 4. The Spectrum of[Cφ*,Cφ] and {Cφ*,Cφ}

Theorem 4.1.
Letφ(z)=zn. Then, for  n=1,
(4.1)σp([Cφ*,Cφ])=σe([Cφ*,Cφ])=σ([Cφ*,Cφ])={0},
and, for n≥2,
(4.2)σp([Cφ*,Cφ])=σe([Cφ*,Cφ])=σ([Cφ*,Cφ])={0,n}.Proof.
LetT=Cφ*Cφ-CφCφ*. Then,
(4.3)T=n(I-PM).
Since any points in the spectrum of a normal operator which are not in the essential spectrum are isolated eigenvalues of finite multiplicity, we first find the eigenvalues.
Ifn≥2, then I-Pℳ is a nontrivial projection and so σ(T)={0,n}.
In the case thatλ=n, for each natural number m which is not a multiple of n, the function wm is an eigenfunction of T, and, hence, λ=n is an eigenvalue of the operator T with infinite multiplicity. For the case λ=0, for each natural number k, wkn is an eigenfunction of T, and, hence, λ=0 is an eigenvalue of the operator T with infinite multiplicity.
The essential spectrum of[Cφ*,Cφ] can be computed directly by using the following:
(4.4)σe([Cφ*,Cφ])=σ(Cφ*Cφ-CφCφ*+B0(D))=σ(nI-CφCφ*+B0(D)).
So if φ(z)=z, then
(4.5)σp([Cφ*,Cφ])=σe([Cφ*,Cφ])=σ([Cφ*,Cφ])={0},
and, if φ(z)=zn and n≥2, then
(4.6)σp([Cφ*,Cφ])=σe([Cφ*,Cφ])=σ([Cφ*,Cφ])={0,n}.Theorem 4.2.
Letφ(z)=zn. Then, for  n=1,
(4.7)σp({Cφ*,Cφ})=σe({Cφ*,Cφ})=σ({Cφ*,Cφ})={2},
for n≥2,
(4.8)σe({Cφ*,Cφ})={2n,n},σp({Cφ*,Cφ})=σ({Cφ*,Cφ})={2,n,2n}.Proof.
LetS=Cφ*Cφ+CφCφ*. Then for each f∈𝒟 and w∈𝕌,
(4.9)(Sf)(w)=nf(w)+n(PMf)(w)-2(n-1)f(0).
Since S is self-adjoint, any points in the spectrum of S which are not in the essential spectrum are eigenvalues of finite multiplicity. So we first find such points.
Letλ∈ℂ be an eigenvalue of the operator S with corresponding eigenvector f∈𝒟. Then, Sf=λf. So we have
(4.10)nf(w)+n(PMf)(w)-2(n-1)f(0)=λf(w).
By putting w=0, it follows that 2nf(0)-2nf(0)+2f(0)=λf(0). If f(0)≠0, then λ=2. Thus,
(4.11)nf(w)+n(PMf)(w)-2(n-1)f(0)=2f(w).
The function f(w)=1 is a nonzero function in 𝒟 that satisfies the equation, and, hence, λ=2 is an eigenvalue of the operator S. If f(0)=0, then
(4.12)n(PMf)(w)=(λ-n)f(w),
and it follows that λ=n or λ=2n. For the case λ=n with n≥2, for each natural number m which is not a multiple of n, the function wm is a nonzero function in 𝒟 that satisfies (4.12), and, hence, λ=n is an eigenvalue of the operator S with infinite multiplicity. In the case that λ=2n, for each natural number k, wkn is a nonzero function in 𝒟 that satisfies (4.12), and, hence, λ=2n is an eigenvalues of the operator S with infinite multiplicity. The essential spectrum of {Cφ*,Cφ} can be computed directly by using the following:
(4.13)σe({Cφ*,Cφ})=σ(Cφ*Cφ+CφCφ*+B0(D))=σ(nI+CφCφ*+B0(D)).
Hence, we conclude that when φ(z)=zn and n≥2,
(4.14)σe({Cφ*,Cφ})={n,2n}.
Also, if φ(z)=z, then
(4.15)σp({Cφ*,Cφ})=σe({Cφ*,Cφ})=σ({Cφ*,Cφ})={2},
and, for n≥2,
(4.16)σp({Cφ*,Cφ})=σ({Cφ*,Cφ})={2,n,2n}.



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