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

Abstract

Let 𝜑(𝑧)=𝑧^𝑛,𝑧∈𝕌, for some positive integer 𝑛 and 𝐶_𝜑 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 𝐶^∗𝜑𝐶_𝜑,𝐶_𝜑𝐶^∗𝜑 and self-commutators of 𝐶_𝜑, which expose that the spectrum and point spectrum coincide. We also find the eigenfunctions of the operators.

1. Introduction

Let 𝜑 be a holomorphic self-map of the unit disk 𝕌∶={𝑧∈ℂ∶|𝑧|<1}. The function 𝜑 induces the composition operator 𝐶_𝜑, defined on the space of holomorphic functions on 𝕌 by 𝐶_𝜑𝑓=𝑓∘𝜑. The restriction of 𝐶_𝜑 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).

Let 𝜑(𝑧)=𝑧^𝑛,𝑧∈𝕌, for some positive integer 𝑛, and 𝐶_𝜑∶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 𝐶^∗𝜑𝐶_𝜑,𝐶_𝜑𝐶^∗𝜑, self-commutator [𝐶^∗𝜑,𝐶_𝜑]=𝐶^∗𝜑𝐶_𝜑−𝐶_𝜑𝐶^∗𝜑 and anti-self-commutator {𝐶^∗𝜑,𝐶_𝜑}=𝐶^∗𝜑𝐶_𝜑+𝐶_𝜑𝐶^∗𝜑, for composition operators 𝐶_𝜑 on the Dirichlet space.

In [10], by using Cowen’s formula for the adjoint of 𝐶_𝜑 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 symbol 𝜑 and essential normality of the composition operator 𝐶_𝜑. 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(𝐵_𝑁) and 𝐻²(𝐵_𝑁) 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 𝐶_𝜑, 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 𝐶^∗𝜓𝐶_𝜑 or 𝐶_𝜑𝐶^∗𝜓 is compact.

In the next section, after giving some background material and presenting formula for the adjoint of 𝐶_𝜑 on 𝒟, we give useful formula for the operators 𝐶^∗𝜑𝐶_𝜑, 𝐶_𝜑𝐶^∗𝜑, [𝐶^∗𝜑,𝐶_𝜑], and {𝐶^∗𝜑,𝐶_𝜑}, when 𝜑 is an arbitrary monomial symbol 𝜑(𝑧)=𝑧^𝑛. In Section 3, we completely determine the point spectrum, spectrum, and essential spectrum of 𝐶^∗𝜑𝐶_𝜑 and 𝐶_𝜑𝐶^∗𝜑. Finally, in Section 4, we determine the same for [𝐶^∗𝜑,𝐶_𝜑] and {𝐶^∗𝜑,𝐶_𝜑}.

2. Preliminaries

Throughout the paper, for a Hilbert space ℋ, 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↦𝑇=𝑇+ℬ₀(ℋ).

For an operator 𝑇∈ℬ(ℋ), the essential norm of 𝑇 is defined by ‖𝑇‖_𝑒∶=inf‖𝑇+𝐾‖∶𝐾∈ℬ₀(ℋ),(2.1) and the essential spectrum 𝜎_𝑒(𝑇) is defined as the spectrum of the image 𝑇 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 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.

The Dirichlet space, which we denote by 𝒟, is the set of all analytic functions 𝑓 on the unit disk 𝕌 for which _𝕌||𝑓^(𝑧)||²𝑑𝐴(𝑧)<∞,(2.2) where 𝑑𝐴 denote the normalized area measure, and equivalently an analytic function 𝑓 is in 𝒟 if ∑^∞𝑛=1𝑛|𝑓(𝑛)|²<∞, where 𝑓(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-map 𝜑 of 𝕌, we define the composition operator 𝐶_𝜑 by 𝐶_𝜑𝑓=𝑓∘𝜑(𝑓∈𝒟).

Martín and Vukotić in [9] express and prove formulas for the adjoint of 𝐶_𝜑 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. Let 𝜑(𝑧)=𝑧^𝑛. For an arbitrary point 𝑤=𝑟𝑒^𝑖𝜃 in 𝕌, writing its nth roots as 𝑟_𝑘,𝑤=𝑟^1/𝑛𝑒^{𝑖(𝜃+2𝑘𝜋)/𝑛}, 𝑘=0,1,…,𝑛−1. The adjoint of 𝐶_𝜑 (viewed as an operator on the Dirichlet space 𝒟) is given by the formula 𝐶^∗𝜑𝑓(𝑤)=^𝑛−1_𝑘=0𝑓𝑟_𝑘,𝑤−(𝑛−1)𝑓(0).(2.3)

Throughout this paper, we denote by ℳ the closed subspace of 𝒟 spanned by the monomials {𝑧^𝑛𝑘∶𝑘=0,1,…} and by 𝑃_ℳ∶𝒟→ℳ the corresponding orthogonal projection onto ℳ.

Remark 2.2. Let 𝜑(𝑧)=𝑧^𝑛 and 𝑟_{𝑘,𝜑(𝑤)}=𝑟^1/𝑛𝑒^{𝑖(𝜃+2𝑘𝜋)/𝑛},𝑘=0,1,…,𝑛−1, be the 𝑛th roots of 𝜑(𝑧)=𝜑(𝑤). For 𝑓∈𝒟 with 𝑓(𝑧)=∑^∞𝑚=0𝑐_𝑚𝑧^𝑚, we have ^𝑛−1_𝑘=0𝑓𝑟_{𝑘,𝜑(𝑤)}=^𝑛−1_∞𝑘=0_𝑚=0𝑐_𝑚𝑟^{𝑚𝑘,𝜑(𝑤)}=^∞_𝑚₌₀𝑐_𝑚𝑛−1_𝑘=0𝑟^{𝑚𝑘,𝜑(𝑤)}=𝑛^∞_𝑗=0𝑐_𝑛𝑗𝑤^𝑛𝑗=𝑛𝑃_ℳ𝑓(𝑤).(2.4)

Before stating our main results, we also need the next results.

Theorem 2.3. Let 𝜑(𝑧)=𝑧^𝑛. Then, 𝐶𝐶^∗𝜑𝐶_𝜑=𝑛𝐼modℬ₀(𝒟),(2.5)_𝜑𝐶^∗𝜑𝑓(𝑤)=𝑛𝑃_ℳ𝑓(𝑤)−(𝑛−1)𝑓(0),(2.6)ℬ₀(𝒟) is the ideal of compact operators on 𝒟.

Proof. By a simple computation and using formula (2.3), it follows that 𝐶𝐶^∗𝜑𝐶_𝜑𝑓(𝑤)=𝑛𝑓(𝑤)−(𝑛−1)𝑓(0),(2.7)_𝜑𝐶^∗𝜑𝑓(𝑤)=^𝑛−1_𝑘=0𝑓𝑟_{𝑘,𝜑(𝑤)}−(𝑛−1)𝑓(0),(2.8) for each 𝑤∈𝕌,𝑓∈𝒟. Thus, 𝐶𝐶^∗𝜑𝐶_𝜑=𝑛𝐼modℬ₀(𝒟),_𝜑𝐶^∗𝜑𝑓(𝑤)=𝑛𝑃_ℳ𝑓(𝑤)−(𝑛−1)𝑓(0).(2.9)

3. Spectrum of 𝐶^∗𝜑𝐶_𝜑 and 𝐶_𝜑𝐶^∗𝜑

Let 𝜑(𝑧)=𝑧^𝑛. In this section, we are going to find the point spectrum, spectrum, essential spectrum, and the eigenfunctions of the operators 𝐶^∗𝜑𝐶_𝜑 and 𝐶_𝜑𝐶^∗𝜑.

Theorem 3.1. Let 𝜑(𝑧)=𝑧^𝑛. Then, 𝜎_𝑒𝐶^∗𝜑𝐶_𝜑={𝑛},𝜎_𝑝𝐶^∗𝜑𝐶_𝜑=𝜎𝐶^∗𝜑𝐶_𝜑={1,𝑛},(3.1) and, for 𝑛≥2, 𝜎_𝑒𝐶_𝜑𝐶^∗𝜑={0,𝑛},𝜎_𝑝𝐶_𝜑𝐶^∗𝜑=𝜎𝐶_𝜑𝐶^∗𝜑={1,0,𝑛},(3.2) and, in the case that 𝑛=1, 𝜎_𝑝𝐶_𝜑𝐶^∗𝜑=𝜎_𝑒𝐶_𝜑𝐶^∗𝜑=𝜎𝐶_𝜑𝐶^∗𝜑={1}.(3.3)

Proof. Since the operator 𝐶^∗𝜑𝐶_𝜑 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 𝐶^∗𝜑𝐶_𝜑. Let 𝜆∈ℂ be an eigenvalue of the operator 𝐶^∗𝜑𝐶_𝜑 with corresponding eigenvector 𝑓∈𝒟. Then, 𝐶^∗𝜑𝐶_𝜑𝑓=𝜆𝑓. By using formula (2.7) for 𝐶^∗𝜑𝐶_𝜑, we have 𝑛𝑓(𝑤)−(𝑛−1)𝑓(0)=𝜆𝑓(𝑤),𝑤∈𝕌.(3.4) 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 𝐶^∗𝜑𝐶_𝜑. If 𝑓(0)=0, then 𝜆=𝑛, and, in this case, the function 𝑤^𝑘,𝑘≥1 is a nonzero function in 𝒟 that satisfies (3.4). Hence, 𝜆=𝑛 is an eigenvalue of the operator 𝐶^∗𝜑𝐶_𝜑 with infinite multiplicity. So 𝜎_𝑝𝐶^∗𝜑𝐶_𝜑=𝜎𝐶^∗𝜑𝐶_𝜑={1,𝑛}.(3.5) Now, let 𝜆∈ℂ be an eigenvalue of the operator 𝐶_𝜑𝐶^∗𝜑 with corresponding eigenvector 𝑓∈𝒟. Then, 𝐶_𝜑𝐶^∗𝜑𝑓=𝜆𝑓. By using formula (2.6) for 𝐶_𝜑𝐶^∗𝜑, we have 𝑛𝑃_ℳ𝑓(𝑤)−(𝑛−1)𝑓(0)=𝜆𝑓(𝑤).(3.6) By putting 𝑤=0, it follows that 𝑛𝑓(0)−(𝑛−1)𝑓(0)=𝜆𝑓(0).(3.7) If 𝑓(0)≠0, then 𝜆=1. Thus, 𝑛𝑃_ℳ𝑓(𝑤)−(𝑛−1)𝑓(0)=𝑓(𝑤).(3.8) Let 𝑓(𝑧)=∑^∞𝑚=0𝑐_𝑚𝑧^𝑚. Then, 𝑛^∞_𝑗=0𝑐_𝑛𝑗𝑤^𝑛𝑗−(𝑛−1)𝑐₀=^∞_𝑚₌₀𝑐_𝑚𝑤^𝑚,𝑤∈𝕌.(3.9) For 𝑛≥2, it follows that 𝑐_𝑚=0, whenever 𝑚≥1. So 𝑓≡1 is an eigenfunction corresponding to 𝜆=1. Thus, 𝜆=1 is an eigenvalue of the operator 𝐶_𝜑𝐶^∗𝜑.
Now suppose that 𝑓(0)=0. Then, 𝑛𝑃_ℳ𝑓=𝜆𝑓 and so 𝜆/𝑛 is an eigenvalue of 𝑃_ℳ. Hence, 𝜆=0 or 𝜆=𝑛.
So, for the case 𝜆=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 𝐶_𝜑𝐶^∗𝜑.
So when 𝑛≥2, the eigenvalues of 𝐶_𝜑𝐶^∗𝜑 are {1,0,𝑛}. In the case that 𝜆=𝑛, for each natural number k, 𝑤^𝑛𝑘 is a nonzero function in 𝒟 that satisfies (3.6), and, hence, the essential spectrum 𝐶_𝜑𝐶^∗𝜑 contains 𝑛. If 𝜆=0 and 𝑛≥2, then we conclude that for each natural number 𝑚 which is not a multiple of 𝑛, 𝑤^𝑚 is a nonzero function in 𝒟 that satisfies (3.6), and, hence, 𝜆=0 is an eigenvalue of the operator 𝐶_𝜑𝐶^∗𝜑 with infinite multiplicity. So the essential spectrum 𝐶_𝜑𝐶^∗𝜑 contains 0. Since 𝜎𝐶_𝜑𝐶^∗𝜑+ℬ₀(𝒟)∪{0}=𝜎𝐶^∗𝜑𝐶_𝜑+ℬ₀(𝒟)∪{0},(3.10) and 𝜎_𝑒(𝐶^∗𝜑𝐶_𝜑)={𝑛}, for 𝑛≥2, we conclude that 𝜎_𝑒𝐶_𝜑𝐶^∗𝜑={0,𝑛}.(3.11) So when 𝜑(𝑧)=𝑧^𝑛 and 𝑛≥2, 𝜎_𝑝𝐶_𝜑𝐶^∗𝜑=𝜎𝐶_𝜑𝐶^∗𝜑={1,0,𝑛},(3.12) and, for 𝑛=1, 𝜎_𝑝𝐶_𝜑𝐶^∗𝜑=𝜎_𝑒𝐶_𝜑𝐶^∗𝜑=𝜎𝐶_𝜑𝐶^∗𝜑={1}.(3.13)

4. The Spectrum of [𝐶^∗𝜑,𝐶_𝜑] and {𝐶^∗𝜑,𝐶_𝜑}

Theorem 4.1. Let 𝜑(𝑧)=𝑧^𝑛. Then, for 𝑛=1, 𝜎_𝑝𝐶^∗𝜑,𝐶_𝜑=𝜎_𝑒𝐶^∗𝜑,𝐶_𝜑=𝜎𝐶^∗𝜑,𝐶_𝜑={0},(4.1) and, for 𝑛≥2, 𝜎_𝑝𝐶^∗𝜑,𝐶_𝜑=𝜎_𝑒𝐶^∗𝜑,𝐶_𝜑=𝜎𝐶^∗𝜑,𝐶_𝜑={0,𝑛}.(4.2)

Proof. Let 𝑇=𝐶^∗𝜑𝐶_𝜑−𝐶_𝜑𝐶^∗𝜑. Then, 𝑇=𝑛𝐼−𝑃_ℳ.(4.3) 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.
If 𝑛≥2, then 𝐼−𝑃_ℳ is a nontrivial projection and so 𝜎(𝑇)={0,𝑛}.
In the case that 𝜆=𝑛, for each natural number 𝑚 which is not a multiple of 𝑛, the function 𝑤^𝑚 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, 𝑤^𝑘𝑛 is an eigenfunction of 𝑇, and, hence, 𝜆=0 is an eigenvalue of the operator 𝑇 with infinite multiplicity.
The essential spectrum of [𝐶^∗𝜑,𝐶_𝜑] can be computed directly by using the following: 𝜎_𝑒𝐶^∗𝜑,𝐶_𝜑==𝜎𝜎𝐶𝑛𝐼^∗𝜑𝐶−_𝜑𝐶−_𝜑𝐶_𝜑𝐶^∗𝜑+ℬ₀(𝒟)𝐶^∗𝜑+ℬ₀(𝒟).(4.4) So if 𝜑(𝑧)=𝑧, then 𝜎_𝑝𝐶^∗𝜑,𝐶_𝜑=𝜎_𝑒𝐶^∗𝜑,𝐶_𝜑=𝜎𝐶^∗𝜑,𝐶_𝜑={0},(4.5) and, if 𝜑(𝑧)=𝑧^𝑛 and 𝑛≥2, then 𝜎_𝑝𝐶^∗𝜑,𝐶_𝜑=𝜎_𝑒𝐶^∗𝜑,𝐶_𝜑=𝜎𝐶^∗𝜑,𝐶_𝜑={0,𝑛}.(4.6)

Theorem 4.2. Let 𝜑(𝑧)=𝑧^𝑛. Then, for 𝑛=1, 𝜎_𝑝𝐶^∗𝜑,𝐶_𝜑=𝜎_𝑒𝐶^∗𝜑,𝐶_𝜑=𝜎𝐶^∗𝜑,𝐶_𝜑={2},(4.7) for 𝑛≥2, 𝜎𝜎_𝑒𝐶^∗𝜑,𝐶_𝜑={2𝑛,𝑛},_𝑝𝐶^∗𝜑,𝐶_𝜑=𝜎𝐶^∗𝜑,𝐶_𝜑={2,𝑛,2𝑛}.(4.8)

Proof. Let 𝑆=𝐶^∗𝜑𝐶_𝜑+𝐶_𝜑𝐶^∗𝜑. Then for each 𝑓∈𝒟 and 𝑤∈𝕌, (𝑆𝑓)(𝑤)=𝑛𝑓(𝑤)+𝑛𝑃_ℳ𝑓(𝑤)−2(𝑛−1)𝑓(0).(4.9) 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.
Let 𝜆∈ℂ be an eigenvalue of the operator 𝑆 with corresponding eigenvector 𝑓∈𝒟. Then, 𝑆𝑓=𝜆𝑓. So we have 𝑛𝑓(𝑤)+𝑛𝑃_ℳ𝑓(𝑤)−2(𝑛−1)𝑓(0)=𝜆𝑓(𝑤).(4.10) By putting 𝑤=0, it follows that 2𝑛𝑓(0)−2𝑛𝑓(0)+2𝑓(0)=𝜆𝑓(0). If 𝑓(0)≠0, then 𝜆=2. Thus, 𝑛𝑓(𝑤)+𝑛𝑃_ℳ𝑓(𝑤)−2(𝑛−1)𝑓(0)=2𝑓(𝑤).(4.11) 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 𝑛𝑃_ℳ𝑓(𝑤)=(𝜆−𝑛)𝑓(𝑤),(4.12) 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 𝑤^𝑚 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, 𝑤^𝑘𝑛 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 {𝐶^∗𝜑,𝐶_𝜑} can be computed directly by using the following: 𝜎_𝑒𝐶^∗𝜑,𝐶_𝜑==𝜎𝜎𝐶𝑛𝐼^∗𝜑𝐶+_𝜑𝐶+_𝜑𝐶_𝜑𝐶^∗𝜑+ℬ₀(𝒟)𝐶^∗𝜑+ℬ₀(𝒟).(4.13) Hence, we conclude that when 𝜑(𝑧)=𝑧^𝑛 and 𝑛≥2, 𝜎_𝑒𝐶^∗𝜑,𝐶_𝜑={𝑛,2𝑛}.(4.14) Also, if 𝜑(𝑧)=𝑧, then 𝜎_𝑝𝐶^∗𝜑,𝐶_𝜑=𝜎_𝑒𝐶^∗𝜑,𝐶_𝜑=𝜎𝐶^∗𝜑,𝐶_𝜑={2},(4.15) and, for 𝑛≥2, 𝜎_𝑝𝐶^∗𝜑,𝐶_𝜑=𝜎𝐶^∗𝜑,𝐶_𝜑={2,𝑛,2𝑛}.(4.16)

Acknowledgments

The authors would like to thank the editor of the Abstract and Applied Analysis and the referee for useful and helpful comments and suggestions.