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International Journal of Mathematics and Mathematical Sciences
VolumeΒ 2012, Article IDΒ 970729, 7 pages
http://dx.doi.org/10.1155/2012/970729
Research Article

Fredholm Weighted Composition Operators on Dirichlet Space

School of Mathematics and Computer Science, Shanxi Normal University, Linfen 041004, China

Received 3 June 2012; Accepted 3 August 2012

Academic Editor: HenrykΒ Hudzik

Copyright Β© 2012 Liankuo Zhao. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

This paper characterizes bounded Fredholm, bounded invertible, and unitary weighted composition operators on Dirichlet space.

1. Introduction

Let 𝐻 be a Hilbert space of analytic functions on the unit disk 𝐷. For an analytic function πœ“ on 𝐷, we can define the multiplication operator π‘€πœ“βˆΆπ‘“β†’πœ“π‘“,π‘“βˆˆπ». For an analytic self-mapping πœ‘ of 𝐷, the composition operator πΆπœ‘ defined on 𝐻 as πΆπœ‘π‘“=π‘“βˆ˜πœ‘,π‘“βˆˆπ». These operators are two classes of important operators in the study of operator theory in function spaces [1–3]. Furthermore, for πœ“ and πœ‘, we define the weighted composition operator πΆπœ“,πœ‘ on 𝐻 as πΆπœ“,πœ‘βˆΆπ‘“βŸΆπœ“(π‘“βˆ˜πœ‘),π‘“βˆˆπ».(1.1)

Recently, the boundedness, compactness, norm, and essential norm of weighted composition operators on various spaces of analytic functions have been studied intensively, see [4–9] and so on. In this paper, we characterize bounded Fredholm weighted composition operators on Dirichlet space of the unit disk.

Recall the Dirichlet space π’Ÿ that consists of analytic function 𝑓 on 𝐷 with finite Dirichlet integral: ξ€œπ·(𝑓)=𝐷||||𝑓′2𝑑𝐴<∞,(1.2) where 𝑑𝐴 is the normalized Lebesgue area measure on 𝐷. It is well known that π’Ÿ is the only mΓΆbius invariant Hilbert space up to an isomorphism [10]. Endow π’Ÿ with norm ξ‚€||||‖𝑓‖=𝑓(0)2+𝐷(𝑓)1/2,π‘“βˆˆπ’Ÿ.(1.3)π’Ÿ is a Hilbert space with inner product βŸ¨π‘“,π‘”βŸ©=𝑓(0)ξ€œπ‘”(0)+π·π‘“ξ…ž(𝑧)π‘”ξ…ž(𝑧)𝑑𝐴(𝑧),𝑓,π‘”βˆˆπ’Ÿ.(1.4) Furthermore π’Ÿ is a reproducing function space with reproducing kernel πΎπœ†1(𝑧)=1+log1βˆ’βˆ’πœ†π‘§,πœ†,π‘§βˆˆπ·.(1.5)

Denote β„³={πœ“βˆΆπœ“isanalyticon𝐷,πœ“π‘“βˆˆπ’Ÿforπ‘“βˆˆπ’Ÿ}. β„³ is called the multiplier space of π’Ÿ. By the closed graph theorem, the multiplication operator π‘€πœ“ defined by πœ“βˆˆβ„³ is bounded on π’Ÿ. For the characterization of the element in β„³, see [11].

For analytic function πœ“ on 𝐷 and analytic self-mapping πœ‘ of 𝐷, the weighted composition operator πΆπœ“,πœ‘ on π’Ÿ is not necessarily bounded. Even the composition operator πΆπœ‘ is not necessarily bounded on π’Ÿ, which is different from the cases in Hardy space and Bergman space. See [12] for more information about the properties of composition operators acting on the Dirichlet space.

The main result of the paper reads as the following.

Theorem 1.1. Let πœ“ and πœ‘ be analytic functions on 𝐷 with πœ‘(𝐷)βŠ‚π·. Then πΆπœ“,πœ‘ is a bounded Fredholm operator on π’Ÿ if and only if πœ“βˆˆβ„³, bounded away from zero near the unit circle, and πœ‘ is an automorphism of 𝐷.

If πœ“(𝑧)=1, then the result above gives the characterization of bounded Fredholm composition operator πΆπœ‘ on π’Ÿ, which was obtained in [12].

As corollaries, in the end of this paper one gives the characterization of bounded invertible and unitary weighted composition operator on π’Ÿ, respectively. Some idea of this paper is derived from [4, 13], which characterize normal and bounded invertible weighted composition operator on the Hardy space, respectively.

2. Proof of the Main Result

In the following, πœ“ and πœ‘ denote analytic functions on 𝐷 with πœ‘(𝐷)βŠ‚π·. It is easy to verify that πœ“βˆˆπ’Ÿ if πΆπœ“,πœ‘ is defined on π’Ÿ.

Proposition 2.1. Let πΆπœ“,πœ‘ be a bounded Fredholm operator on π’Ÿ. Then πœ“ has at most finite zeroes in 𝐷 and πœ‘ is an inner function.

Proof. If πΆπœ“,πœ‘ is a bounded Fredholm operator, then there exist a bounded operator 𝑇 and a compact operator 𝑆 on π’Ÿ such that π‘‡ξ€·πΆπœ“,πœ‘ξ€Έβˆ—=𝐼+𝑆,(2.1) where 𝐼 is the identity operator.
Since ξ€·πΆπœ“,πœ‘ξ€Έβˆ—πΎπ‘€ξ«πΆ(𝑧)=βˆ—πœ“,πœ‘πΎπ‘€,𝐾𝑧=𝐾𝑀,πΆπœ“,πœ‘πΎπ‘§ξ¬=βŸ¨πΎπ‘€,πœ“πΎπ‘§βˆ˜πœ‘βŸ©=πœ“(𝑀)𝐾𝑧=(πœ‘(𝑀))πœ“(𝑀)πΎπœ‘(𝑀)(𝑧),(2.2) we have ||||β€–β€–πΎβ€–π‘‡β€–πœ“(𝑀)πœ‘(𝑀)‖‖‖‖𝐾𝑀‖‖β‰₯β€–β€–π‘‡ξ€·πΆπœ“,πœ‘ξ€Έβˆ—π‘˜π‘€β€–β€–β‰₯β€–β€–π‘˜π‘€β€–β€–βˆ’β€–β€–π‘†π‘˜π‘€β€–β€–β€–β€–=1βˆ’π‘†π‘˜π‘€β€–β€–,(2.3) where π‘˜π‘€=𝐾𝑀/‖𝐾𝑀‖ is the normalization of reproducing kernel function 𝐾𝑀.
Since 𝑆 is compact and π‘˜π‘€ weakly converges to 0 as |𝑀|β†’1, β€–π‘†π‘˜π‘€β€–β†’0 as |𝑀|β†’1. It follows that there exists constant π‘Ÿ, 0<π‘Ÿ<1, such that β€–π‘†π‘˜π‘€β€–<1/2 for all 𝑀 with π‘Ÿ<|𝑀|<1. Inequality (2.3) shows that ||||πœ“(𝑀)‖‖𝐾𝑀‖‖β‰₯12β€–β€–πΎβ€–π‘‡β€–πœ‘(𝑀)β€–β€–,π‘Ÿ<|𝑀|<1,(2.4) which implies that πœ“ has no zeroes in {π‘€βˆˆπ·,π‘Ÿ<|𝑀|<1}, and, hence, πœ“ has at most finite zeroes in {π‘€βˆˆπ·,|𝑀|β‰€π‘Ÿ}.
Since π‘˜π‘€ weakly converges to 0 as |𝑀|β†’1, βŸ¨πœ“,π‘˜π‘€βŸ©β†’0 as |𝑀|β†’1, that is, πœ“(𝑀)β€–β€–πΎπ‘€β€–β€–βŸΆ0,|𝑀|⟢1.(2.5) It follows from (2.4) that β€–πΎπœ‘(𝑀)β€–=(1+log(1/(1βˆ’|πœ‘(𝑀)|2)))1/2β†’βˆž and hence |πœ‘(𝑀)|β†’1 as |𝑀|β†’1, that is, πœ‘ is an inner function.

For the proof of the following lemma, we cite Carleson's formula for the Dirichlet integral [14].

Let π‘“βˆˆπ’Ÿ, 𝑓=𝐡𝑆𝐹 be the canonical factorization of 𝑓 as a function in the Hardy space, where ∏𝐡=βˆžπ‘—=1(βˆ’π‘Žπ‘—/|π‘Žπ‘—|)((π‘Žπ‘—βˆ’π‘§)/(1βˆ’βˆ’π‘Žπ‘—π‘§)), is a Blaschke product, 𝑆 is the singular part of 𝑓 and 𝐹 is the outer part of 𝑓. Then ξ€œπ·(𝑓)=Tβˆžξ“π‘›=1𝑃𝛼𝑛(||||πœ‰)𝑓(πœ‰)2||||π‘‘πœ‰+2πœ‹T2||||πœβˆ’πœ‰2||||𝑓(πœ‰)2||||π‘‘πœ‡(𝜁)π‘‘πœ‰+2πœ‹T𝑒2𝑒(𝜁)βˆ’π‘’2𝑒(πœ‰)ξ€Έ(𝑒(𝜁)βˆ’π‘’(πœ‰))||||πœβˆ’πœ‰2||||π‘‘πœ||||2πœ‹π‘‘πœ‰,2πœ‹(2.6) where T is the unit circle, 𝑒(πœ‰)=log|𝑓(πœ‰)|, 𝑃𝛼(πœ‰) is the Poisson kernel, and πœ‡ is the singular measure corresponding to 𝑆.

Lemma 2.2. Let πΆπœ“,πœ‘ be a bounded operator on π’Ÿ, πœ“=𝐡𝐹 with 𝐡 a finite Blaschke product. Then 𝐢𝐹,πœ‘ is bounded.

Proof. Let 𝑀𝐡 be the multiplication operator on π’Ÿ. Then πΆπœ“,πœ‘=𝑀𝐡𝐢𝐹,πœ‘. Since 𝐡 is a finite Blaschke product, by the Carleson's formula, we have 𝐷(πœ“(π‘“βˆ˜πœ‘))=𝐷(𝐡𝐹(π‘“βˆ˜πœ‘))β‰₯𝐷(𝐹(π‘“βˆ˜πœ‘)),π‘“βˆˆπ’Ÿ.(2.7)
Since ‖𝑓‖2=|𝑓(0)|2+𝐷(𝑓),π‘“βˆˆπ’Ÿ, by the inequality above it is easy to verify that 𝐢𝐹,πœ‘ is bounded on π’Ÿ if πΆπœ“,πœ‘ is bounded.

Lemma 2.3. Let 𝐹 be an analytic function on 𝐷 with zero-free. If 𝐢𝐹,πœ‘ is a bounded Fredholm operator on π’Ÿ, then πœ‘ is univalent.

Proof. If πœ‘(π‘Ž)=πœ‘(𝑏) for π‘Ž,π‘βˆˆπ· with π‘Žβ‰ π‘, by a similar reasoning as [1, Lemma 3.26], there exist infinite sets {π‘Žπ‘›} and {𝑏𝑛} in 𝐷 which is disjoint such that πœ‘(π‘Žπ‘›)=πœ‘(𝑏𝑛). Hence, 𝐢𝐹,πœ‘ξ€Έβˆ—βŽ›βŽœβŽœβŽπΎπ‘Žπ‘›πΉξ€·π‘Žπ‘›ξ€Έβˆ’πΎπ‘π‘›πΉξ€·π‘π‘›ξ€ΈβŽžβŽŸβŽŸβŽ =0,(2.8) which contradicts to that kernel of (𝐢𝐹,πœ‘)βˆ— is finite dimensional.

Corollary 2.4. If πΆπœ“,πœ‘ is a bounded Fredholm operator on π’Ÿ, then πœ‘ is an automorphism of 𝐷 and πœ“βˆˆβ„³.

Proof. By Proposition 2.1, πœ“ has the factorization of 𝐡𝐹 with 𝐡 a finite Blaschke product and 𝐹 zero free in 𝐷. By Lemma 2.2, 𝐢𝐹,πœ‘ is a bounded operator on π’Ÿ. Since πΆπœ“,πœ‘=𝑀𝐡𝐢𝐹,πœ‘ and 𝑀𝐡 is a Fredholm operator, 𝐢𝐹,πœ‘ is a Fredholm operator also. By Proposition 2.1 and Lemma 2.3, πœ‘ is an univalent inner function, it follows from [1, Corollary 3.28] that πœ‘ is an automorphism of 𝐷.
Since πΆπœ“,πœ‘πΆπœ‘βˆ’1=π‘€πœ“, π‘€πœ“ is a bounded multiplication operator on π’Ÿ, which implies that πœ“βˆˆβ„³.

The following lemmas is well-known. It is easy to verify by the fact π‘€βˆ—πœ“πΎπ‘€=πœ“(𝑀)𝐾𝑀 also.

Lemma 2.5. Let πœ“βˆˆβ„³. Then π‘€πœ“ is an invertible operator on π’Ÿ if and only if πœ“ is invertible in β„³.

Lemma 2.6. Let πœ“βˆˆβ„³. Then π‘€πœ“ is a Fredholm operator on π’Ÿ if and only if πœ“ is bounded away from the unit circle.

Now we give the proof of Theorem 1.1.

Proof of Theorem 1.1. If πΆπœ“,πœ‘ is a bounded Fredholm operator on π’Ÿ, by Corollary 2.4, πœ“βˆˆβ„³ and πœ‘ is an automorphism of 𝐷. Since πΆπœ‘ is invertible, π‘€πœ“ is a Fredholm operator. So πœ“ is bounded away form the unit circle follows from Lemma 2.6.
On the other hand, if πœ“βˆˆβ„³ and bounded away from the unit circle, then π‘€πœ“ is a bounded Fredholm operator on π’Ÿ. If πœ‘ is an automorphism of 𝐷, then πΆπœ‘ is invertible. Hence πΆπœ“,πœ‘=π‘€πœ“πΆπœ‘ is a bounded Fredholm operator on π’Ÿ.

As corollaries, in the following, we characterize bounded invertible and unitary weighted composition operators on π’Ÿ.

Corollary 2.7. Let πœ“ and πœ‘ be analytic functions on 𝐷 with πœ‘(𝐷)βŠ‚π·. Then πΆπœ“,πœ‘ is a bounded invertible operator on π’Ÿ if and only if πœ“βˆˆβ„³, invertible in β„³, and πœ‘ is an automorphism of 𝐷.

Proof. Since a bounded invertible operator is a bounded Fredholm operator, the proof is similar to the proof of Theorem 1.1.

Corollary 2.8. Let πœ“ and πœ‘ be analytic functions on 𝐷 with πœ‘(𝐷)βŠ‚π·. πΆπœ“,πœ‘ is a bounded operator on π’Ÿ. Then πΆπœ“,πœ‘ is a unitary operator if and only if πœ“ is a constant with |πœ“|=1 and πœ‘ is a rotation of 𝐷.

Proof. If πΆπœ“,πœ‘ is a unitary operator, then it must be an invertible operator. By Corollary 2.7, πœ‘ is an automorphism of 𝐷 and πœ“ is invertible in β„³.
Let 𝑛 be nonnegative integer, 𝑒𝑛(𝑧)=𝑧𝑛,π‘§βˆˆπ·. A unitary is also an isometry, so we have β€–β€–πΆβ€–πœ“β€–=πœ“,πœ‘π‘’0β€–β€–=‖‖𝑒0β€–β€–=1,(2.9)β€–πœ“πœ‘π‘›β€–β€–πΆβ€–=πœ“,πœ‘π‘’π‘›β€–β€–=‖‖𝑒𝑛‖‖=βˆšπ‘›,𝑛β‰₯1.(2.10)
Let π›Όβˆˆπ· such that πœ‘(𝛼)=0. Since πœ‘ is an automorphism of 𝐷, πœ‘π‘› is a finite Blaschke product with zero 𝛼 of order 𝑛. By Carleson’s formula for Dirichlet integral, we have 𝐷(πœ“πœ‘π‘›ξ€œ)=𝑛T𝑃𝛼||||(πœ‰)πœ“(πœ‰)2||||π‘‘πœ‰2πœ‹+𝐷(πœ“).(2.11) Hence, 𝑛=β€–πœ“πœ‘π‘›β€–2=||πœ“(0)πœ‘(0)𝑛||2+𝐷(πœ“πœ‘π‘›)=||πœ“(0)πœ‘(0)𝑛||2ξ€œ+𝑛T𝑃𝛼||||(πœ‰)πœ“(πœ‰)2||||π‘‘πœ‰2πœ‹+𝐷(πœ“),𝑛β‰₯1.(2.12) That is, ||1=πœ“(0)πœ‘(0)𝑛||2𝑛+ξ€œT𝑃𝛼||||(πœ‰)πœ“(πœ‰)2||||π‘‘πœ‰+2πœ‹π·(πœ“)𝑛,𝑛β‰₯1.(2.13) Let π‘›β†’βˆž, then ∫1=T𝑃𝛼(πœ‰)|πœ“(πœ‰)|2(|π‘‘πœ‰|/2πœ‹).
By (2.12), we have 𝐷(πœ“)=0 and |πœ“(0)πœ‘(0)|=0. By (2.9), we obtain πœ“ is a constant with |πœ“|=1, which implies that πœ‘(0)=0, that is, πœ‘ is a rotation of 𝐷.
The sufficiency is easy to verify.

Remark 2.9. The key step in the proof of the main result is to analyze zeros of the symbol πœ“ and univalency of πœ‘. The following result pointed out by the referee gives a simple characterization of the symbols πœ“ and πœ‘ for the bounded Fredholm operator πΆπœ“,πœ‘ on π’Ÿ.

Proposition 2.10. Let πœ“ and πœ‘ be analytic functions on 𝐷 with πœ‘(𝐷)βŠ‚π·. πΆπœ“,πœ‘ is a bounded Fredholm operator on π’Ÿ. Then πœ“ has only finitely many zeros in 𝐷 and πœ‘ is univalent.

Proof. If πœ“(π‘Ž)=0 for π‘Žβˆˆπ·, then πΆβˆ—πœ“,πœ‘πΎπ‘Ž=πœ“(π‘Ž)πΎπœ‘(π‘Ž)=0, which implies that πΎπ‘Ž is in the kernel of πΆβˆ—πœ“,πœ‘. Thus if πœ“ had infinitely many zeros, the kernel of πΆβˆ—πœ“,πœ‘ would be infinite dimensional and hence this operator would not be Fredholm.
If πœ‘(π‘Ž)=πœ‘(𝑏) for π‘Ž,π‘βˆˆπ· with π‘Žβ‰ π‘, by a similar reasoning as [1, Lemma 3.26], there exist infinite sets {π‘Žπ‘›} and {𝑏𝑛} in 𝐷 which is disjoint such that πœ‘(π‘Žπ‘›)=πœ‘(𝑏𝑛). Since πœ“ has only finitely many zeros in 𝐷, we can choose infinitely many π‘Žπ‘› and 𝑏𝑛 such that πœ“(π‘Žπ‘›)β‰ 0, πœ“(𝑏𝑛)β‰ 0. Hence, ξ€·πΆπœ“,πœ‘ξ€Έβˆ—βŽ›βŽœβŽœβŽπΎπ‘Žπ‘›πœ“ξ€·π‘Žπ‘›ξ€Έβˆ’πΎπ‘π‘›πœ“ξ€·π‘π‘›ξ€ΈβŽžβŽŸβŽŸβŽ =0.(2.14) Since πΆπœ“,πœ‘ is a Fredholm operator, πœ‘ must be univalent.

Acknowledgments

Thanks are for referee for many helpful suggestions which promote the author to think the related issues deeply. This work is supported by YSF of Shanxi (2010021002-2) and NSFC (11201274).

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