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Abstract and Applied Analysis
VolumeΒ 2011Β (2011), Article IDΒ 146758, 15 pages
http://dx.doi.org/10.1155/2011/146758
Research Article

Unital Compact Homomorphisms between Extended Analytic Lipschitz Algebras

Department of Mathematics, Faculty of Science, Arak University, Arak 38156-8-8349, Iran

Received 22 May 2011; Accepted 10 August 2011

Academic Editor: Malisa R.Β Zizovic

Copyright Β© 2011 Davood Alimohammadi and Maliheh Mayghani. 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

Let 𝑋 and 𝐾 be compact plane sets with πΎβŠ†π‘‹. We define 𝐴(𝑋,𝐾)={π‘“βˆˆπΆ(𝑋)βˆΆπ‘“|𝐾∈𝐴(𝐾)}, where 𝐴(𝐾)={π‘”βˆˆπΆ(𝑋)βˆΆπ‘” is analytic on int(𝐾)}. For π›Όβˆˆ(0,1], we define Lip(𝑋,𝐾,𝛼)={π‘“βˆˆπΆ(𝑋)βˆΆπ‘π›Ό,𝐾(𝑓)=sup{|𝑓(𝑧)βˆ’π‘“(𝑀)|/|π‘§βˆ’π‘€|π›ΌβˆΆπ‘§,π‘€βˆˆπΎ,𝑧≠𝑀}<∞} and Lip𝐴(𝑋,𝐾,𝛼)=𝐴(𝑋,𝐾)∩Lip(𝑋,𝐾,𝛼). It is known that Lip𝐴(𝑋,𝐾,𝛼) is a natural Banach function algebra on 𝑋 under the norm ||𝑓||Lip(𝑋,𝐾,𝛼)=||𝑓||𝑋+𝑝𝛼,𝐾(𝑓), where ||𝑓||𝑋=sup{|𝑓(π‘₯)|∢π‘₯βˆˆπ‘‹}. These algebras are called extended analytic Lipschitz algebras. In this paper we study unital homomorphisms from natural Banach function subalgebras of Lip𝐴(𝑋1,𝐾1,𝛼1) to natural Banach function subalgebras of Lip𝐴(𝑋2,𝐾2,𝛼2) and investigate necessary and sufficient conditions for which these homomorphisms are compact. We also determine the spectrum of unital compact endomorphisms of Lip𝐴(𝑋,𝐾,𝛼).

1. Introduction and Preliminaries

We let β„‚, 𝔻={π‘§βˆˆβ„‚βˆΆ|𝑧|<1}, 𝔻={π‘§βˆˆβ„‚βˆΆ|𝑧|≀1}, 𝔻(πœ†,π‘Ÿ)={π‘§βˆˆβ„‚βˆΆ|π‘§βˆ’πœ†|<π‘Ÿ}, and 𝔻(πœ†,π‘Ÿ)={π‘§βˆˆβ„‚βˆΆ|π‘§βˆ’πœ†|β‰€π‘Ÿ} denote the field of complex numbers, the open unit disc, the closed unit disc, and the open and closed discs with center at πœ† and radius π‘Ÿ, respectively. We also denote 𝔻(0,π‘Ÿ) by π”»π‘Ÿ.

Let 𝐴 and 𝐡 be unital commutative semisimple Banach algebras with maximal ideal spaces β„³(𝐴) and β„³(𝐡). A homomorphism π‘‡βˆΆπ΄β†’π΅ is called unital if 𝑇1𝐴=1𝐡. If 𝑇 is a unital homomorphism from 𝐴 into 𝐡, then 𝑇 is continuous and there exists a norm-continuous map πœ‘βˆΆβ„³(𝐡)β†’β„³(𝐴) such that ξ‚Šξπ‘‡π‘“=π‘“βˆ˜πœ‘ for all π‘“βˆˆπ΄, where ̂𝑔 is the Gelfand transform 𝑔. In fact, πœ‘ is equal the adjoint of π‘‡βˆ—βˆΆπ΅βˆ—β†’π΄βˆ— restricted to β„³(𝐡). Note that π‘‡βˆ— is a weak*-weak* continuous map from π΅βˆ— into π΄βˆ—. Thus πœ‘ is a continuous map from β„³(𝐡) with the Gelfand topology into β„³(𝐴) with the Gelfand topology.

Let 𝐴 be a unital commutative semisimple Banach algebra, and let 𝑇 be an endomorphism of 𝐴, a homomorphism from 𝐴 into 𝐴. We denote the spectrum of 𝑇 by 𝜎(𝑇) and define𝜎(𝑇)={πœ†βˆˆβ„‚βˆΆπœ†πΌβˆ’π‘‡isnotinvertible}.(1.1)

For a compact Hausdorff space 𝑋, we denote by 𝐢(𝑋) the Banach algebra of all continuous complex-valued functions on 𝑋.

Definition 1.1. Let 𝑋 be a compact Hausdorff space. A Banach function algebra on 𝑋 is a subalgebra 𝐴 of 𝐢(𝑋) which contains 1𝑋, the constant function 1 on 𝑋, separates the points of 𝑋, and is a unital Banach algebra with an algebra norm β€–β‹…β€–. If the norm of a Banach function algebra on 𝑋 is ‖⋅‖𝑋, the uniform norm on 𝑋, it is called a uniform algebra on 𝑋.

Let 𝐴 and 𝐡 be Banach function algebras on 𝑋 and π‘Œ, respectively. If πœ‘βˆΆπ‘Œβ†’π‘‹ is a continuous mapping such that π‘“βˆ˜πœ‘βˆˆπ΅ for all π‘“βˆˆπ΄ and if π‘‡βˆΆπ΄β†’π΅ is defined by 𝑇𝑓=π‘“βˆ˜πœ‘, then 𝑇 is a unital homomorphism, which is called the induced homomorphism from 𝐴 into 𝐡 by πœ‘. In particular, if π‘Œ=𝑋 and 𝐡=𝐴, then 𝑇 is called the induced endomorphism of 𝐴 by the self-map πœ‘ of 𝑋.

Let 𝐴 be a Banach function algebra on a compact Hausdorff space 𝑋. For π‘₯βˆˆπ‘‹, the map 𝑒π‘₯βˆΆπ΄β†’β„‚, defined by 𝑒π‘₯(𝑓)=𝑓(π‘₯), is an element of β„³(𝐴) and is called the evaluation homomorphism on 𝐴 at π‘₯. This fact implies that 𝐴 is semisimple and ‖𝑓‖𝑋≀‖𝑓‖ℳ(𝐴) for all π‘“βˆˆπ΄. Note that the map π‘₯↦𝑒π‘₯βˆΆπ‘‹β†’β„³(𝐴) is a continuous one-to-one mapping. If this map is onto, we say that 𝐴 is natural.

Proposition 1.2. Let 𝑋 and π‘Œ be compact Hausdorff spaces, and let 𝐴 and 𝐡 be natural Banach function algebras on 𝑋 and π‘Œ, respectively. Then every unital homomorphism π‘‡βˆΆπ΄β†’π΅ is induced by a unique continuous map πœ‘βˆΆπ‘Œβ†’π‘‹. In particular, if 𝑋 is a compact plane set and the coordinate function 𝑍 belongs to 𝐴, then πœ‘=𝑇𝑍 and so πœ‘βˆˆπ΅.

Proof. Let π‘‡βˆΆπ΄β†’π΅ be a unital homomorphism. Since 𝐴 and 𝐡 are unital commutative semisimple Banach algebras, there exists a continuous map πœ“βˆΆβ„³(𝐡)β†’β„³(𝐴) such that ξ‚Šξπ‘‡π‘“=π‘“βˆ˜πœ“ for all π‘“βˆˆπ΄. The naturality of the Banach function algebra 𝐴 on 𝑋 implies that the map π½π΄βˆΆπ‘‹β†’β„³(𝐴), defined by 𝐽𝐴(π‘₯)=𝑒π‘₯, is a homeomorphism and so π½π΄βˆ’1βˆΆβ„³(𝐴)→𝑋 is continuous. Since 𝐡 is a Banach function algebra on π‘Œ, the map π½π΅βˆΆπ‘Œβ†’β„³(𝐡), defined by 𝐽𝐡(𝑦)=𝑒𝑦, is continuous. We now define the map πœ‘βˆΆπ‘Œβ†’π‘‹ by πœ‘=π½π΄βˆ’1βˆ˜πœ“βˆ˜π½π΅. Clearly, πœ‘ is continuous. Let π‘“βˆˆπ΄. Since ξ‚Šξ€·π‘’(𝑇𝑓)(𝑦)=𝑇𝑓𝑦=ξ‚€ξξ‚ξ€·π½π‘“βˆ˜πœ“π΅ξ€Έ=(𝑦)π‘“βˆ˜π½π΄ξ‚=𝑓𝑒(πœ‘(𝑦))πœ‘(𝑦)ξ€Έ=π‘’πœ‘(𝑦)(=𝑓)=𝑓(πœ‘(𝑦))(π‘“βˆ˜πœ‘)(𝑦),(1.2) for all π‘¦βˆˆπ‘Œ, we have 𝑇𝑓=π‘“βˆ˜πœ‘. Therefore, 𝑇 is induced by πœ‘.
Now, let 𝑋 be a compact plane set, and let π‘βˆˆπ΄. Then πœ‘=π‘βˆ˜πœ‘=𝑇𝑍, and so πœ‘βˆˆπ΅.

Corollary 1.3. Let 𝑋 be a compact Hausdorff space, and let 𝐴 be a natural Banach function algebra on 𝑋. Then every unital endomorphism 𝑇 of 𝐴 is induced by a unique continuous self-map πœ‘ of 𝑋. In particular, if 𝑋 is a compact plane set and 𝐴 contains the coordinate function 𝑍, then πœ‘=𝑇𝑍 and so πœ‘βˆˆπ΄.

Definition 1.4. Let 𝑋 be a compact plane set which is connected by rectifiable arcs, and let 𝛿(𝑧,𝑀) be the geodesic metric on 𝑋, the infimum of the length of the arcs joining 𝑧 and 𝑀. 𝑋 is called uniformly regular if there exists a constant 𝐢 such that, for all 𝑧,π‘€βˆˆπ‘‹, 𝛿(𝑧,𝑀)≀𝐢|π‘§βˆ’π‘€|.

The following lemma occurs in [1] but it is important and we will be using it in the sequel.

Lemma 1.5 (see [1, Lemma 1.5]). Let 𝐻 and 𝐾 be two compact plane sets with π»βŠ†int(𝐾). Then there exists a finite union of uniformly regular sets in int(𝐾) containing 𝐻, namely π‘Œ, and then a positive constant 𝐢 such that for every analytic complex-valued function 𝑓 on int(𝐾) and any 𝑧,π‘€βˆˆπ», ||𝑓||ξ€·(𝑧)βˆ’π‘“(𝑀)≀𝐢|π‘§βˆ’π‘€|β€–π‘“β€–π‘Œ+β€–β€–π‘“ξ…žβ€–β€–π‘Œξ€Έ.(1.3)

Let 𝑋 be a compact plane set. We denote by 𝐴(𝑋) the algebra of all continuous complex-valued functions on 𝑋 which are analytic on int(𝑋), the interior of 𝑋, and call it the analytic uniform algebra on 𝑋. It is known that 𝐴(𝑋) is a natural uniform algebra on 𝑋.

Let 𝑋 and 𝐾 be compact plane sets such that πΎβŠ†π‘‹. We define 𝐴(𝑋,𝐾)={π‘“βˆˆπΆ(𝑋)βˆΆπ‘“|𝐾∈𝐴(𝐾)}. Clearly, 𝐴(𝑋,𝐾)=𝐴(𝑋) if 𝐾=𝑋, and 𝐴(𝑋,𝐾)=𝐢(𝑋) if int(𝐾) is empty. We know that 𝐴(𝑋,𝐾) is a natural uniform algebra on 𝑋 (see [2]) and call it the extended analytic uniform algebra on 𝑋 with respect to 𝐾.

Let (𝑋,𝑑) be a compact metric space. For π›Όβˆˆ(0,1], we denote by Lip(𝑋,𝛼) the algebra of all complex-valued functions 𝑓 for which 𝑝𝛼,𝑋(𝑓)=sup{|𝑓(𝑧)βˆ’π‘“(𝑀)|/𝑑𝛼(𝑧,𝑀)βˆΆπ‘§,π‘€βˆˆπ‘‹,𝑧≠𝑀}<∞. For π‘“βˆˆLip(𝑋,𝛼), we define the 𝛼-Lipschitz norm 𝑓 by ‖𝑓‖Lip(𝑋,𝛼)=‖𝑓‖𝑋+𝑝𝛼,𝑋(𝑓). Then (Lip(𝑋,𝛼),β€–β‹…β€–Lip(𝑋,𝛼)) is a unital commutative Banach algebra. For π›Όβˆˆ(0,1), we denote by lip(𝑋,𝛼) the algebra of all complex-valued functions 𝑓 on 𝑋 for which |𝑓(𝑧)βˆ’π‘“(𝑀)|/𝑑𝛼(𝑧,𝑀)β†’0 as 𝑑(𝑧,𝑀)β†’0. Then lip(𝑋,𝛼) is a unital closed subalgebra of Lip(𝑋,𝛼). These algebras are called Lipschitz algebras of order 𝛼 and were first studied by Sherbert in [3, 4]. We know that the Lipschitz algebras Lip(𝑋,𝛼) and lip(𝑋,𝛼) are natural Banach function algebras on 𝑋.

Let (𝑋,𝑑) be a compact metric space, and let 𝐾 be a compact subset of 𝑋. For π›Όβˆˆ(0,1], we denote by Lip(𝑋,𝐾,𝛼) the algebra of all complex-valued functions 𝑓 on 𝑋 for which 𝑝𝛼,𝐾(𝑓)=sup{|𝑓(𝑧)βˆ’π‘“(𝑀)|/𝑑𝛼(𝑧,𝑀)βˆΆπ‘§,π‘€βˆˆπΎ,𝑧≠𝑀}<∞. In fact, Lip(𝑋,𝐾,𝛼)={π‘“βˆˆπΆ(𝑋)βˆΆπ‘“|𝐾∈Lip(𝐾,𝛼)}. For π‘“βˆˆLip(𝑋,𝐾,𝛼), we define ‖𝑓‖Lip(𝑋,𝐾,𝛼)=‖𝑓‖𝑋+𝑝𝛼,𝐾(𝑓). Then Lip(𝑋,𝐾,𝛼) under the algebra norm β€–β‹…β€–Lip(𝑋,𝐾,𝛼) is a unital commutative Banach algebra. Moreover, Lip(𝑋,𝛼) is a subalgebra of Lip(𝑋,𝐾,𝛼); Lip(𝑋,𝐾,𝛼)=Lip(𝑋,𝛼) if 𝑋⧡𝐾 is finite, and Lip(𝑋,𝐾,𝛼)=𝐢(𝑋) if 𝐾 is finite. For π›Όβˆˆ(0,1), we denote by lip(𝑋,𝐾,𝛼) the algebra of all complex-valued functions 𝑓 on 𝑋 for which |𝑓(𝑧)βˆ’π‘“(𝑀)|/𝑑𝛼(𝑧,𝑀)β†’0 as 𝑑(𝑧,𝑀)β†’0 with 𝑧,π‘€βˆˆπΎ. In fact, lip(𝑋,𝐾,𝛼)={π‘“βˆˆπΆ(𝑋)βˆΆπ‘“|𝐾∈lip(𝐾,𝛼)}. Clearly, lip(𝑋,𝐾,𝛼) is a closed unital subalgebra of Lip(𝑋,𝐾,𝛼). Moreover, lip(𝑋,𝛼) is a subalgebra of lip(𝑋,𝐾,𝛼); lip(𝑋,𝐾,𝛼)=lip(𝑋,𝛼) if 𝑋⧡𝐾 is finite, and lip(𝑋,𝐾,𝛼)=𝐢(𝑋) if 𝐾 is finite. The Banach algebras Lip(𝑋,𝐾,𝛼) and lip(𝑋,𝐾,𝛼) are Banach function algebras on 𝑋 and were first introduced by Honary and Moradi in [5].

Let 𝑋 be a compact plane set. We define Lip𝐴(𝑋,𝛼)=Lip(𝑋,𝛼)∩𝐴(𝑋) for π›Όβˆˆ(0,1] and lip𝐴(𝑋,𝛼)=Lip(𝑋,𝛼)∩𝐴(𝑋) for π›Όβˆˆ(0,1). These algebras are called analytic Lipschitz algebras. We know that analytic Lipschitz algebras Lip𝐴(𝑋,𝛼) and lip𝐴(𝑋,𝛼) under the norm β€–β‹…β€–Lip(𝑋,𝛼) are natural Banach function algebras on 𝑋 (see [6]).

Let 𝑋 and 𝐾 be compact plane sets with πΎβŠ†π‘‹. We define Lip𝐴(𝑋,𝐾,𝛼)=Lip(𝑋,𝐾,𝛼)∩𝐴(𝑋,𝐾) for π›Όβˆˆ(0,1] and lip𝐴(𝑋,𝐾,𝛼)=lip(𝑋,𝐾,𝛼)∩𝐴(𝑋,𝐾) for π›Όβˆˆ(0,1). Then Lip𝐴(𝑋,𝐾,𝛼) and lip(𝑋,𝐾,𝛼) are closed unital subalgebras of Lip(𝑋,𝐾,𝛼) and lip(𝑋,𝐾,𝛼) under the norm β€–β‹…β€–Lip(𝑋,𝐾,𝛼), respectively. Moreover, Lip𝐴(𝑋,𝐾,𝛼)=Lip𝐴(𝑋,𝛼)[lip𝐴(𝑋,𝐾,𝛼)=Lip𝐴(𝑋,𝛼)] if 𝐾=𝑋, and Lip𝐴(𝑋,𝐾,𝛼)=Lip(𝑋,𝐾,𝛼)[lip𝐴(𝑋,𝐾,𝛼)=lip(𝑋,𝐾,𝛼)] if int(𝐾) is empty.

The algebras Lip𝐴(𝑋,𝐾,𝛼) and lip(𝑋,𝐾,𝛼) are called extended analytic Lipschitz algebras and were first studied by Honary and Moradi in [5]. They showed that the extended analytic Lipschitz algebras Lip𝐴(𝑋,𝐾,𝛼) and lip𝐴(𝑋,𝐾,𝛼) under the norm β€–β‹…β€–Lip(𝑋,𝐾,𝛼) are natural Banach function algebras on 𝑋 [5, Theorem 2.4].

Behrouzi and Mahyar in [1] studied endomorphisms of some uniform subalgebras of 𝐴(𝑋) and some Banach function subalgebras of Lip𝐴(𝑋,𝛼) and investigated some necessary and sufficient conditions for these endomorphisms to be compact, where 𝑋 is a compact plane set and π›Όβˆˆ(0,1].

In Section 2, we study unital homomorphisms from natural Banach function subalgebras of Lip𝐴(𝑋1,𝐾1,𝛼1) to natural Banach function subalgebras of Lip𝐴(𝑋2,𝐾2,𝛼2) and investigate necessary and sufficient conditions for which these homomorphisms are compact. In Section 3, we determine the spectrum of unital compact endomorphisms of Lip𝐴(𝑋,𝐾,𝛼).

2. Unital Compact Homomorphisms

We first give a sufficient condition for which a continuous map πœ‘βˆΆπ‘‹2→𝑋1 induces a unital homomorphism 𝑇 from a subalgebra 𝐡1 of 𝐴(𝑋1,𝐾1) into a subalgebra 𝐡2 of 𝐴(𝑋2,𝐾2).

Proposition 2.1. Let 𝑋𝑗 and 𝐾𝑗 be compact plane sets with int(𝐾𝑗)β‰ βˆ… and πΎπ‘—βŠ†π‘‹π‘—, and let 𝐡𝑗 be a subalgebra of 𝐴(𝑋𝑗,𝐾𝑗) which is a natural Banach function algebra on 𝑋𝑗 under an algebra norm ‖⋅‖𝑗, where π‘—βˆˆ{1,2}. If πœ‘βˆˆπ΅2 with πœ‘(𝑋2)βŠ†int(𝐾1), then πœ‘ induces a unital homomorphism π‘‡βˆΆπ΅1→𝐡2. Moreover, if π‘βˆˆπ΅1, then πœ‘=𝑇𝑍.

Proof. The naturality of Banach function algebra 𝐡2 on 𝑋2 implies that 𝜎𝐡2(β„Ž)=β„Ž(𝑋2), where 𝜎𝐴(β„Ž) is the spectrum of β„Žβˆˆπ΄ in the Banach algebra 𝐴. Let π‘“βˆˆπ΅1. Since πœ‘βˆˆπ΅2, πœ‘(𝑋2)βŠ†int(𝐾1), and 𝑓 is analytic on int(𝐾1), we conclude that 𝑓 is analytic on an open neighborhood of 𝜎𝐡2(πœ‘). By using the Functional Calculus Theorem [2, Theorem  5.1 in Chapter I], there exists π‘”βˆˆπ΅2 such that ̂𝑔=π‘“βˆ˜ξπœ‘ on β„³(𝐡2). It follows that 𝑔(𝑧)=𝑒𝑧𝑒(𝑔)=̂𝑔𝑧𝑒=π‘“ξπœ‘π‘§ξ€·π‘’ξ€Έξ€Έ=𝑓𝑧(ξ€Έπœ‘)=𝑓(πœ‘(𝑧))=(π‘“βˆ˜πœ‘)(𝑧),(2.1) for all π‘§βˆˆπ‘‹2 and so 𝑔=π‘“βˆ˜πœ‘. Therefore, π‘“βˆ˜πœ‘βˆˆπ΅2. This implies that the map π‘‡βˆΆπ΅1→𝐡2 defined by 𝑇𝑓=π‘“βˆ˜πœ‘ is a unital homomorphism from 𝐡1 into 𝐡2, which is induced by πœ‘. Now let π‘βˆˆπ΅1. Then πœ‘=𝑇𝑍 by Proposition 1.2.

Corollary 2.2. Let 𝑋 and 𝐾 be compact plane sets with int(𝐾)β‰ βˆ… and πΎβŠ†π‘‹. Let 𝐡 be a subalgebra of 𝐴(𝑋,𝐾) which is a natural Banach function algebra on 𝑋 under an algebra norm ‖⋅‖𝐡. If πœ‘βˆˆπ΅ with πœ‘(𝑋)βŠ†int(𝐾), then πœ‘ induces a unital endomorphism 𝑇 of 𝐡. Moreover, if π‘βˆˆπ΅, then πœ‘=𝑇𝑍.

Proposition 2.3. Suppose that π›Όπ‘—βˆˆ(0,1], π‘§π‘—βˆˆβ„‚, 0<π‘Ÿπ‘—<𝑅𝑗, 𝐺𝑗=𝔻(𝑧𝑗,𝑅𝑗), Ω𝑗=𝔻(𝑧𝑗,π‘Ÿπ‘—), 𝑋𝑗=𝐺𝑗, and 𝐾𝑗=Ω𝑗, where π‘—βˆˆ{1,2}. Then for each 𝜌∈(π‘Ÿ1,𝑅1] there exists a continuous map πœ‘πœŒβˆΆπ‘‹2→𝑋1 with πœ‘πœŒ(𝑋2)=𝔻(𝑧1,𝜌) such that πœ‘πœŒβˆˆLip𝐴(𝑋2,𝐾2,𝛼2) and πœ‘πœŒ does not induce any homomorphism from Lip𝐴(𝑋1,K1,𝛼1) to Lip𝐴(𝑋2,𝐾2,𝛼2).

Proof. Let 𝜌∈(π‘Ÿ1,𝑅1]. We define the map πœ‘πœŒβˆΆπ‘‹2→𝑋1 by πœ‘πœŒβŽ§βŽͺβŽͺ⎨βŽͺβŽͺβŽ©π‘§(𝑧)=1+πœŒξ€·π‘§βˆ’π‘§2ξ€Έπ‘Ÿ2||π‘§βˆ’π‘§2||β‰€π‘Ÿ2,𝑧1+πœŒξ€·π‘§βˆ’π‘§1ξ€Έ||π‘§βˆ’π‘§2||π‘Ÿ2<||π‘§βˆ’π‘§2||≀𝑅2.(2.2) Clearly, πœ‘πœŒ is a continuous mapping, πœ‘πœŒ(𝑋2)=𝔻(𝑧1,𝜌), and πœ‘πœŒβˆˆLip𝐴(𝑋2,𝐾2,𝛼2). We now define the function π‘“πœŒβˆΆπ‘‹1β†’β„‚ by π‘“πœŒβŽ§βŽͺβŽͺ⎨βŽͺβŽͺβŽ©πœŒξ€·(𝑧)=π‘§βˆ’π‘§1ξ€Έπ‘Ÿ1||π‘§βˆ’π‘§1||β‰€π‘Ÿ1,πœŒξ€·π‘§βˆ’π‘§1ξ€Έ||π‘§βˆ’π‘§1||π‘Ÿ1<||π‘§βˆ’π‘§1||≀𝑅1.(2.3) Then, π‘“πœŒβˆˆLip𝐴(𝑋1,𝐾1,𝛼1). Since 0<π‘Ÿ1π‘Ÿ2/𝜌<π‘Ÿ2 and ξ€·π‘“πœŒβˆ˜πœ‘πœŒξ€ΈβŽ§βŽͺβŽͺ⎨βŽͺβŽͺ⎩𝜌(𝑧)=2π‘Ÿ1π‘Ÿ2ξ€·π‘§βˆ’π‘§2ξ€Έ||π‘§βˆ’π‘§2||β‰€π‘Ÿ1π‘Ÿ2𝜌,πœŒξ€·π‘§βˆ’π‘§2ξ€Έ||π‘§βˆ’π‘§2||π‘Ÿ1π‘Ÿ2𝜌<||π‘§βˆ’π‘§2||≀𝑅2,(2.4) we conclude that π‘“πœŒβˆ˜πœ‘πœŒβˆ‰Lip𝐴(𝑋2,𝐾2,𝛼2). Therefore, πœ‘πœŒ does not induce any homomorphism from Lip𝐴(𝑋1,𝐾1,𝛼1) to Lip𝐴(𝑋2,𝐾2,𝛼2). Hence, the proof is complete.

Corollary 2.4. Suppose that π›Όβˆˆ(0,1], πœ†βˆˆβ„‚, 0<π‘Ÿ<𝑅, 𝐺=𝔻(πœ†,𝑅), Ξ©=𝔻(πœ†,π‘Ÿ), 𝑋=𝐺, and 𝐾=Ξ©. Then for each 𝜌∈(π‘Ÿ,𝑅], there exists a continuous self-map πœ‘πœŒ of 𝑋 with πœ‘πœŒ(𝑋)=𝔻(πœ†,𝜌) such that πœ‘πœŒβˆˆLip𝐴(𝑋,𝐾,𝛼) and πœ‘πœŒ does not induce any endomorphism of Lip𝐴(𝑋,𝐾,𝛼).

We now give a sufficient condition for a unital homomorphism from a subalgebra 𝐡1 of Lip𝐴(𝑋1,𝐾1,𝛼1) into a subalgebra 𝐡2 of Lip𝐴(𝑋2,𝐾2,𝛼2) to be compact.

Theorem 2.5. Suppose that π›Όπ‘—βˆˆ(0,1], 𝑋𝑗 and 𝐾𝑗 are compact plane sets with int(𝐾𝑗)β‰ βˆ… and πΎπ‘—βŠ†π‘‹π‘—, and 𝐡𝑗 is a subalgebra of Lip𝐴(𝑋𝑗,𝐾𝑗,𝛼𝑗) which is a natural Banach function algebra on 𝑋𝑗 under the norm β€–β‹…β€–Lip(𝑋𝑗,𝐾𝑗,𝛼𝑗), where π‘—βˆˆ{1,2}. Let πœ‘βˆΆπ‘‹2→𝑋1 be a continuous mapping. If πœ‘ is constant or πœ‘βˆˆπ΅2 with πœ‘(𝑋2)βŠ†int(𝐾1), then πœ‘ induces a unital compact homomorphism π‘‡βˆΆπ΅1→𝐡2.

Proof. If πœ‘βˆΆπ‘‹2→𝑋1 is constant, then the map π‘‡βˆΆπ΅1→𝐡2 defined by 𝑇𝑓=π‘“βˆ˜πœ‘ is a unital homomorphism from 𝐡1 into 𝐡2 with dim𝑇(𝐡1)≀1, and so it is compact.
Let πœ‘βˆΆπ‘‹2→𝑋1 be a nonconstant mapping with πœ‘βˆˆπ΅2 and πœ‘(𝑋2)βŠ†Ξ©1. Then the map π‘‡βˆΆπ΅1→𝐡2 defined by 𝑇𝑓=π‘“βˆ˜πœ‘ is a unital homomorphism from 𝐡1 to 𝐡2 by Proposition 2.1. To prove the compactness of 𝑇, let {𝑓𝑛}βˆžπ‘›=1 be a bounded sequence in 𝐡1 with ‖𝑓𝑛‖Lip(𝑋1,𝐾1,𝛼1)≀1 for all π‘›βˆˆβ„•. This implies that {𝑓𝑛|𝐾1}βˆžπ‘›=1 is a bounded sequence in 𝐢(𝐾1) which is equicontinuous on (𝐾1,𝑑𝛼11). By Arzela-Ascoli’s theorem, {𝑓𝑛}βˆžπ‘›=1 has a subsequence {𝑓𝑛𝑗}βˆžπ‘—=1 such that {𝑓𝑛𝑗|𝐾1}βˆžπ‘—=1 is convergent in 𝐢(𝐾1). Since 𝑓𝑛𝑗|𝐾1∈𝐴(𝐾1) for all π‘—βˆˆβ„•, {𝑓𝑛𝑗|𝐾1}βˆžπ‘—=1 is convergent in 𝐴(𝐾1). By Montel's theorem, the sequences {𝑓𝑛𝑗}βˆžπ‘—=1 and {π‘“ξ…žπ‘›π‘—}βˆžπ‘—=1 are uniformly convergent on the compact subsets of int(𝐾1). Since πœ‘(𝑋2) and 𝐾1 are compact sets in the complex plane and πœ‘(𝑋2)βŠ†int(𝐾1), by using Lemma 1.5, we deduce that there exists a finite union of uniformly regular sets in int(𝐾1) containing πœ‘(𝑋2), namely π‘Œ, and then a positive constant 𝐢 such that for every analytic complex-valued function 𝑓 on int(𝐾1) and any 𝑧,π‘€βˆˆπœ‘(𝑋2)||𝑓||ξ€·(𝑧)βˆ’π‘“(𝑀)≀𝐢|π‘§βˆ’π‘€|β€–π‘“β€–π‘Œ+β€–β€–π‘“ξ…žβ€–β€–π‘Œξ€Έ.(2.5) Therefore, there exists a positive constant 𝐢 such that |||𝑓𝑛𝑗(πœ‘(𝑧))βˆ’π‘“π‘›π‘—|||||||‖‖𝑓(πœ‘(𝑀))β‰€πΆπœ‘(𝑧)βˆ’πœ‘(𝑀)π‘›π‘—β€–β€–π‘Œ+β€–β€–π‘“ξ…žπ‘›π‘—β€–β€–π‘Œξ‚,(2.6) for all π‘—βˆˆβ„• and any 𝑧,π‘€βˆˆπ‘‹2. Let 𝑗,π‘˜βˆˆβ„•. Then, for all 𝑧,π‘€βˆˆπΎ2 with πœ‘(𝑧)β‰ πœ‘(𝑀), we have |||π‘“ξ‚€ξ‚€π‘›π‘—ξ‚βˆ’ξ‚€π‘“βˆ˜πœ‘π‘›π‘˜π‘“βˆ˜πœ‘ξ‚ξ‚(𝑧)βˆ’ξ‚€ξ‚€π‘›π‘—ξ‚βˆ’ξ‚€π‘“βˆ˜πœ‘π‘›π‘˜|||βˆ˜πœ‘ξ‚ξ‚(𝑀)|π‘§βˆ’π‘€|𝛼2=|||ξ‚€π‘“π‘›π‘—βˆ’π‘“π‘›π‘˜ξ‚ξ‚€π‘“(πœ‘(𝑧))βˆ’π‘›π‘—βˆ’π‘“π‘›π‘˜ξ‚|||(πœ‘(𝑀))||||β‹…||||πœ‘(𝑧)βˆ’πœ‘(𝑀)πœ‘(𝑧)βˆ’πœ‘(𝑀)|π‘§βˆ’π‘€|𝛼2≀𝐢𝑝𝛼2,𝐾2‖‖𝑓(πœ‘)π‘›π‘—βˆ’π‘“π‘›π‘˜β€–β€–π‘Œ+β€–β€–π‘“ξ…žπ‘›π‘—βˆ’π‘“ξ…žπ‘›π‘˜β€–β€–π‘Œξ‚.(2.7) The above inequality is certainly true for all 𝑧,π‘€βˆˆπΎ2 with 𝑧≠𝑀 and πœ‘(𝑧)=πœ‘(𝑀). Therefore, 𝑝𝛼2,𝐾2π‘“ξ‚€ξ‚€π‘›π‘—ξ‚βˆ’ξ‚€π‘“βˆ˜πœ‘π‘›π‘˜βˆ˜πœ‘ξ‚ξ‚β‰€πΆπ‘π›Ό2,𝐾2‖‖𝑓(πœ‘)π‘›π‘–βˆ’π‘“π‘›π‘—β€–β€–π‘Œ+β€–β€–π‘“ξ…žπ‘›π‘–βˆ’π‘“ξ…žπ‘›π‘—β€–β€–π‘Œξ‚,(2.8) and so β€–β€–ξ‚€π‘“π‘›π‘—ξ‚βˆ’ξ‚€π‘“βˆ˜πœ‘π‘›π‘˜ξ‚β€–β€–βˆ˜πœ‘Lip(𝑋2,𝐾2,𝛼2)≀1+𝐢𝑝𝛼2,𝐾2‖‖𝑓(πœ‘)π‘›π‘—βˆ’π‘“π‘›π‘˜β€–β€–π‘Œ+β€–β€–π‘“ξ…žπ‘›π‘—βˆ’π‘“ξ…žπ‘›π‘˜β€–β€–π‘Œξ‚.(2.9)
Since π‘Œ is a compact subset of int(𝐾1), we deduce that the sequences {𝑓𝑛𝑗}βˆžπ‘—=1 and {π‘“ξ…žπ‘›π‘—}βˆžπ‘—=1 are convergent uniformly on π‘Œ. Therefore, {π‘“π‘›π‘—βˆ˜πœ‘}βˆžπ‘—=1 is a Cauchy sequence on Lip(𝑋2,𝐾2,𝛼2), that is {𝑇𝑓𝑛𝑗}βˆžπ‘—=1 is convergent in Lip(𝑋2,𝐾2,𝛼2). Hence, 𝑇 is compact.

Corollary 2.6. Suppose that π›Όβˆˆ(0,1], 𝑋 and 𝐾 are compact plane sets with int(𝐾)β‰ βˆ…, and πΎβŠ†π‘‹. Let 𝐡 be a subalgebra of Lip𝐴(𝑋,𝐾,𝛼) which is a natural Banach function algebra on 𝑋 with the norm β€–β‹…β€–Lip(𝑋2,𝐾2,𝛼2), and let πœ‘ be a self-map of 𝑋. If πœ‘ is constant or πœ‘βˆˆπ΅ with πœ‘(𝑋)βŠ†int(𝐾), then πœ‘ induces a unital compact endomorphism of 𝐡.

Definition 2.7.   (a)A sector in 𝔻(𝑧0,π‘Ÿ) at a point πœ”βˆˆπœ•π”»(𝑧0,π‘Ÿ) is the region between two straight lines in 𝔻(𝑧0,π‘Ÿ) that meet at πœ” and are symmetric about the radius to πœ”. (b)If 𝑓 is a complex-valued function on 𝔻(𝑧0,π‘Ÿ) and πœ”βˆˆπœ•π”»(𝑧0,π‘Ÿ), then ∠limπ‘§β†’πœ”π‘“(𝑧)=𝐿 means that 𝑓(𝑧)→𝐿 as π‘§β†’πœ” through any sector at πœ”. When this happens, we say that 𝐿 is angular (or non-tangential) limit of 𝑓 at πœ”. (c)An analytic map πœ‘βˆΆπ”»(𝑧0,π‘Ÿ)β†’π”»πœŒ has an angular derivation at a point πœ”βˆˆπœ•π”»π‘Ÿ(𝑧0,π‘Ÿ) if for some πœ‚βˆˆπœ•π”»πœŒβˆ limπ‘§β†’πœ”πœ‚βˆ’π‘“(𝑧)πœ”βˆ’π‘§(2.10) exists (finitely). We call the limit the angular derivative of πœ‘ at πœ” and denote it by βˆ πœ‘ξ…ž(πœ”).

Lemma 2.8. Let 0<π‘Ÿβ‰€1, and let πœ‘βˆΆπ”»(𝑧0,π‘Ÿ)β†’π”»πœŒ be an analytic function and πœ“βˆΆπ”»β†’π”» defined by πœ“(𝑧)=(1/𝜌)πœ‘(𝑧0+π‘Ÿπ‘§). Then πœ‘ has angular derivation at πœ”βˆˆπœ•π”»(𝑧0,π‘Ÿ) if and only if πœ“ has angular derivation at (πœ”βˆ’π‘§0)/π‘Ÿβˆˆπœ•π”». Moreover, βˆ πœ‘ξ…žπ‘Ÿ(πœ”)=πœŒβˆ πœ“ξ…žξ‚€πœ”βˆ’π‘§0π‘Ÿξ‚.(2.11)

The following result is a modification of Julia-Caratheodory’s theorem. For further details and proof of Julia-Caratheodory’s theorem, see [7, pages 295–300].

Theorem 2.9. Take 0<π‘Ÿβ‰€1. Let πœ‘βˆΆπ”»(𝑧0,π‘Ÿ)→𝔻 be a nonconstant analytic function and πœ”βˆˆπœ•π”»(𝑧0,π‘Ÿ). Then the following are equivalent: (i)liminfπ‘§β†’πœ”(β€–πœ‘β€–π”»π‘Ÿβˆ’|πœ‘(𝑧)|)/(π‘Ÿβˆ’|𝑧|)=𝛿<∞, (ii)∠limπ‘§β†’πœ”(πœ‚βˆ’πœ‘(𝑧))/(πœ”βˆ’π‘§) exists for some πœ‚βˆˆπœ•π”», (iii)∠limπ‘§β†’πœ”πœ‘ξ…ž(𝑧) exists and ∠limπ‘§β†’πœ”πœ‘(𝑧)=πœ‚βˆˆπœ•π”».

The boundary point πœ‚ in (ii) and (iii) is the same, and 𝛿>0 in (i). Also the limit of the difference quotients in (ii) coincides with the limit of the derivative in (iii), and both are equal to πœ”πœ‚π›Ώ.

Note that the existence of the angular derivative πœ‘ at πœ”βˆˆπœ•π”»(𝑧0,π‘Ÿ), according to Theorem 2.9, is equivalent to liminfπ‘§β†’πœ”(β€–πœ‘β€–π”»(𝑧0,π‘Ÿ)βˆ’|πœ‘(𝑧)|)/(π‘Ÿβˆ’|π‘§βˆ’π‘§0|)<∞. In this case the angular derivative of πœ‘ at πœ” is nonzero.

Proposition 2.10. Let 𝑋 be a compact plane set, and let 𝔻(𝑧0,π‘Ÿ)βŠ†π‘‹ and 𝐾=𝔻(𝑧0,π‘Ÿ). Suppose that π‘βˆˆπœ•π”»(𝑧0,π‘Ÿ) and πœ‘βˆˆLip𝐴(𝑋,𝐾,1) is a nonconstant function such that |πœ‘(𝑐)|=β€–πœ‘β€–π”»(𝑧0,π‘Ÿ). Then the angular derivative of πœ‘ at 𝑐 exists and is nonzero.

Proof. Let Ξ“={π‘§βˆˆπ”»(𝑧0,π‘Ÿ)∢|π‘§βˆ’π‘|/(π‘Ÿβˆ’|π‘§βˆ’π‘§0|)<2}. For every π‘§βˆˆΞ“ we have β€–πœ‘β€–π”»(𝑧0,π‘Ÿ)βˆ’||||πœ‘(𝑧)||π‘Ÿβˆ’π‘§βˆ’π‘§0||=||||βˆ’||||πœ‘(𝑐)πœ‘(𝑧)||π‘Ÿβˆ’π‘§βˆ’π‘§0||≀|π‘§βˆ’π‘|||π‘Ÿβˆ’π‘§βˆ’π‘§0||||||πœ‘(𝑧)βˆ’πœ‘(𝑐)|π‘§βˆ’π‘|<2𝑝1,𝐾(πœ‘).(2.12) Therefore, liminfπ‘§β†’πœ”(β€–πœ‘β€–π”»(𝑧0,π‘Ÿ)βˆ’|πœ‘(𝑧)|)/(π‘Ÿβˆ’|π‘§βˆ’π‘§0|)<∞, and, by Theorem 2.9, the proof is complete.

Definition 2.11.   (a)A plane set 𝑋 at π‘βˆˆπœ•π‘‹ has an internal circular tangent if there exists a disc 𝐷 in the complex plane such that π‘βˆˆπœ•π· and 𝐷⧡{𝑐}βŠ†int(𝑋). (b)A plane set 𝑋 is called strongly accessible from the interior if it has an internal circular tangent at each point of its boundary. Such sets include the closed unit disc 𝔻 and 𝔻(𝑧0⋃,π‘Ÿ)β§΅π‘›π‘˜=1𝔻(π‘§π‘˜,π‘Ÿπ‘˜), where closed discs 𝔻(π‘§π‘˜,π‘Ÿπ‘˜) are mutually disjoint in 𝔻(𝑧0,π‘Ÿ). (c)A compact plane set 𝑋 has peak boundary with respect to π΅βŠ†πΆ(𝑋) if for each π‘βˆˆπœ•π‘‹ there exists a nonconstant function β„Žβˆˆπ΅ such that β€–β„Žβ€–π‘‹=β„Ž(𝑐)=1.

Example 2.12. The closed unit disc 𝔻 has peak boundary with respect to 𝐴(𝔻) because, if π‘βˆˆπœ•π”», then the function β„ŽβˆΆπ”»β†’πΆ defined by β„Ž(𝑧)=(1/2)(1+𝑐𝑧) belongs to 𝐴(𝔻) and satisfies β€–β„Žβ€–π”»=β„Ž(𝑐)=1.

Let 𝑋 be a compact plane set. The algebra 𝑅(𝑋) consists of all functions in 𝐢(𝑋) which can be approximated by rational functions with poles off 𝑋. It is known that 𝑅(𝑋) is a natural uniform algebra on 𝑋.

Example 2.13. Let 𝑋 be a compact plane set such that ℂ⧡𝑋 is strongly accessible from the interior. If 𝑅(𝑋)βŠ†π΅βŠ†πΆ(𝑋), then 𝑋 has a peak boundary with respect to 𝐡.

Proof. Let 𝑧0βˆˆβ„‚β§΅π‘‹. Since ℂ⧡𝑋 is strongly accessible from the interior, for each π‘βˆˆπœ•(ℂ⧡𝑋), there exists a 𝛿>0 such that |π‘βˆ’π‘§0|=𝛿 and 𝔻(𝑧0,𝛿)βŠ†int(ℂ⧡𝑋). Now, we define the function β„ŽβˆΆπ‘‹β†’β„‚ by π›Ώβ„Ž(𝑧)=2ξ€·π‘βˆ’π‘§0ξ€Έξ€·π‘§βˆ’π‘§0ξ€Έ.(2.13) Then β„Žβˆˆπ΅, β€–β„Žβ€–π‘‹=β„Ž(𝑐)=1.

Theorem 2.14. Let 𝑋1 be a compact plane set such that 𝐺1=int(𝑋1) is connected, and 𝐺1=𝑋1. Suppose that 𝑋1 has peak boundary with respect to Lip𝐴(𝑋1,1). Let Ξ©1βŠ†πΊ1 be a bounded connected open set in the complex plane, and let 𝐾1=Ξ©1. Let Ξ©2 be a bounded connected open set in the complex plane, and let 𝐾2=Ξ©2 such that 𝐾2 is strongly accessible from the interior. Suppose that 𝑋2 is a compact plane set such that 𝐾2βŠ†π‘‹2. If π‘‡βˆΆLip𝐴(𝑋1,𝐾1,1)β†’Lip𝐴(𝑋2,𝐾2,1) is a unital compact homomorphism, then 𝑇 is induced by a continuous mapping πœ‘βˆΆπ‘‹2→𝑋1 such that πœ‘ is constant on 𝐾2 or πœ‘(𝐾2)βŠ†πΊ1=int(𝑋1). Moreover, πœ‘=𝑇𝑍.

Proof. Since Lip𝐴(𝑋1,𝐾1,1) and Lip𝐴(𝑋2,𝐾2,1) are, respectively, natural Banach function algebras on 𝑋1 and 𝑋2, π‘‡βˆΆLip𝐴(𝑋1,𝐾1,1)β†’Lip𝐴(𝑋2,𝐾2,1) is a unital homomorphism, 𝑋1 is a compact plane set, and π‘βˆˆLip𝐴(𝑋1,𝐾1,1), we conclude that 𝑇 is induced by πœ‘=𝑇𝑍 and so πœ‘βˆˆLip𝐴(𝑋2,𝐾2,1) by Proposition 1.2. Suppose that πœ‘ is nonconstant on Ξ©2. Since πœ‘ is analytic on Ξ©2, we deduce that πœ‘(Ξ©2) is an open subset of 𝑋1 and so πœ‘(Ξ©2)βŠ†πΊ1. We now show that πœ‘(𝐾2)βŠ†πΊ1. Suppose that πœ‘(𝐾2)ΜΈβŠ†πΊ1. Then there exists π‘βˆˆπœ•πΎ2 such that πœ‘(𝑐)βˆˆπœ•π‘‹1. Since 𝑋1 has peak boundary with respect to Lip𝐴(𝑋1,1), there exists a nonconstant function β„ŽβˆˆLip𝐴(𝑋1,1) such that β€–β„Žβ€–π‘‹1=β„Ž(πœ‘(𝑐))=1. We now define the sequence {𝑓𝑛}βˆžπ‘›=1 of complex-valued functions on 𝑋1 by 𝑓𝑛(𝑧)=(1/𝑛)β„Žπ‘›(𝑧). Let π‘›βˆˆβ„•. Then ‖‖𝑓𝑛‖‖𝑋1=1π‘›ξ€·β€–β„Žβ€–π‘‹1𝑛=1𝑛,𝑝(2.14)1,𝐾1𝑓𝑛||β„Ž=sup𝑛(𝑧)βˆ’β„Žπ‘›||(𝑀)𝑛|π‘§βˆ’π‘€|βˆΆπ‘§,π‘€βˆˆπΎ1ξ‚Όξ‚»||||,𝑧≠𝑀≀supβ„Ž(𝑧)βˆ’β„Ž(𝑀)|π‘§βˆ’π‘€|βˆΆπ‘§,π‘€βˆˆπΎ1ξ‚Όξ‚»||||,𝑧≠𝑀≀supβ„Ž(𝑧)βˆ’β„Ž(𝑀)|π‘§βˆ’π‘€|βˆΆπ‘§,π‘€βˆˆπ‘‹1ξ‚Ό,𝑧≠𝑀≀𝑝1,𝑋1(β„Ž).(2.15) Thus ‖‖𝑓𝑛‖‖Lip(𝑋1,𝐾1,1)≀1𝑛+𝑝1,𝑋1(β„Ž)≀1+𝑝1,𝑋1(β„Ž),(2.16) by (2.14) and (2.15). This implies that {𝑓𝑛}βˆžπ‘›=1 is a bounded sequence in Lip𝐴(𝑋1,𝐾1,1). The compactness of homomorphism 𝑇 implies that there exists a subsequence {𝑓𝑛𝑗}βˆžπ‘—=1 of {𝑓𝑛}βˆžπ‘›=1 and a function 𝑔 in Lip𝐴(𝑋2,𝐾2,1) such that limπ‘—β†’βˆžβ€–β€–π‘‡π‘“π‘›π‘—β€–β€–βˆ’π‘”Lip(𝑋2,𝐾2,1)=0.(2.17) This implies that limπ‘—β†’βˆžβ€–β€–π‘‡π‘“π‘›π‘—β€–β€–βˆ’π‘”π‘‹2=0.(2.18) On the other hand, we have ‖𝑇𝑓𝑛𝑗‖𝑋2≀1/𝑛𝑗 for all π‘—βˆˆβ„• by (2.14). Hence, limπ‘—β†’βˆžβ€–β€–π‘‡π‘“π‘›π‘—β€–β€–π‘‹2=0.(2.19) By (2.18) and (2.19), 𝑔=0. Therefore, by (2.17) we have limπ‘—β†’βˆžβ€–β€–π‘‡π‘“π‘›π‘—β€–β€–Lip(𝑋2,𝐾2,1)=0.(2.20) This implies that limπ‘—β†’βˆžπ‘1,𝐾2ξ‚€π‘“π‘›π‘—ξ‚βˆ˜πœ‘=0.(2.21) Assume that πœ€>0. By (2.21), there exists a natural number 𝑁 such that for each π‘—βˆˆβ„• with 𝑗β‰₯π‘βŽ§βŽͺ⎨βŽͺ⎩|||𝑓supπ‘›π‘—ξ‚ξ‚€π‘“βˆ˜πœ‘(𝑧)βˆ’π‘›π‘—ξ‚|||βˆ˜πœ‘(𝑀)|π‘§βˆ’π‘€|βˆΆπ‘§,π‘€βˆˆπΎ2⎫βŽͺ⎬βŽͺ⎭,𝑧≠𝑀<πœ€.(2.22) In particular, ξ‚»||sup((β„Žβˆ˜πœ‘)(𝑧))π‘›π‘βˆ’((β„Žβˆ˜πœ‘)(𝑀))𝑛𝑁||𝑛𝑁|π‘§βˆ’π‘€|βˆΆπ‘§,π‘€βˆˆπΎ2ξ‚Ό,𝑧≠𝑀<πœ€.(2.23) This implies that 1𝑛𝑁||sup((β„Žβˆ˜πœ‘)(𝑧))π‘›π‘βˆ’((β„Žβˆ˜πœ‘)(𝑐))𝑛𝑁|||π‘§βˆ’π‘|βˆΆπ‘§βˆˆπΎ2ξ‚Ό,𝑧≠𝑐<πœ€.(2.24) Since π‘βˆˆπœ•πΎ2 and 𝐾2 is strongly accessible from the interior, there exists an open disc 𝐷=𝔻(𝑧0,π‘Ÿ) such that π‘βˆˆπœ•π· and 𝐷⧡{𝑐}βŠ†int(𝐾2). Since πœ‘ is analytic on int(𝐷)βŠ†int(𝐾2) and β„Ž is analytic on πœ‘(𝐷)βŠ†int(𝑋1), we deduce that β„Žβˆ˜πœ‘ is analytic on int(𝐷). On the other hand, we can easily show that 𝑝1,𝐷(β„Žβˆ˜πœ‘)≀𝑝1,𝑋1(β„Ž)𝑝1,𝐾2(πœ‘)<∞.(2.25) Therefore, β„Žβˆ˜πœ‘βˆˆLip𝐴(𝑋2,𝐷,1). Since β€–β„Žβ€–π‘‹1=β„Ž(πœ‘(𝑐))=1, we conclude that (β„Žβˆ˜πœ‘)(𝑐)=β€–β„Žβˆ˜πœ‘β€–π·=1.(2.26) We claim that β„Žβˆ˜πœ‘ is constant on 𝐷. If β„Žβˆ˜πœ‘ is nonconstant on 𝐷, then, by Proposition 2.10, ∠(β„Žβˆ˜πœ‘)ξ…ž(𝑐) exists and is nonzero and since (β„Žβˆ˜πœ‘)𝑛𝑁(𝑐)=1, (β„Žβˆ˜πœ‘)𝑛𝑁(𝑐)βˆˆπœ•π·. If Ξ“ is a sector in 𝐷 at π‘βˆˆπœ•π·, then 1π‘›π‘βˆ limπ‘§β†’π‘π‘§βˆˆΞ“||||(β„Žβˆ˜πœ‘)𝑛𝑁(𝑧)βˆ’(β„Žβˆ˜πœ‘)𝑛𝑁(𝑐)||||π‘§βˆ’π‘β‰€πœ€(2.27) by (2.24). Thus 1π‘›π‘βˆ ξ€·(β„Žβˆ˜πœ‘)π‘›π‘ξ€Έξ…ž(𝑐)β‰€πœ€.(2.28) But βˆ ξ€·(β„Žβˆ˜πœ‘)π‘›π‘ξ€Έξ…ž(𝑐)=𝑛𝑁(β„Žβˆ˜πœ‘)π‘›π‘βˆ’1(𝑐)β‹…βˆ (β„Žβˆ˜πœ‘)ξ…ž(𝑐).(2.29) Hence, by (2.28) we have ∠(β„Žβˆ˜πœ‘)ξ…ž1(𝑐)=π‘›π‘βˆ ξ€·(β„Žβˆ˜πœ‘)π‘›π‘ξ€Έξ…ž(𝑐)β‰€πœ€.(2.30) Since πœ€ is assumed to be a positive number, we conclude that ∠(β„Žβˆ˜πœ‘)ξ…ž(𝑐)=0, contradicting to ∠(β„Žβˆ˜πœ‘)ξ…ž(𝑐)β‰ 0. Hence, our claim is justified. Since πœ‘ is nonconstant on 𝐾2, πœ‘ is a nonconstant analytic function on connected open set 𝐷. This implies that πœ‘(𝐷) is a connected open set in the complex plane. This implies that β„Ž is constant on connected open set 𝐺1. The continuity of β„Ž on 𝑋1=𝐺1 follows that β„Ž is constant on 𝐺1=𝑋1. This contradicting to β„Ž is nonconstant on 𝑋1. Therefore, πœ‘(𝐾2)βŠ†πΊ1.

Corollary 2.15. Let 𝑋 be a compact plane set such that 𝐺=int(𝑋) is connected and 𝐺=𝑋. Let Ξ©βŠ†πΊ be a bounded connected open set in the complex plane, and let 𝐾=Ξ©. Suppose that 𝐾 is strongly accessible from the interior and 𝑋 has peak boundary with respect to Lip𝐴(𝑋,1). If 𝑇 is a unital compact endomorphism of Lip𝐴(𝑋,𝐾,1), then 𝑇 is induced by a continuous self-map πœ‘ of 𝑋 such that πœ‘ is constant on 𝐾 or πœ‘(𝐾)βŠ†πΊ=int(𝑋). Moreover, πœ‘=𝑇𝑍.

Lemma 2.16. Let 𝐺 and Ξ© be bounded connected open sets in the complex plane with Ξ©βŠ†πΊ, and let 𝑋=𝐺 and 𝐾=Ξ©. Then for each π‘βˆˆπΊβ§΅πΎ there exists a function π‘“π‘βˆˆLip𝐴(𝑋,𝐾,1) such that 𝑓𝑐 is not analytic at 𝑐.

Proof. Let π‘βˆˆπΊβ§΅πΎ. Then there is a positive number π‘Ÿ such that {π‘§βˆˆβ„‚βˆΆ|π‘§βˆ’π‘|β‰€π‘Ÿ}βŠ†πΊβ§΅πΎ.(2.31) We now define the function π‘“π‘βˆΆπ‘‹β†’β„‚ by 𝑓𝑐(⎧βŽͺ⎨βŽͺβŽ©π‘§)=π‘§βˆ’π‘π‘§βˆˆπ‘‹,|π‘§βˆ’π‘|β‰₯π‘Ÿ,(1+π‘Ÿ)(π‘§βˆ’π‘)1+|π‘§βˆ’π‘|π‘§βˆˆπ‘‹,|π‘§βˆ’π‘|<π‘Ÿ.(2.32) It is easily seen that π‘“π‘βˆˆLip𝐴(𝑋,𝐾,1) and 𝑓𝑐 is not analytic at 𝑐.

Definition 2.17. Let 𝑋 and 𝐾 be compact plane sets such that πΎβŠ†π‘‹. We say that 𝐾 has peak 𝐾-boundary with respect to π΅βŠ†π΄(𝑋,𝐾) if for each π‘βˆˆπœ•πΎ there is a function β„Žβˆˆπ΅ such that β„Ž is nonconstant on 𝐾 and β€–β„Žβ€–π‘‹=β„Ž(𝑐)=1.

Example 2.18. Let π‘Ÿβˆˆ(0,1] and 𝐾=π”»π‘Ÿ. Suppose that Lip𝐴(𝔻,𝐾,1)βŠ†π΅βŠ†π΄(𝔻,𝐾). Then 𝐾 has peak 𝐾-boundary with respect to 𝐡.

Proof. We first assume that π‘Ÿ=1. If for each π‘βˆˆπœ•π”» the function β„ŽβˆΆπ”»β†’β„‚ defined by β„Ž(𝑧)=(1/2)(1+𝑐𝑧), then β„Žβˆˆπ΅, β„Ž is nonconstant on 𝐾=𝔻 and β„Ž(𝑐)=1=β€–β„Žβ€–π”».
We now assume that 0<π‘Ÿ<1. For each π‘βˆˆπœ•πΎ, set 𝑧0=(1+π‘Ÿ)𝑐/π‘Ÿ. Then 𝑧0βˆˆβ„‚β§΅π”». Define the function β„ŽβˆΆπ”»β†’β„‚ by ⎧βŽͺβŽͺ⎨βŽͺβŽͺβŽ©βˆ’π‘Ÿβ„Ž(𝑧)=π‘ξ€·π‘§βˆ’π‘§0ξ€Έπ‘§βˆˆ||𝔻,π‘§βˆ’π‘§0||βˆ’π‘Ÿ||β‰₯1,π‘§βˆ’π‘§0||π‘ξ€·π‘§βˆ’π‘§0ξ€Έπ‘§βˆˆ||𝔻,π‘§βˆ’π‘§0||<1.(2.33) It is easily seen that β„ŽβˆˆLip𝐴(𝔻,𝐾,1) and β€–β„Žβ€–π”»=1=β„Ž(𝑐).

Lemma 2.19. Let Ξ© be a connected open set in the complex plane, and let πœ‘ be a one-to-one analytic function on Ξ©. If 𝑓 is a continuous complex-valued function on πœ‘(Ξ©) and π‘“βˆ˜πœ‘ is analytic on Ξ©, then 𝑓 is an analytic function on πœ‘(Ξ©).

Proof. By [8, Theorem 7.5 and Corollary 7.6 in Chapter IV], we deduce that πœ‘(Ξ©) is a connected open set in the complex plane, πœ‘ξ…ž(𝑧)β‰ 0 for all π‘§βˆˆΞ©, and πœ‘βˆ’1βˆΆπœ‘(Ξ©)β†’Ξ© is an analytic function on πœ‘(Ξ©). Since 𝑓=π‘“βˆ˜πœ‘βˆ˜πœ‘βˆ’1 on πœ‘(Ξ©), we conclude that 𝑓 is analytic on πœ‘(Ξ©).

Theorem 2.20. Let 𝑋1 be a compact plane set such that 𝐺1=int(𝑋1) is connected and 𝐺1=𝑋1. Suppose that 𝐾1 is a compact subset of 𝑋1 such that Ξ©1=int(𝐾1) is connected, 𝐾1=Ξ©1, and 𝐾1 has peak 𝐾1-boundary with respect to Lip𝐴(𝑋1,𝐾1,1). Let Ξ©2 be a bounded connected open set in the complex plane, and let 𝐾2=Ξ©2 such that 𝐾2 is strongly accessible from the interior. Suppose that 𝑋2 is a compact plane set such that 𝐾2βŠ†π‘‹2. If π‘‡βˆΆLip𝐴(𝑋1,𝐾1,1)β†’Lip𝐴(𝑋2,𝐾2,1) is a unital compact homomorphism and 𝑇𝑍 is one-to-one on Ξ©2, then 𝑇 is induced by a continuous mapping πœ‘βˆΆπ‘‹2→𝑋1 such that πœ‘=𝑇𝑍 and πœ‘(𝐾2)βŠ†Ξ©1=int(𝐾1).

Proof. Since Lip𝐴(𝑋1,𝐾1,1) and Lip𝐴(𝑋2,𝐾2,1) are, respectively, natural Banach function algebras on 𝑋1 and 𝑋2, π‘‡βˆΆLip𝐴(𝑋1,𝐾1,1)β†’Lip𝐴(𝑋2,𝐾2,1) is a unital homomorphism, 𝑋1 is a compact plane set, and π‘βˆˆLip𝐴(𝑋1,𝐾1,1), we conclude that 𝑇 is induced by πœ‘=𝑇𝑍 and so πœ‘βˆˆLip𝐴(𝑋2,𝐾2,1) by Proposition 1.2.
To prove πœ‘(𝐾2)βŠ†Ξ©1, we first show that πœ‘(Ξ©2)βŠ†πΎ1. Since πœ‘ is a one-to-one analytic mapping on Ξ©2, we conclude that πœ‘(Ξ©2) is an open set in the complex plane. This follows that πœ‘(Ξ©2)βŠ†int(𝑋1)=𝐺1 since πœ‘(Ξ©2)βŠ†π‘‹1. Suppose that πœ‘(Ξ©2)ΜΈβŠ†πΎ1. Then there exists πœ†βˆˆΞ©2 such that πœ‘(πœ†)∈𝐺1⧡𝐾1. By Lemma 2.16, there exists a function π‘“βˆˆLip𝐴(𝑋1,𝐾1,1) such that 𝑓 is not analytic at πœ‘(πœ†). But π‘“βˆ˜πœ‘=π‘‡π‘“βˆˆLip𝐴(𝑋2,𝐾2,1), so that π‘“βˆ˜πœ‘ is analytic on Ξ©2. Since 𝑓 is continuous on πœ‘(Ξ©2) and πœ‘ is a one-to-one analytic function on Ξ©2, we conclude that 𝑓 is analytic on πœ‘(Ξ©2) by Lemma 2.19. This contradicts to the fact 𝑓 is not analytic at πœ‘(πœ†)βˆˆπœ‘(Ξ©2). Therefore, πœ‘(Ξ©2)βŠ†πΎ1 so πœ‘(Ξ©2)βŠ†int(𝐾1)=Ξ©1 since πœ‘(Ξ©2) is an open set in the complex plane. Since πœ‘ is continuous on 𝐾2, πœ‘(Ξ©2)βŠ†Ξ©1, 𝐾2=Ξ©2, and 𝐾1=Ξ©1, we can easily show that πœ‘(𝐾2)βŠ†πΎ1. We now show that πœ‘(𝐾2)βŠ†Ξ©1. Suppose that πœ‘(𝐾2)ΜΈβŠ†Ξ©1. Then there exists π‘βˆˆπœ•πΎ2 such that πœ‘(𝑐)βˆˆπœ•πΎ1. Since 𝐾1 has peak 𝐾1-boundary with respect to Lip𝐴(𝑋1,𝐾1,1), there exists a function β„ŽβˆˆLip𝐴(𝑋1,𝐾1,1) such that β„Ž is not constant on 𝐾1 and β€–β„Žβ€–π‘‹1=β„Ž(πœ‘(𝑐))=1.(2.34) Applying the similar argument used in the proof of Theorem 2.14, we can prove that β„Ž is constant on 𝐾1. This contradiction shows that πœ‘(𝐾2)βŠ†Ξ©1.

Corollary 2.21. Let 𝑋 be a compact plane set such that 𝐺=int(𝑋) is connected and 𝐺=𝑋. Let 𝐾 be a compact subset of 𝑋 such that Ξ©=int(𝐾) is connected and 𝐾=Ξ©. Suppose that 𝐾 has peak 𝐾-boundary with respect to Lip𝐴(𝑋,𝐾,1) and 𝐾 is strongly accessible from the interior. If 𝑇 is a unital compact endomorphism of Lip𝐴(𝑋,𝐾,1) and 𝑇𝑍 is a one-to-one mapping on Ξ©, then 𝑇 is induced by a continuous self-map πœ‘ of 𝑋 such that πœ‘=𝑇𝑍 and πœ‘(𝐾)βŠ†Ξ©=int(𝐾).

3. Spectrum of Unital Compact Endomorphisms

In this section we determine the spectrum of a unital compact endomorphism of a subalgebra of Lip𝐴(𝑋,𝐾,𝛼) which is a natural Banach function algebra with the norm β€–β‹…β€–Lip(𝑋,𝐾,𝛼).

The following result is a modification of [9, Theorem 1.7] for unital compact endomorphisms of natural Banach function algebras.

Theorem 3.1. Let 𝑋 be a compact Hausdorff space and 𝐡 a natural Banach function algebra on 𝑋. If 𝑇 is a unital compact endomorphism of 𝐡 induced by a self-map πœ‘βˆΆπ‘‹β†’π‘‹, then β‹‚βˆžπ‘›=0πœ‘π‘›(𝑋) is finite, and if 𝑋 is connected, β‹‚βˆžπ‘›=0πœ‘π‘›(𝑋) is singleton where πœ‘π‘› is the 𝑛th iterate of πœ‘, that is, πœ‘0(π‘₯)=π‘₯ and πœ‘π‘›(π‘₯)=πœ‘(πœ‘π‘›βˆ’1(π‘₯)). If β‹‚βˆžπ‘›=0πœ‘π‘›(𝑋)={π‘₯0}, then π‘₯0 is a fixed point for πœ‘. In fact, if ⋂𝐹=βˆžπ‘›=0πœ‘π‘›(𝑋), then πœ‘(𝐹)=𝐹.

Theorem 3.2. Suppose that 𝑋 is a compact plane set with int(𝑋)β‰ βˆ…, Ξ© is a connected open set in the complex plane with Ξ©βŠ†int(𝑋), and 𝐾=Ξ©. Let 𝐡 be a subalgebra of 𝐴(𝑋,𝐾) containing the coordinate function 𝑍 which is a natural Banach function algebra on 𝑋 with an algebra norm ‖⋅‖𝐡. Let 𝑇 be a unital compact endomorphism of 𝐡 induced by a self-map πœ‘ of 𝑋. If πœ‘(𝑋)βŠ†int(𝐾) and 𝑧0 is a fixed point of πœ‘, then πœ‘πœŽ(𝑇)={0,1}βˆͺξ€½ξ€·ξ…žξ€·π‘§0𝑛.βˆΆπ‘›βˆˆβ„•(3.1)

Proof. Clearly 0 and also 1∈𝜎(𝑇) since 𝑇(1𝑋)=1𝑋. If πœ‘ is constant then the proof is complete. Let πœ†βˆˆπœŽ(𝑇)⧡{0,1}. The compactness of 𝑇 implies that there exists π‘“βˆˆπ΅β§΅{0} such that 𝑇𝑓=π‘“βˆ˜πœ‘=πœ†π‘“. Since πœ‘(𝑧0)=𝑧0∈int(𝐾), 𝑓(𝑧0)=0. We claim that 𝑓(𝑗)(𝑧0)β‰ 0 for some π‘—βˆˆβ„•. If 𝑓(𝑛)(𝑧0)=0 for all π‘›βˆˆβ„•, then 𝑓=0 on an open disc with center 𝑧0 and so on Ξ©. By maximum modules principle, it follows that 𝑓=0 on 𝑋 since πœ‘(𝑋)βŠ†Ξ©, πœ†βˆˆβ„‚β§΅{0}, and πœ†π‘“(𝑧)=𝑓(πœ‘(𝑧)) for all π‘§βˆˆπ‘‹. This contradicts to 𝑓≠0. Hence, our claim is justified. Let π‘š=min{π‘›βˆˆβ„•βˆΆπ‘“(𝑛)(𝑧0)β‰ 0}. Then 𝑓(π‘˜)(𝑧0)=0 for all π‘˜βˆˆ{0,…,π‘›βˆ’1} and 𝑓(π‘š)(𝑧0)β‰ 0. By π‘š times differentiation of π‘“βˆ˜πœ‘=πœ†π‘“, we have (πœ‘ξ…ž(𝑧0))π‘šπ‘“(π‘š)(πœ‘(𝑧0))=πœ†π‘“(π‘š)(𝑧0), and therefore πœ†=(π‘“ξ…ž(𝑧0))π‘š. Then 𝜎(𝑇)⧡{0,1}βŠ†{(πœ‘ξ…ž(𝑧0))π‘›βˆΆπ‘›βˆˆβ„•}.
Conversely, first we show that, if πœ†βˆˆπœŽ(𝑇) with |πœ†|=1, then πœ†=1. Let πœ†βˆˆπœŽ(𝑇) and |πœ†|=1. The compactness of 𝑇 implies that there exists π‘”βˆˆπ΅β§΅{0} such that π‘”βˆ˜πœ‘=πœ†π‘”. It follows that |π‘”βˆ˜πœ‘|=|𝑔|. Since πœ‘(𝐾)βŠ†int(𝐾)=Ξ© and 𝑔 is analytic on the connected open set Ξ©, we conclude that 𝑔 is constant on Ξ© by maximum modules principle. Since πœ‘(𝑋)βŠ†Ξ©, π‘”βˆ˜πœ‘=πœ†π‘”, and πœ†βˆˆβ„‚β§΅{0}, we deduce that 𝑔 is constant on 𝑋. Applying again π‘”βˆ˜πœ‘=πœ†π‘” implies that πœ†=1 since π‘”βˆˆπ΅β§΅{0}.
We now claim that πœ‘ξ…ž(𝑧0)∈𝜎(𝑇). If πœ‘ξ…ž(𝑧0)βˆ‰πœŽ(𝑇), then there exists a nonzero linear operator π‘†βˆΆπ΅β†’π΅ such that ξ€·π‘‡βˆ’πœ‘ξ…žξ€·π‘§0𝐼𝑆=𝐼.(3.2) Since π‘βˆ’π‘§01π‘‹βˆˆπ΅, β„Ž=𝑆(π‘βˆ’π‘§01𝑋)∈𝐡 and so β„Žβˆ˜πœ‘βˆ’πœ‘ξ…žξ€·π‘§0ξ€Έβ„Ž=π‘βˆ’π‘§01𝑋,(3.3) by (3.2). By differentiation at 𝑧0, we have 0=β„Žξ…žξ€·πœ‘ξ€·π‘§0πœ‘ξ€Έξ€Έξ…žξ€·π‘§0ξ€Έβˆ’πœ‘ξ…žξ€·π‘§0ξ€Έβ„Žξ…žξ€·π‘§0ξ€Έ=1,(3.4) this is a contradiction. Hence, our claim is justified.
We now show that (πœ‘ξ…ž(𝑧0))π‘›βˆˆπœŽ(𝑇) for all π‘›βˆˆβ„•. If πœ‘ξ…ž(𝑧0)=0 or |πœ‘ξ…ž(𝑧0)|=1, the proof is complete. Suppose that πœ‘ξ…ž(𝑧0)β‰ 0 and |πœ‘ξ…ž(𝑧0)|β‰ 1. If (πœ‘ξ…ž(𝑧0))π‘—βˆ‰πœŽ(𝑇) for some π‘—βˆˆβ„• with 𝑗>1, then there exists a nonzero linear operator π‘†π‘—βˆΆπ΅β†’π΅ such that ξ‚€ξ€·πœ‘π‘‡βˆ’ξ…žξ€·π‘§0𝑗𝐼𝑆𝑗=𝐼.(3.5) Since (π‘βˆ’π‘§01𝑋)π‘—βˆˆπ΅, β„Žπ‘—=𝑆𝑗(π‘βˆ’π‘§01𝑋)π‘—βˆˆπ΅ and so β„Žπ‘—ξ€·πœ‘βˆ˜πœ‘βˆ’ξ…žξ€·π‘§0ξ€Έξ€Έπ‘—β„Žπ‘—=ξ€·π‘βˆ’π‘§01𝑋𝑗,(3.6) by (3.5). By π‘—βˆ’1 times differentiation at 𝑧0, we have β„Žπ‘—ξ€·π‘§0ξ€Έ=β„Žξ…žπ‘—ξ€·π‘§0ξ€Έ=β‹―=β„Žπ‘—(π‘—βˆ’1)𝑧0ξ€Έ=0,(3.7) and by 𝑗 times differentiation at 𝑧0, we have ξ€·πœ‘0=ξ…žξ€·π‘§0ξ€Έ