Spectra of weighted composition operators with automorphic symbols

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Spectra of weighted composition operators with automorphic symbols Olli Hyvärinen 1 , Mikael Lindström ∗ , Ilmari Nieminen 2 , Erno Saukko 3 Department of Mathematical Sciences, P.O. Box 3000, FI-90014 University of Oulu, Finland Received 11 December 2012; accepted 4 June 2013

Communicated by D. Voiculescu

Abstract Let ϕ be an automorphism of the open unit disc D. For such ϕ, we investigate the spectra of invertible weighted composition operators uCϕ acting on a wide class of analytic function spaces; this class contains, p for example, Hardy spaces H p (D), weighted Bergman spaces Aα (D), and weighted Banach spaces of ∞ H -type. We present new techniques for deducing the spectrum and for calculating the spectral radius of p uCϕ . We also characterize the Fredholmness of weighted composition operators on H p (D) and Aα (D). © 2013 Elsevier Inc. All rights reserved. Keywords: Weighted composition operator; Spectrum; Automorphism; Hardy spaces; Weighted Bergman spaces; Weighted Banach spaces of H ∞ -type; Fredholm operator

1. Introduction We denote by H (D) the family of all analytic functions on the open unit disc D of the complex plane C. Let ϕ : D → D be an analytic map and u ∈ H (D). These maps induce via composition and multiplication a linear weighted composition operator uCϕ which is defined on H (D) by * Corresponding author.

E-mail addresses: [email protected] (O. Hyvärinen), [email protected] (M. Lindström), [email protected] (I. Nieminen), [email protected] (E. Saukko). 1 The research of the first author has been supported by a grant from the Emil Aaltonen Foundation. 2 The research of the third author has been supported by a grant from the Väisälä Foundation. 3 The research of the fourth author has been supported by the Finnish National Doctoral Programme in Mathematics and its Applications. 0022-1236/$ – see front matter © 2013 Elsevier Inc. All rights reserved. http://dx.doi.org/10.1016/j.jfa.2013.06.003

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(uCϕ )f = u(f ◦ ϕ). There are two particularly interesting special cases of such operators: on one hand, taking u = 1 gives the composition operator Cϕ , and on the other, putting ϕ = id, the identity function of D, gives the multiplication operator Mu . Weighted composition operators are fundamental objects of study in analysis that arise naturally in many situations. A classical result due to Forelli [9] states that all surjective isometries of the Hardy space H p (D), 1 < p < ∞, p = 2, are weighted composition operators. Kolaski [20] gave a characterization of all surjective isometries of the standard weighted Bergman p space Aα (D) similar to that of Forelli’s. Moreover, weighted composition operators have arisen in the study of commutants of multiplication operators, and they play a role in the theory of dynamical systems as well, just to mention a few examples. Good general references of composition operators on classical spaces of analytic functions on the unit disc are the books written by Cowen and MacCluer [6], and Shapiro [26]. The problem of relating operator theoretic properties (e.g., compactness, and spectrum) of uCϕ to function theoretic properties of u and ϕ has been a subject of great interest for quite some time; see, e.g., [1,2,4,7,19]. For an automorphic symbol ϕ, Kamowitz [18] determined the spectrum of uCϕ on the disc algebra A(D) and, more recently, Gunatillake [14] carried out a similar project for invertible weighted composition operators on the Hardy–Hilbert space H 2 (D). For a non-automorphic symbol ϕ with a fixed point in D, Aron and Lindström [1] completely described the spectrum of a weighted composition operator uCϕ acting on the weighted Banach spaces of H ∞ -type. In the case of a composition operator Cϕ acting on H 2 (D), Cowen proved in the remarkable paper [5] (see also [6]) several deep and interesting results concerning the spectrum and the essential spectrum of Cϕ . In this paper, by using ideas of Kamowitz and Gunatillake as a starting point, we compute the spectrum of the invertible weighted composition operator uCϕ for an automorphic symbol ϕ on a wide class of analytic function spaces; this class contains, for example, Hardy spaces, weighted Bergman spaces, and weighted Banach spaces of H ∞ -type. The analysis of the spectral behavior of uCϕ is typically case based, with the cases depending upon the type of the symbol ϕ, that is, elliptic, parabolic or hyperbolic automorphism. As we will see, parabolic and hyperbolic cases are the most interesting ones having the Denjoy–Wolff point of the automorphism on the boundary of D. In fact, for these two cases we present new techniques for determining the spectral radius and the spectrum of uCϕ . One aim of our paper is to complement Gunatillake’s work by generalizing his results, since at the end of his paper [14, p. 860] he indicates that he was not able to determine the spectrum of uCϕ on weighted Bergman spaces if ϕ is either a parabolic or a hyperbolic automorphism of D. However, our results are not merely generalizations but also improvements of his work, since we get better results for a hyperbolic ϕ even on the space H 2 (D). Our results related to this will be given in Section 4. In Section 3, we characterize the invertibility and Fredholmness of uCϕ on the most important analytic function spaces for a general analytic selfmap ϕ of D and arbitrary u ∈ H (D). An analogous characterization for composition operators on a variety of analytic function spaces have been carried out in [11,10,15,23]. Very recently, Bourdon [3] has obtained more general results on invertibility of weighted composition operators acting on sets of analytic functions without norm or linear structure. Altogether, we show in this paper that it is possible to design a unified approach to determine the spectra of weighted composition operators on a huge class of analytic function spaces. The main results of this paper are stated in Theorems 4.3 and 4.9, which concern parabolic and hyperbolic symbols ϕ, respectively. A summary of our investigation is given at the end of the paper, see Corollary 5.1.

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2. Preliminaries In this section we introduce the general class A of analytic function spaces on which we study weighted composition operators. We will give examples of spaces which are contained (and examples of those which are not contained) in this class. However, let us first fix notation. 2.1. Some matters of notation and terminology Let ϕ be an analytic selfmap of D. For all n ∈ N we denote the n-th iterate of ϕ by ϕn , that is, ϕn := ϕ ◦ · · · ◦ ϕ

(composed n times);

for n = 0 we set ϕ0 := id, the identity function of D. If ϕ is an automorphism, then the same is true for ϕ −1 and we denote ϕ−n := (ϕ −1 )n . It is very easy to check that for n  1 we have     (uCϕ )n f (z) = u(z) · · · u ϕn−1 (z) f ϕn (z) for all f ∈ H (D) and z ∈ D. Hence (uCϕ )n = u(n) Cϕn , where u(n) :=

n−1 

u ◦ ϕm

m=0

belongs to H (D). For n = 0 we shall find it convenient to set u(0) = 1. For z ∈ D, let ϕz denote the Möbius transformation of D, ϕz (w) :=

z−w , 1 − zw

for all w ∈ D.

The pseudo-hyperbolic distance between the points z and w in D is defined by (z, w) = |ϕz (w)|. Recall that the pseudo-hyperbolic distance is invariant under automorphisms of D, meaning that (ϕ(z), ϕ(w)) = (z, w) whenever ϕ is an automorphism of D. It is well-known that automorphisms of D naturally fall into three distinct classes which can be characterized by the configuration of their fixed points; see [6, Chapter 7]. Recall that a nontrivial automorphism ϕ of D (i.e., ϕ is not the identity function of D) is called • elliptic if ϕ has a unique fixed point in D; • parabolic if ϕ has a unique fixed point in ∂D; and • hyperbolic if ϕ has two distinct fixed points in ∂D. Spectral properties of Cϕ — and hence also those of uCϕ — depend to a great extent on the fixed point configuration of ϕ. We shall consider the three cases separately in Section 4. Concerning the boundary fixed points of ϕ, one result of particular importance is the celebrated Denjoy–Wolff Theorem (see [6, Theorem 2.51]). If ϕ is either a parabolic or a hyperbolic automorphism of D, this theorem guarantees that there is a (unique) fixed point a ∈ ∂D such that lim ϕn (z) = a

n→∞

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uniformly on compact subsets of D; the point a is called the Denjoy–Wolff point of ϕ. Moreover, • if ϕ is parabolic, then ϕ  (a) = 1; and • if ϕ is hyperbolic and its other fixed point is b ∈ ∂D, then 0 < ϕ  (a) < 1 and ϕ  (b) = 1/ϕ  (a). Here ϕ  denotes the angular derivative of ϕ, which coincides with the usual derivative whenever ϕ is an automorphism. Let us state, for a later reference, one useful formula which relates the iterates of ϕ to its angular derivative at a, the Denjoy–Wolff point of ϕ:  1/n  lim 1 − ϕn (0) = ϕ  (a),

n→∞

(2.1)

where ϕ is either a parabolic or a hyperbolic automorphism of D. For a proof, see [6, pp. 251–252]. Let X be a Banach space, and denote the space of all bounded linear operators on X by L(X). Recall that an operator T ∈ L(X) is said to be Fredholm if both the dimension of its kernel and the codimension of its range are finite. This occurs if and only if both Ker T and Ker T ∗ are finite dimensional. Equivalently, T is Fredholm if and only if T is invertible modulo compact operators, that is, there is a bounded operator S ∈ L(X) such that both T S − I and ST − I are compact on X. It is well-known that an operator T is Fredholm if and only if its adjoint T ∗ is likewise Fredholm; see [24]. Let us recall definitions for a couple of convenient spaces which are used throughout the paper. The space H ∞ (D) is the Banach space of all bounded analytic functions on D with the norm f ∞ = sup{|f (z)|; z ∈ D}. The disc algebra A(D) is a subspace of H ∞ (D) consisting of those analytic functions which are continuous on the closed unit disc D. Throughout the paper we shall use the following convenient notation. For two non-negative quantities A(t) and B(t) we denote A(t)  B(t) if there exists some positive constant c, not depending on t , so that A(t)  cB(t) for all t . If A(t)  B(t)  A(t), then we denote A(t) ≈ B(t). 2.2. Framework Let A be a Banach space of analytic functions on the unit disc which contains constant functions; let · A denote its norm. As usually, we denote the Banach dual of A by A∗ . The evaluation functional at z ∈ D, denoted by δz : A → C, is defined by δz (f ) = f (z) for all f ∈ A. Let us state a couple of natural conditions (and their weaker variants, distinguished by prime) for A-type spaces. (C1) There is a positive constant s such that for each f ∈ A and each z ∈ D we have |f (z)|  f A (1 − |z|2 )−s and for any z ∈ D there is some fz ∈ A with fz A  1 such that fz (z)(1 − |z|2 )s = 1. (C1) There is a positive constant s such that for each f ∈ A and each z ∈ D we have |f (z)|  f A (1 − |z|2 )−s and also that δz A∗ → ∞ as |z| → 1. (C2) There is a positive constant s such that Cϕ  (1 − |ϕ(0)|2 )−s whenever ϕ is an automorphism of D. (C3) For each u ∈ H ∞ (D) we have Mu  u ∞ . (C4) Polynomials are dense in A.

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2.3. Remarks Let us point out some immediate observations about the conditions (C1)–(C4). (i) It follows from (C1) that there are positive constants α and s such that 1 α  δz A∗  . (1 − |z|2 )s (1 − |z|2 )s In particular, δz A∗ → ∞ as |z| → 1, and so the condition (C1) is, indeed, weaker than (C1). (ii) Since A contains constant functions, it follows from (C3) that H ∞ (D) ⊂ A. However, the (D) for some positive s. Thus A-type spaces are condition (C1) guarantees that A ⊆ Hv∞ s (D). large spaces in the sense that H ∞ (D) ⊂ A ⊆ Hv∞ s (iii) If δ0 : A → C is bounded and condition (C2) is valid, then for each f ∈ A and each z ∈ D we have |f (z)|  f A (1 − |z|2 )−s . Indeed, let ϕz (w) = (z − w)/(1 − zw), w ∈ D. Then         f (z) = f ◦ ϕz (0)  Cϕ (f )  f A 1 − |z|2 −s . z A 2.4. Examples p

Hardy spaces H p (D) and weighted Bergman spaces Aα (D) are the most important examples of A spaces. p

(a) The standard weighted Bergman space Aα (D), p  1, α > −1, is the set of all analytic functions on D such that   p  α p f p = f (z) 1 − |z|2 dA(z) < ∞, Aα

D

where dA(z) is the normalized area measure on D. All conditions (C1), (C2), (C3) and (C4) are valid with s = (α + 2)/p (see [16] and [27]). (b) The Hardy spaces H p (D), 1  p < ∞, are defined by  H p (D) = f ∈ H (D);

p f H p

1 = lim r→1 2π

2π   iθ p f re  dθ < ∞ , 0

and the conditions (C1), (C2), (C3) and (C4) are fulfilled with s = 1/p (see [8] and [27]). (D), 0 < p < ∞, are defined (c) The standard weighted Banach spaces of analytic functions Hv∞ p by

   ∞ = sup vp (z)f (z) < ∞ , Hv∞ (D) = f ∈ H (D); f H vp p z∈D

where vp (z) := (1 − |z|2 )p is the standard weight. For w ∈ D, the function fw (z) = ((1 − |w|2 )/(1 − wz)2 )p belongs to H ∞ (D), which yields δz (Hv∞p )∗ = 1/(1 − |z|2 )p . Thus

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condition (C1) holds with s = p. Moreover, it is easy to see that condition (C3) is valid for Hv∞ (D). Using that p (1 − |z|)s  (1 − |ϕ(z)|)s



1 + |ϕ(0)| 1 − |ϕ(0)|

s > 0,

one obtains f ◦ ϕ Hv∞s  (1 − |ϕ(0)|2 )−s f Hv∞s , and so condition (C2) also holds. However, the condition (C4) fails to hold for Hv∞ (D). p 0 ∞ The closed subspace Hvp (D) of Hvp (D) is defined by

   f (z) = 0 . Hv0p (D) = f ∈ Hv∞ ; lim v (z) p p |z|→1

The space Hv0p (D) clearly fulfills the conditions (C1), (C2), (C3), (C4) and Hv∞ (D) is its p bidual space. (d) The weighted Dirichlet spaces Dα2 (D), α  0, consist of all analytic functions f ∈ H (D) satisfying  2 f 2D 2 = f (0) +



α

  2   f (z) 1 − |z| α dA(z) < ∞.

D

For α > 1 the space Dα2 (D) coincides with A2α−2 (D), and for α = 1 we obtain D12 (D) = H 2 (D). The conditions (C1) , (C2) and (C4) are fulfilled with s = α/2 whenever α > 0, but Dα2 (D) fails to satisfy the condition (C3). (e) The classical Bloch spaces are defined by

      B(D) = f ∈ H (D); f B = f (0) + sup 1 − |z| f  (z) < ∞ z∈D

and

    B0 (D) = f ∈ B(D); lim 1 − |z| f  (z) = 0 . |z|→1

The space B(D) has a predual and B(D) is the bidual of B0 (D). For any f ∈ B(D) it holds that |f (z)|  f B log(2/(1 − |z|2 )) (see [27]). To see that condition (C1) is satisfied by both spaces, consider for each w ∈ ∂D, the function fw (z) = log(1 − wz) that is in B(D), actually in BMOA. Then δz B∗ ≈ log(2/(1 − |z|2 )). Moreover, the unit ball of B0 (D) is dense (with respect to the compact-open topology) in the unit ball of B(D), as can be seen by approximating any function by its Taylor series. Therefore (C1) is also valid for B0 (D). Finally, condition (C4) holds for B0 (D), but it fails to hold for B(D). (f) The disc algebra A(D) is a simple example of a space which fails to satisfy conditions (C1), (C1) and (C3). The space S p (D) = {f ∈ H (D); f  ∈ H p (D)}, where p  1, also fails to satisfy the condition (C3). Indeed, by [8, Theorem 3.11], we know that S p (D) ⊂ A(D), and so it is strictly contained in H ∞ (D).

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3. Fredholm weighted composition operators In this section we obtain necessary and sufficient conditions in terms of u and ϕ for the weighted composition operator uCϕ to be invertible on A-type spaces. This shall be done by first characterizing the Fredholmness of uCϕ , and then using these arguments to yield the desired result; hence the title for this section. Along the way we show that if the Banach space A satisfies only the two conditions (C1) and (C4) and uCϕ : A → A is assumed to be Fredholm, then ϕ is necessarily an automorphism. This class of Banach spaces of analytic functions is huge, and it contains many important analytic p function spaces. In particular, the spaces Aα (D), H p (D), Dα2 (D), Hv0p (D), and B0 (D) belong to this class. Lemma 3.1. Suppose A satisfies the conditions (C1) and (C4). Then the following holds. (a) The map z → δz from D into A∗ is continuous. (b) For all f ∈ A we have lim

|z|→1

δz (f ) = 0. δz A∗

That is, δz / δz A∗ weak∗ -converges to zero as |z| → 1. Proof. Both of these results have been proven in [11], but for the sake of completeness we provide some arguments. . It follows from the Closed Graph Theorem that one (a) By condition (C1) , we obtain A ⊂ Hv∞ s can find a β > 0 such that f Hv∞s  β f A for all f ∈ A. Now we use the following result from [22] (see also [16]): there is a constant βs < ∞ (depending only on s) such that     f (z) − f (w)  βs f H ∞ max (z, w) , (z, w) vs (1 − |z|)s (1 − |w|)s for all f ∈ Hv∞ and all z, w ∈ D, where (z, w) denotes the pseudo-hyperbolic distance s between z and w. Thus   (z, w) (z, w) δz − δw A∗  βs β max , (1 − |z|)s (1 − |w|)s for all z, w ∈ D. This finishes the proof. (b) Condition (C1) guarantees that for any polynomial P , |δz (P )| sup{|P (w)|; w ∈ D} |z|→1  −−−→ 0. δz A∗ δz A∗ Then the claim follows by using condition (C4).

2

Lemma 3.2. Let A be a Banach space such that for any z ∈ D the point-evaluation functional δz : A → C is bounded. If uCϕ : A → A is Fredholm, then u has at most finitely many zeros in D.

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Proof. Let a ∈ D and f ∈ A be arbitrary. Since 

   f, (uCϕ )∗ δa = u(a)f ϕ(a) ,

we conclude that δa ∈ Ker(uCϕ )∗ whenever u(a) = 0. To finish the proof it suffices to notice that the family of functionals, {δz ; u(z) = 0}, is linearly independent and is contained in Ker(uCϕ )∗ , which is finite dimensional by assumption. 2 The proof of our next lemma is a modification of [6, Lemma 3.2.6]. Lemma 3.3. Let A be a Banach space such that for any z ∈ D the point-evaluation functional δz : A → C is bounded. If uCϕ : A → A is Fredholm, then ϕ is one-to-one. Proof. Suppose not; then we can find distinct points, say a and b, in D such that ϕ(a) = c = ϕ(b). Pick disjoint open discs Ua and Ub which contain a and b, respectively, and notice that c ∈ ϕ(Ua ) ∩ ϕ(Ub ). Let us first consider the case of a non-constant ϕ. It follows from the Open Mapping Theorem that U := ϕ(Ua ) ∩ ϕ(Ub ) is a non-empty open set, and so we can pick a sequence (cn )∞ n=1 of distinct points in U . But then, by the construction of U , we can also find sequences (an )∞ n=1 and (bn )∞ n=1 of distinct points in Ua and Ub , respectively, such that ϕ(an ) = cn = ϕ(bn ) for each n. Clearly {an } ∩ {bn } = ∅ (since Ua ∩ Ub = ∅), and by Lemma 3.2 we know that there exists an N ∈ N such that u(an ) = 0 and u(bn ) = 0 whenever n  N . It follows from the above facts and the proof of Lemma 3.2 that for n  N and f ∈ A,     0 = f ϕ(an ) − f ϕ(bn ) = u(an )

    1 1 f ϕ(an ) − u(bn ) f ϕ(bn ) u(an ) u(bn ) 

 δbn δan = f , (uCϕ )∗ − . u(an ) u(bn )

∗ Thus { u(aann ) − u(bbnn ) }∞ n=N is contained in Ker(uCϕ ) . The desired contradiction follows by noting δ

δ

∗ that the set { u(aann ) − u(bbnn ) }∞ n=N is linearly independent, and hence Ker(uCϕ ) cannot be finite dimensional. So ϕ is one-to-one, if ϕ is non-constant. If ϕ was constant, say, ϕ(z) = c for all z ∈ D, then uCϕ = uδc and therefore δ

δ

  (z − c)k ; k = 1, 2, . . . ⊂ Ker(uCϕ ). Then again Ker(uCϕ ) would not be finite dimensional, and so we are done.

2

Lemma 3.4. Let A be a Banach space satisfying the conditions (C1) and (C4). If uCϕ : A → A is Fredholm, then ϕ is onto. Proof. Let us assume, towards a contradiction, that ϕ is not onto. Then we can find w0 ∈ ∂ϕ(D) ∩ D and such a sequence (zn ) of points in D that ϕ(zn ) → w0 as n → ∞. We also get that |zn | → 1 when n → ∞. Put ln := δzn / δzn A∗ ∈ A∗ . By Lemma 3.1(b), ln → 0 weakly∗ in A∗ when n → ∞ and every ln has norm one. Now,

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(uCϕ )∗ ln =

9

u(zn )δϕ(zn ) . δzn A∗

By Lemma 3.1(a), δϕ(zn ) → δw0 in A∗ as n → ∞. Since u ∈ A we conclude by Lemma 3.1(b) that (uCϕ )∗ ln A∗ → 0 when n → ∞. On the other hand, since uCϕ is Fredholm, there are operators S and K on A, with K compact, such that (uCϕ )S = I − K. Hence S ∗ (uCϕ )∗ = I − K ∗ , and so  ∗    S (uCϕ )∗ ln − ln 

A∗

→ 0,

when n → ∞. Thus we get (S ∗ (uCϕ )∗ )ln A∗ → 1 when n → ∞. As this is a contradiction, we conclude that ϕ is onto. 2 Remark 3.5. By analyzing the proofs of Lemmas 3.1, 3.3 and 3.4 we notice that if the Banach space A satisfies only the two conditions (C1) and (C4) and uCϕ : A → A is assumed to be Fredholm, then ϕ is necessarily an automorphism. Theorem 3.6. Let A satisfy the conditions (C1) , (C2) and (C4). The weighted composition operator uCϕ is Fredholm on A if and only if Mu is Fredholm and ϕ is an automorphism of the unit disc. Proof. Assume first that uCϕ is Fredholm. By the preceding two lemmas, ϕ is an automorphism. Then Cϕ is bounded by (C2), and it is easy to check that Cϕ−1 = Cϕ −1 . Therefore, the operator Mu = (uCϕ )Cϕ−1 ∈ L(A). Furthermore, Ker Mu = Ker(uCϕ ) and ∗  Ker Mu∗ = Ker (uCϕ )Cϕ−1 = Ker(uCϕ )∗ . Thus Mu is Fredholm. Suppose that Mu is a Fredholm operator and ϕ is an automorphism. Since uCϕ = Mu Cϕ ∈ L(A), Ker Mu = Ker uCϕ and Ker Mu∗ = Ker(uCϕ )∗ , the claim follows. 2 Theorem 3.6 gives us a very nice characterization for invertible weighted composition operators, similar to that obtained by Gunatillake [14, Corollary 2.0.1]. Also, very recently, Bourdon [3] obtained a characterization of invertible weighted composition operators under very general conditions (for example, his result applies to the spaces S p (D) of functions whose derivative is in H p (D), to any weighted Hardy space H 2 (β), to the Lipschitz spaces Lipα (D)). Corollary 3.7. Let A satisfy the conditions (C1), (C2) and (C4). The operator uCϕ is invertible on A if and only if u is bounded and bounded away from zero on the unit disc and ϕ is an automorphism of the unit disc. The inverse operator of uCϕ : A → A is also a weighted composition operator and it has the form (uCϕ )−1 =

1 C −1 . u ◦ ϕ −1 ϕ

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Proof. If uCϕ is invertible and therefore Fredholm, then by the above result ϕ is an automorphism of the unit disc. Moreover, it follows that (uCϕ )∗ is bounded from below. Thus we find a β > 0 such that |u(z)| δϕ(z) A∗ β δz A∗

for all z ∈ D,

from which we conclude, by (C1), that for any z ∈ D, s   u(z)  (1 − |ϕ(z)|)  (1 − |z|)s



1 − |ϕ(0)| 1 + |ϕ(0)|

s > 0.

We still need to show that u is necessarily bounded. It is clear from the proof of Theorem 3.6 that the multiplication operator Mu is bounded on A. Condition (C1) guarantees that for each z ∈ D there is a function fz ∈ A with fz A  1 such that fz (z)(1 − |z|2 )s = 1, and also that |u(z)fz (z)|  ufz A (1 − |z|2 )−s . This implies |u(z)|  Mu , and hence u is bounded on D, as claimed. Conversely, assume that u is bounded and bounded away from zero on the unit disc and that ϕ is an automorphism of the unit disc. Put v = u◦ϕ1 −1 and notice that the same is true for v and ϕ −1 . Therefore, vCϕ −1 : A → A is bounded and it is straightforward to check that vCϕ −1 = (uCϕ )−1 . 2

Remark 3.8. In the rest of the paper we only consider weighted composition operators uCϕ : A → A which are induced by an automorphic symbol ϕ. If A satisfies conditions (C1) and (C2), then such an operator uCϕ is invertible if and only if u is both bounded and bounded away from zero on D, regardless of the fact whether A satisfies the condition (C4). 4. Spectra of weighted composition operators In this section we determine the spectrum of invertible uCϕ for an automorphic symbol ϕ. On one hand, our aim is to provide general proofs for the obtained results instead of relying on space-specific techniques or repeating the same arguments on different spaces; on the other hand, we aim to optimize the proofs so that it becomes apparent what is really needed and what is irrelevant. For this purpose Kamowitz’s excellent paper [18] (which seems to have received surprisingly little attention) serves as a good starting point. Indeed, some of the fundamental ideas of the upcoming theorems stem from the work of Kamowitz. We split our discussion into three cases: first we consider parabolic, then hyperbolic, and finally elliptic ϕ. For the main results of this paper, we need to assume that the space A satisfies the conditions (C1), (C2) and (C3). The most important spaces satisfying these conditions are p (D). Hardy spaces H p (D), weighted Bergman spaces Aα (D), and weighted Banach spaces Hv∞ p However, before treating the parabolic case we establish a lower bound for the spectral radius of uCϕ which holds for any automorphism ϕ of D. Lemma 4.1. Suppose A is a Banach space which satisfies the condition (C1). Let u ∈ A(D) and let ϕ be an automorphism of D with a fixed point a ∈ D. If uCϕ is bounded on A, then

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r(uCϕ ) 

11

|u(a)| . |ϕ  (a)|s

If ϕ is either elliptic or parabolic, then |ϕ  (a)| = 1, and hence r(uCϕ )  |u(a)|. Proof. Suppose first that a ∈ D (i.e., ϕ is elliptic). Clearly, ϕn (a) = a for all n and u(n) Cϕn 

  n |u(n) (a)| δϕn (a) A∗  = u(n) (a) = u(a) , δa A∗

so r(uCϕ )  |u(a)|. Let us now consider the case a ∈ ∂D. Let 0 < ε < 1 and notice that by (C1), u(n) Cϕn 



s  |u(n) (εa)| δϕn (εa) A∗  1 − |εa|  u(n) (εa) . δεa A∗ 1 − |ϕn (εa)|

Since |u(n) (εa)| → |u(a)|n as ε → 1, and lim

ε→1

1 − |εa| 1 1 , = = 1 − |ϕn (εa)| |ϕn (a)| |ϕ  (a)|n

we conclude that r(uCϕ ) = limn u(n) Cϕn 1/n  |u(a)/ϕ  (a)s |.

2

4.1. The parabolic case Let us now consider the case where ϕ is a parabolic automorphism of D. We start by determining the spectral radius of uCϕ . Lemma 4.2. Suppose A is a Banach space which satisfies the conditions (C1), (C2) and (C3). Let uCϕ be invertible on A where ϕ is a parabolic automorphism of D, and let a ∈ ∂D be the unique fixed point of ϕ. If u ∈ A(D), then   1/n lim u(n) ∞ = u(a)

n→∞

  and r(uCϕ ) = u(a).

Proof. First notice that for |z| = 1 the mapping arg(z) → arg(ϕ(z)) is injective and so monotonic. Let ε > 0. Since |u(a)| > 0 and u is continuous at a, it follows that there is an open arc Va ⊂ ∂D which contains a and satisfies |u(z)| < (1 + ε)|u(a)| whenever z ∈ Va . Since a ∈ Va , there exists an m ∈ N such that ϕm (z) ∈ Va for all z ∈ ∂D \ Va . Then for any z ∈ ∂D and for any n ∈ N, at most m elements from {z, ϕ(z), . . . , ϕn (z)} are not contained in Va . For any n > m we have   n−m     u(n) ∞ = maxu(n) (z)  u m , ∞ (1 + ε) u(a) |z|=1

1/n

from which it follows that limn u(n) ∞  (1 + ε)|u(a)|. This holds for any ε > 0; hence 1/n limn u(n) ∞  |u(a)|. But as a ∈ ∂D, it is evident that u(n) ∞  |u(a)|n , and thus 1/n u(n) ∞ → |u(a)| as n → ∞.

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Let us then establish the assertion concerning the spectral radius of uCϕ . We have, by Lemma 4.1, that r(uCϕ )  |u(a)|, and so it suffices to show the reverse inequality. Note that   (uCϕ )n  = Mu Cϕ  Mu Cϕ  u(n) ∞ Cϕ , n n n (n) (n) and Cϕn  (1 − |ϕn (0)|)−s by conditions (C3) and (C2), respectively. Since −s/n  −s   → ϕ  (a) = 1 1 − ϕn (0) as n → ∞ by formula (2.1), one concludes that  1/n   r(uCϕ ) = lim (uCϕ )n   u(a), n→∞

and so we are done.

2

Now we can describe the spectrum of each invertible uCϕ on A when u ∈ A(D) and ϕ is a parabolic automorphism of D. Our method of proof is more general than the one presented by Gunatillake [14, Theorem 3.3.1] when proving the corresponding result for the Hardy space H 2 (D). His proof relies on inner functions on H 2 (D) whereas our proof is based on interpolation sequences for H ∞ (D). Recall that the model of iteration (see [6, Section 2.4]) for a parabolic automorphism ϕ belongs to the halfplane/translation case (see [6, p. 71]). If z0 ∈ D is fixed, then the sequence of iterates ϕn (z0 ), n = 0, 1, 2, . . . , is an interpolating sequence for H ∞ (D); see [5, Proposition 4.9]. Our first main result goes as follows: Theorem 4.3. Suppose A is a Banach space which satisfies the conditions (C1), (C2) and (C3). Let uCϕ be invertible on A where ϕ is a parabolic automorphism of D, and let a ∈ ∂D be the unique fixed point of ϕ. If u ∈ A(D), then    σ (uCϕ ) = λ ∈ C; |λ| = u(a) . Proof. Fix a point z0 ∈ D. Then the sequence (zm )m , where zm = ϕm (z0 ), is an interpolating sequence for H ∞ (D). Hence, by the Open Mapping Theorem, there exist a constant c > 0 and a sequence (fm )m ⊂ H ∞ (D) such that for all m we have fm ∞  c and    1, if k = m, fm ϕk (zm ) = (4.1) 0, if k = m. By (C1), for each m we can find a function gm ∈ A with gm A  1 such that  2 s   gm ϕm (zm ) 1 − ϕm (zm ) = 1.

(4.2)

Let hm := fm gm . Then hm ∈ A and by (C3), hm A = Mfm (gm ) A  fm ∞  c for all m. Pick any λ ∈ C with |λ| = r(uCϕ ) and notice that |λ| = |u(a)| by Lemma 4.2. It follows from (C1) that for all n ∈ N we have       (λ − uCϕ )2n    (λ − uCϕ )2n hn (zn ) 1 − |zn |2 s . (4.3) In view of (4.1) we can write

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2n    2n 2n−k  (λ − uCϕ )2n hn (zn ) = (−uCϕ )k hn (zn ) λ k



k=0

=

2n  2n

  λ2n−k (−1)k u(k) (zn )hn ϕk (zn )

k

  2n = (−1)n λn u(n) (zn )gn ϕn (zn ) . n k=0



(4.4)

In order to emphasize the similarity of treatment in both parabolic and hyperbolic cases, we denote w(z) := u(z)/ϕ  (z)s . Then w ∈ A(D) is bounded away from zero and w(n) (z) = u(n) (z)/ϕn (z)s . As one puts together the formulas (4.2), (4.3) and (4.4) and applies the Schwarz– Pick Lemma, the function w(n) comes to play:       (λ − uCϕ )2n   2n λn u(n) (zn )gn ϕn (zn )  1 − |zn |2 s n

s   2n  n 1 − |zn |2 2n  n . = λ u(n) (zn ) = λ w (z ) n (n) n n 1 − |ϕn (zn )|2 Since zn tends to a, we have w(zn ) → w(a) as n → ∞, and so obtain 1/2n  1/2 |u(a)|1/2  1/2  lim w(n) (zn ) = w(a) =  s/2 = u(a) . n→∞ ϕ (a) Hence,  1/2n  1/2 r(λ − uCϕ ) = lim (λ − uCϕ )2n   2|λ|1/2 u(a) = 2r(uCϕ ), n→∞

where we have used the well-known fact (see [18, Lemma 1.2]) that 1/2n 2n = 2. n→∞ n lim

Therefore, for any λ satisfying |λ| = r(uCϕ ), we have that r(λ − uCϕ )  2r(uCϕ ). But the Spectral Mapping Theorem states that σ (λ − uCϕ ) = {λ − μ; μ ∈ σ (uCϕ )}, and so it follows that −λ ∈ σ (uCϕ ). Thus, {λ; |λ| = r(uCϕ )} ⊆ σ (uCϕ ). For the reverse inclusion, notice that (uCϕ )−1 = u◦ϕ1 −1 Cϕ −1 , where ϕ −1 is also a parabolic

automorphism and a is the unique fixed point of ϕ −1 . If λ ∈ σ (uCϕ ), then λ−1 ∈ σ ((uCϕ )−1 ), and so, by Lemma 4.2, |λ|−1  |u(a)|−1 . Hence |u(a)|  |λ|. But as |λ|  r(uCϕ ) = |u(a)|, one concludes that σ (uCϕ ) ⊆ {λ; |λ| = r(uCϕ )}, and hence the proof is finished. 2 4.2. The hyperbolic case Let us start again by estimating the spectral radius of uCϕ for a hyperbolic automorphism ϕ of D.

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Lemma 4.4. Suppose A is a Banach space which satisfies the conditions (C1), (C2) and (C3). Let uCϕ be invertible on A where ϕ is a hyperbolic automorphism of D with fixed points a (attractive) and b (repulsive) in ∂D. If u ∈ A(D), then     1/n lim u(n) ∞ = max u(a), u(b)

n→∞

and       |u(a)| |u(b)| max  s ,  s  r(uCϕ )  max u(a), u(b) ϕ  (a)−s . ϕ (a) ϕ (b) Proof. We prove first the estimate for the spectral radius of uCϕ . The lower bound is due to Lemma 4.1, and so it suffices to establish the validity of the upper bound. Since a is the attractive fixed point of ϕ, it follows that for each pair (Va , Vb ) of open arcs on ∂D with a ∈ Va and b ∈ Vb there exists some m such that ϕn (z) ∈ Va for all z ∈ ∂D \ (Va ∪ Vb ) whenever n  m. Thus for each z ∈ ∂D and for each n ∈ N at most m elements from {z, ϕ(z), . . . , ϕn (z)} are not contained in Va ∪ Vb . Since uCϕ is invertible, it follows that u is bounded away from zero. Hence, by continuity of u, for any given ε > 0 we can find open arcs Va and Vb so that |u(z)|  (1 + ε) max{|u(a)|, |u(b)|} whenever z ∈ Va ∪ Vb . Note that, by condition (C3),   (uCϕ )n  = Mu Cϕ  Mu Cϕ  u(n) ∞ Cϕ . n n n (n) (n) For n > m we have   n−m         u(n) ∞ = maxu(n) (z)  u m , ∞ (1 + ε) max u(a) , u(b) |z|=1

(4.5)

and by (C2), Cϕn  (1 − |ϕn (0)|)−s . Since (1 − |ϕn (0)|)−s/n → ϕ  (a)−s as n → ∞, we conclude that     r(uCϕ )  (1 + ε) max u(a), u(b) ϕ  (a)−s . This holds for any ε > 0; hence the second claim follows. As for the first statement, one obtains from inequality (4.5) that     1/n lim u(n) ∞  (1 + ε) max u(a), u(b) .

n→∞

Therefore         1/n max u(a), u(b)  lim u(n) ∞  max u(a), u(b) , n→∞

and we are done.

2

Remark 4.5. If |u(a)|  |u(b)|, then r(uCϕ ) = |u(a)/ϕ  (a)s |, by Lemma 4.4.

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|u(b)| The following theorem shows that the formula r(uCϕ ) = max{ ϕ|u(a)|  (a)s , ϕ  (b)s } holds always on p (D). For the space H 2 (D), Gunatillake [14] obtained each of the spaces H p (D), Aα (D), and Hv∞ p only upper and lower estimates of the spectral radius of uCϕ using complicated calculations, see [14, Lemma 3.4.2] and the proof of [14, Lemma 3.4.3]. His result should be compared with our Lemma 4.4 which holds for any A-type space.

Theorem 4.6. Let A be any of the spaces H p (D), Aα (D), or Hv∞ (D). Let ϕ be a hyperbolic p automorphism of D with the Denjoy–Wolff point a and the other fixed point b. Suppose u ∈ A(D) is bounded away from zero on D. Then p

  |u(a)| |u(b)| r(uCϕ ) = max  s ,  s , ϕ (a) ϕ (b) where s is a constant depending on the space. |u(b)| Proof. By Lemma 4.4, we only need to show that r(uCϕ )  max{ ϕ|u(a)|  (a)s , ϕ  (b)s }. We proceed as follows. For any n ∈ N we have

 n−1          (uCϕ )n  =  u ◦ ϕj Cϕn    j =0

  n−1   n−1     u◦ϕ  s j    ϕ ◦ ϕ · C = j ϕ n .   (ϕ  ◦ ϕj )s j =0

j =0

    s Notice that n−1 j =0 ϕ ◦ ϕj = (ϕn ) . Further, let us denote w(z) := u(z)/ϕ (z) and observe that w ∈ A(D) is also bounded away from zero. By condition (C3), we obtain     s  (uCϕ )n 1/n  w(n) 1/n  ϕ Cϕ 1/n . ∞ n

n

Since Lemma 4.4 guarantees that       |u(a)| |u(b)| 1/n lim w(n) ∞ = max w(a), w(b) = max  s ,  s , n→∞ ϕ (a) ϕ (b) it is enough to show that limn (ϕn )s Cϕn 1/n  1. p We discuss each of the cases H p (D), Aα (D), and Hv∞ (D) separately. p (a) We start with the Hardy space H p (D) with 1  p < ∞. Now s = 1/p. For any f ∈ H p (D) we have p   1/p  ϕ Cϕn f H p = n



∂D

    f ◦ ϕn (z)p ϕ  (z) |dz|. n

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Since ϕ maps the boundary of D onto itself, one obtains by substituting ζ = ϕn (z) that    1/p   p  ϕ f (ζ )p |dζ | = f p p .  C f = ϕn n H Hp ∂D

In particular, (ϕn )1/p Cϕn = 1 for each n. p (b) Next we consider the weighted Bergman space Aα (D) with 1  p < ∞ and α > −1. Now p s = α+2 p . For any f ∈ Aα (D) we have    α+2   ϕ p (f ◦ ϕn )p p = n A



α

  α+2     ϕ (z) f ◦ ϕn (z)p 1 − |z|2 α dA(z) n

D



=

        f ◦ ϕn (z)p ϕ  (z)α 1 − |z|2 α ϕ  (z)2 dA(z). n n

D

Since ϕn is an automorphism, the Schwarz–Pick Lemma gives us 2   (ϕn ) (z) = 1 − |ϕn (z)| . 1 − |z|2

This fact together with the substitution ζ = ϕn (z) yields      α+2      ϕ p (f ◦ ϕn )p p = f (ζ )p 1 − |ζ |2 α dA(ζ ) = f p p , n A A α

α

D α+2

and so we obtain (ϕn ) p Cϕn = 1 for all n. (c) Let us then deal with the space Hv∞ (D) with 0 < p < ∞. Now s = p. It follows again from p (D) the Schwarz–Pick Lemma together with the substitution ζ = ϕn (z) that for any f ∈ Hv∞ p we have   p       ϕ Cϕ f  = sup(ϕn ) (z)p f ϕn (z)  1 − |z|2 p n n z∈D

   2 p  = supf ϕn (z)  1 − ϕn (z) = f Hv∞p . z∈D

In particular, (ϕn )p Cϕn = 1 for each n. Hence, we conclude that the formula     |u(a)| |u(b)| n 1/n  r(uCϕ ) = lim (uCϕ ) = max  s ,  s n→∞ ϕ (a) ϕ (b) is valid on each of the spaces H p (D), Aα (D), and Hv∞ (D). p p

2

The claims (a) and (b) in the following corollary are easily obtained from Lemma 4.4 and Theorem 4.6, respectively.

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Corollary 4.7. Let ϕ be a hyperbolic automorphism of D with the Denjoy–Wolff point a and the other fixed point b. Suppose u ∈ A(D) is bounded away from zero on D. (a) Let A be any space satisfying the conditions (C1), (C2) and (C3). Then any λ ∈ σ (uCϕ ) satisfies         min u(a), u(b) ϕ  (a)s  |λ|  max u(a), u(b) ϕ  (a)−s . (b) Let A be any of the spaces H p (D), Aα (D), or Hv∞ (D). If λ ∈ σ (uCϕ ), then p p

    |u(a)| |u(b)| |u(a)| |u(b)| min  s ,  s  |λ|  max  s ,  s . ϕ (a) ϕ (b) ϕ (a) ϕ (b) Proof. Let us give a proof for (a). Proof for (b) is similar with the exception that we know exactly the spectral radius of uCϕ , and so we get a better bound for the spectrum, too. The Spectral Mapping Theorem guarantees that if λ ∈ σ (uCϕ ), then λ−1 ∈ σ ((uCϕ )−1 ). Since −1 ϕ is also a hyperbolic automorphism of D with fixed points a (repulsive) and b (attractive), Lemma 4.4 yields that    −1  −s 1 ϕ  (b)s 1 1 (b) =  max , ϕ . |λ| |u(a)| |u(b)| min{|u(a)|, |u(b)|} As ϕ is hyperbolic, we have ϕ  (b) = 1/ϕ  (a), and hence we obtain     |λ|  min u(a), u(b) ϕ  (a)s , as claimed.

2

(D), More can be said about σ (uCϕ ) if uCϕ acts on any of the spaces H p (D), Aα (D) or Hv∞ p as we will see in the following two theorems. More precisely, we will completely describe the spectrum of uCϕ whenever |u(b)/ϕ  (b)s |  |u(a)/ϕ  (a)s |. Let ϕ : D → D be a hyperbolic automorphism with the Denjoy–Wolff point a and the repulsive fixed point b. Then there is an automorphism φ of D with φ(1) = a and φ(−1) = b, so that ψ(z) := (φ −1 ◦ ϕ ◦ φ)(z) = (z + r)/(1 + rz), where r = (1 − ϕ  (a))/(1 + ϕ  (a)) and 0 < r < 1. If v := u ◦ φ, then uCϕ and vCψ are similar and have the same spectrum. p

(D). Let ϕ be a hyperbolic Theorem 4.8. Let A be any of the spaces Aα (D), H p (D), or Hv∞ p automorphism of D whose attractive fixed point is a and the repulsive one is b. If u ∈ A(D) is bounded away from zero and |u(a)/ϕ  (a)s | = |u(b)/ϕ  (b)s |, then p

  σ (uCϕ ) = λ ∈ C; |λ| = r(uCϕ ) . Proof. In view of Corollary 4.7 we only need to show that the claimed circle is contained in σ (uCϕ ). This can be done by mimicking the proof of Theorem 4.3 with the exception that we need a different argument to find a suitable interpolation sequence. So, fix a point z0 ∈ D such that |ϕ  (z0 )| < 1. Put zn = ϕn (z0 ). The Schwarz–Pick Lemma gives that

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   1 + |ϕn (z0 )| 1 − |ϕn+1 (z0 )| 1 − |ϕn+1 (z0 )|2 1 + |ϕn (z0 )| = = ϕ  ϕn (z0 )  → ϕ  (a) < 1 2 1 − |ϕn (z0 )| 1 + |ϕn+1 (z0 )| 1 − |ϕn (z0 )| 1 + |ϕn+1 (z0 )| as n → ∞, and thus we conclude by [8, Theorem 9.2] that (zn )n is an interpolating sequence for H ∞ (D). The claim follows as one repeats the same arguments as in the proof of Theorem 4.3. Indeed, let w(z) := u(z)/ϕ  (z)s . Then w ∈ A(D) is bounded away from zero and w(n) (z) = u(n) (z)/ϕn (z)s . Take any λ ∈ C with |λ| = r(uCϕ ) and notice that |λ| = |w(a)| by Theorem 4.6. Since zn → a, it follows that w(zn ) → w(a) as n → ∞, yielding  1/2n  1/2 lim w(n) (zn ) = w(a) = r(uCϕ )1/2 .

n→∞

One proceeds as in the proof of Theorem 4.3 and obtains  1/2n  1/2 r(λ − uCϕ ) = lim (λ − uCϕ )2n   2|λ|1/2 w(a) = 2r(uCϕ ), n→∞

and therefore concludes that λ ∈ σ (uCϕ ), which gives the claim.

2

The following theorem is one of our main results. It both generalizes (is valid on a larger class of spaces) and improves (gives a stronger result even on H 2 (D)) the corresponding main result of Gunatillake’s paper [14, Theorem 3.5.1]. Our proof relies on the fact that (1 − z)β−s belongs p to each of the spaces H p (D), Aα (D), and Hv∞ (D) whenever β > 0. p β−s ∞ ∈ Hvs (D) for β > 0. Moreover, by [6, Lemma 7.3], we Indeed, it is trivial that (1 − z) know that (1 − z)t ∈ H 2 (D) for t > −1/2 and (1 − z)t ∈ A2α (D) for t > −(α + 2)/2. Since 2/p (1 − z)t H p = (1 − z)tp/2 H 2 , we get that (1 − z)t ∈ H p (D) for t > −1/p, or in our case, p whenever β > 0. A similar argument shows that (1 − z)t ∈ Aα (D) for t > −(α + 2)/p, and so in our case, whenever β > 0. Theorem 4.9. Let A be any of the spaces Aα (D), H p (D), or Hv∞ (D). Let ϕ be a hyperbolic p automorphism of D with the attractive fixed point a and the repulsive fixed point b. If u ∈ A(D) is bounded away from zero and |u(b)/ϕ  (b)s |  |u(a)/ϕ  (a)s |, then p

  |u(b)| |u(a)| σ (uCϕ ) = λ ∈ C;  s  |λ|   s . ϕ (b) ϕ (a) Before we give a proof, let us observe that the above theorem gives the optimal result for a composition operator Cϕ ; see [6, Theorem 7.4]. Proof of Theorem 4.9. If |u(b)/ϕ  (b)s | = |u(a)/ϕ  (a)s |, then the claim follows from Theorem 4.8, and so we may assume that |u(b)/ϕ  (b)s | < |u(a)/ϕ  (a)s |. Moreover, by Corollary 4.7 it is enough to show that the claimed annulus is contained in σ (uCϕ ). We may assume without loss of generality that ϕ(z) = (z + r)/(1 + rz) with 0 < r < 1. Then a = 1 and b = −1. Let |u(−1)/ϕ  (−1)s | < R < |u(1)/ϕ  (1)s | and notice that there exists β > 0 such that R = |u(1)|ϕ  (1)β−s . As in the proof of Theorem 4.8, we can find a point z0 ∈ D so that (ϕn (z0 ))n is an interpolating sequence for H ∞ (D). Hence we can find a function h ∈ H ∞ (D) satisfying

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   1/k, if n = 2k , k = 1, 2, . . . , h ϕn (z0 ) = 0, otherwise. Let g(z) := h(z)(1 − z)β−s , and notice that g ∈ A because (1 − z)β−s ∈ A whenever β > 0. Let us now define Fλ (z) :=

∞  g(z) 1  u(n) (z)  g ϕ (z) . + n λ λ λn

(4.6)

n=1

One can then check that λFλ (z) − u(z)Fλ (ϕ(z)) = g(z) holds pointwise in D, provided that the sum is well-defined. p (D) We will soon verify (by considering each of the spaces Aα (D), H p (D), and Hv∞ p separately) that Fλ ∈ A whenever |λ| > R. Assuming (for now) that this is the case, then (λ − uCϕ )Fλ = g. If there is some λ with |λ| > R and λ ∈ / σ (uCϕ ), then we have Fλ = (λ − uCϕ )−1 g. Next we define H (w) := F1/w (z0 ) = g(z0 )w + w

∞  u(2k ) (z0 )  k=1

k

β−s 2k w . 1 − ϕ2k (z0 )

Since   k    u(2k ) (z0 )  β−s 1/2   lim  = u(1)ϕ  (1)β−s = R, 1 − ϕ2k (z0 )  k→∞ k we conclude that the radius of convergence of H is R −1 . Moreover, H is defined by a Hadamard gap series, so every point on {w; |w| = R −1 } is a singular point of H ; see [13, Theorem 9.2.1]. Next we show that each λ with |λ| = R belongs to σ (uCϕ ). / σ (uCϕ ). Then λ → (λ − uCϕ )−1 g(z0 ) is analytic in a neighborSuppose |λ0 | = R and λ0 ∈ hood V of λ0 and it coincides with Fλ (z0 ) whenever λ ∈ V and |λ| > R. But then λ → Fλ (z0 ) is regular at λ0 , and so H (w) = F1/w (z0 ) is also regular at w = 1/λ0 . Since this is impossible, we conclude that {λ; |λ| = R} ⊂ σ (uCϕ ). This holds for any |u(−1)/ϕ  (−1)s | < R < |u(1)/ϕ  (1)s |, and so 

    λ ∈ C; u(−1)/ϕ  (−1)s   |λ|  u(1)/ϕ  (1)s  ⊂ σ (uCϕ ).

Hence the proof is finished as soon as we have shown that Fλ ∈ A for |λ| > R. Although we prove this by considering each space separately, the technique is almost the same in all three cases. So let us first introduce notation. Since u and ϕ  are continuous on D and |u(−1)/ϕ  (−1)s | < R, there is a small neighborhood of −1, say U ⊂ D, such that |u(z)/ϕ  (z)s | < R for all z ∈ U . Denote U0 := U ∩ D and put Un := ϕn−1 (U0 ) for each n. Notice that if z ∈ Uk−1 \ Uk , then  n    R |ϕn (z)|s , u(n) (z) < R k |ϕk (z)|s |u(ϕk (z)) · · · u(ϕn−1 (z))|,

if n  k, if n > k,

and that the second case can be written as |u(n) (z)| < R k |ϕk (z)|s |u(n−k) (ϕk (z))|.

(4.7)

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20

p

We start with the weighted Bergman space Aα (D). Fix λ with |λ| > R. In order to show that p Fλ ∈ Aα (D) we need to estimate the integrals p

Jn :=

    u(n) (z)    p    1 − |z|2 α dA(z) g ϕ (z) n  λn 

D

    and verify that ∞ n=1 Jn < ∞. For each n we write D = Un + D\Un and apply the estimate (4.7) to both of them. For the first integral, we proceed similarly as in Theorem 4.6 and obtain via the change of variable formula that     u(n) (z)    β−s p    1 − |z|2 α dA(z)  λn h ϕn (z) 1 − ϕn (z)  Un

 

R |λ| R |λ|

np 

  s       ϕ (z) 1 − ϕn (z) β−s p 1 − |z|2 α dA(z) n

np

Un

  (1 − z)β−s p p , Aα

(4.8)

p

which is finite since (1 − z)β−s ∈ Aα (D).     For the second part, we start by writing D\Un = Un−1 \Un + · · · + U0 \U1 + D\U0 . We apply the estimate (4.7) to each of the integrals and then proceed similarly as in the proof of Theorem 4.6. So for each k we obtain by substituting ζ = ϕk (z) that    u(n) (z)   β−s p    1 − |z|2 α dA(z)  λn 1 − ϕn (z) 

 Uk−1 \Uk



 

R |λ|

kp



Uk−1 \Uk

    α+2  u(n−k) (ϕk (z))   β−s p  ϕ (z)   1 − |z|2 α dA(z) 1 − ϕn (z) k   n−k λ



   u(n−k) (ζ )   β−s p  R kp   1 − |ζ |2 α dA(ζ ). 1 − ϕ (ζ ) n−k   n−k |λ| λ D\U0

Hence,     u(n) (z)   β−s p    1 − |z|2 α dA(z) 1 − ϕ (z) n  λn 

D\Un



  n   u(k) (ζ )   β−s p  R (n−k)p   1 − |ζ |2 α dA(ζ ). 1 − ϕk (ζ )   λk  |λ| k=0

D\U0

Pick an ε > 0 such that R + ε < |λ|. Then there exists an N ∈ N such that for any ζ ∈ D \ U0 we have

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   u(k) (ζ )  β−s 1/k R + ε   < 1 − ϕk (ζ ) <1  λk  |λ| whenever k  N . So for k  N ,

R |λ|



(n−k)p  

 u(k) (ζ )   β−s p  R + ε np 2 α   1 − ϕk (ζ )  λk  1 − |ζ | dA(ζ )  |λ| D\U0

and for 0  k < N , 

    u(k) (ζ )   β−s p  u ∞ kp  2 α Cϕ (1 − z)β−s p p .   1 − ϕ 1 − |ζ | (ζ ) dA(ζ )  k k  λk  Aα |λ|

D\U0

It follows from (4.8) and the above estimates that for n  N , p

Jn 

R |λ|

np +

N−1  k=0

R n−k |λ|n

p +

n  R + ε np k=N

|λ|

n

R+ε |λ|

np .

Thus, ∞  n=1

∞  R+ε n Jn  n < ∞, |λ| n=1

p

since R + ε < |λ|, and so Fλ ∈ Aα (D) whenever |λ| > R. p One can check similarly that  Fλ ∈ H (D).  Indeed, one regards the sets Un as the corresponding subsets of ∂D and writes ∂D = Un + ∂D\Un . Then  np      u(n) (z)  β−s p R (1 − z)β−s p p   1 − ϕ (z) |dz|  n  λn  H |λ|

Un

    again following Theorem 4.6. Writing ∂D\Un = Un−1 \Un + · · · + U0 \U1 + ∂D\U0 allows one to observe that     u(n) (z)  β−s p   |dz|  λn 1 − ϕn (z)  ∂D\Un



 n  R (n−k)p |λ| k=0

   u(k) (ζ )  β−s p   |dζ |. 1 − ϕ (ζ ) k  λk 

∂D\U0

By repeating the same estimates as in the Bergman case one obtains the claim. We still need to verify that Fλ ∈ Hv∞ (D) = Hv∞ (D). Let Un ’s be as in the Bergman case and p s observe that

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22

   u(n) (z)  β−s 1/n |u(1)|  β−s R   = 1 − ϕn (z) lim sup = ϕ (1) < 1.  n→∞ z∈D\U  λn |λ| |λ| 0

(4.9)

A glance at (4.9) makes it obvious that (4.6) converges uniformly on D \ U0 . Next we verify that (4.6) converges also on U0 . So, let z ∈ Uk−1 \ Uk for some k. Following again the proof of Theorem 4.6 and applying the estimate (4.7) give for n  k that    n   u(n) (z)    β−s  β−s  R   s  2 s    1 − |z|2 s 1 − |z| 1 − ϕ 1 − ϕ (z)  (z) (z) ϕ n n n  λn    |λ| n  R  (1 − z)β−s  ∞ .  Hv s |λ| Thus we obtain  ∞    u(n) (z)    β−s    1 − |z|2 s h ϕ (z) 1 − ϕ (z) n n  λn  n=1



∞ ∞     u(n) (z)   β−s  R n   1 − |z|2 s .  +  λn 1 − ϕn (z)  |λ| n=1

(4.10)

n=k+1

But for n > k the estimate (4.7) yields |u(n) (z)|  R k |ϕk (z)|s |u(n−k) (ϕk (z))|, and it follows again from the Schwarz–Pick Lemma that   s  2 s s  R k ϕk (z) 1 − |z|2 = R k 1 − ϕk (z)  R k . Denote ζ := ϕk (z) ∈ D \ U0 and notice that we can estimate the tail of (4.10) by   k  ∞  ∞    u(n) (z)   u(n) (ζ )   β−s  β−s  R 2 s     1 − ϕ 1 − ϕ (z)  (ζ ) 1 − |z| n n  λn   λn  |λ|

n=k+1

n=1

 ∞    u(n) (ζ )  β−s   , 1 − ϕn (ζ )   λn  n=1

which we know converges uniformly on D \ U0 by (4.9). This estimate is independent of the particular set Uk , and therefore Fλ ∈ Hv∞ (D). This finishes our proof. 2 s Remark 4.10. Before closing the hyperbolic case, let us point out a couple of observations concerning the point spectrum of uCϕ where ϕ(z) = (z + r)/(1 + rz). Proofs for these results are easy modifications of the corresponding results found in Gunatillake’s paper [14, Subsection 3.5], and so we omit them. If u ∈ A(D) is bounded away from zero, it satisfies |u(1)| > |u(−1)| and u is also bounded, then it can be shown (see [14, Lemma 3.5.1]) that

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n=0

and

∞  n=1

23

u(−1) u(ϕ−n (z))

define non-zero bounded analytic functions on D. Simple modifications of Gunatillake’s results [14, Lemmas 3.5.3 and 3.5.4] then yield the following result. p t Let A be any of the spaces Aα (D), H p (D), or Hv∞ (D), so that the function ( 1+z 1−z ) ∈ A p for |t| < s (see [6, Lemma 7.3], again). Let ϕ be a hyperbolic automorphism of D with fixed points a (attractive) and b (repulsive) in ∂D. Suppose that u ∈ A(D) and u is bounded. If 1 < |u(a)/u(b)| < ϕ  (a)−2s , then      σ (uCϕ ) = λ; u(b)/ϕ  (b)s   |λ|  u(a)/ϕ  (a)s  and every interior point of this annulus is an eigenvalue of infinite multiplicity for uCϕ . 4.3. The elliptic case Now we deduce the spectrum of each uCϕ on A when ϕ is an elliptic automorphism of D. Although the elliptic case is dealt with standard techniques, it is included for the sake of completeness. Theorem 4.11. Let A be a Banach space satisfying the conditions (C1), (C2) and (C3). Suppose that u ∈ A(D) and ϕ is an automorphism of D such that there is a positive integer j with ϕj (z) = z for all z ∈ D. If m is the smallest such integer, then σ (uCϕ ) = {λ; λm = u(m) (z), z ∈ D}. Proof. It is clear from the proof of Corollary 3.7 that 0 ∈ σ (uCϕ ) if and only if u is not bounded away from zero on D. Suppose that λ = 0, λm = u(m) (z0 ) for some z0 ∈ D, which is not a fixed point for ϕ. Let g ∈ A(D) ⊂ A satisfy     g(z0 ) = 1 and g ϕ(z0 ) = · · · = g ϕm−1 (z0 ) = 0. If g ∈ Im(uCϕ − λ), then there exists f ∈ A such that   u(z)f ϕ(z) − λf (z) = g(z)

for all z ∈ D.

Thus we get the following equations:   ⎧ )f ϕ(z ) − λf (z0 ) = 1, u(z ⎪ 0 0 ⎪       ⎪ ⎪ ⎪ u ϕ(z0 ) f ϕ2 (z0 ) − λf ϕ(z0 ) = 0, ⎪ ⎪ ⎨ .. . ⎪ ⎪       ⎪ ⎪ ⎪ u ϕm−2 (z0 ) f ϕm−1 (z0 ) − λf ϕm−2 (z0 ) = 0, ⎪ ⎪     ⎩ u ϕm−1 (z0 ) f (z0 ) − λf ϕm−1 (z0 ) = 0. From the last m − 1 equations we get

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24

  u(ϕ(z0 )) · · · u(ϕm−1 (z0 ))f (z0 ) f ϕ(z0 ) = . λm−1 By replacing f (ϕ(z0 )) in the first equation, we obtain u(m) (z0 )f (z0 ) − λf (z0 ) = 1, λm−1 which is a contradiction. Therefore Im(uCϕ − λ) = A, so uCϕ − λ is not invertible. Hence λ ∈ σ (uCϕ ). Since u ∈ A(D) and σ (uCϕ ) is compact, we conclude that 

   λ; λm = u(m) (z), z ∈ D = λ; λm = u(m) (z), z ∈ D, ϕ(z) = z ⊂ σ (uCϕ ).

For the reverse inclusion, we notice that (uCϕ )m = Mu(m) : A → A is bounded. Then the Spectral Mapping Theorem yields that σ (uCϕ )m = σ (Mu(m) ). Let G := {λ; λm = u(m) (z), z ∈ D}. If λ ∈ C \ G, then ψ(z) = 1/(λm − u(m) (z)) ∈ H ∞ (D) and we obtain by (C3) that Mψ is bounded / σ (uCϕ )m . on A. Moreover, it can be seen that Mψ = (λm − Mu(m) )−1 , which yields that λm ∈ Thus σ (uCϕ ) ⊂ G, and the proof of the theorem is complete. 2 The following result from Kamowitz’s paper [18, Lemma 4.2] (see also [14, Lemma 3.2.1]) is needed in the proof of our next lemma. Lemma 4.12. Suppose that u ∈ A(D) has no zeros on ∂D and let μ = e2πθi with irrational θ . Then      u(z)u(μz) · · · u μn−1 z 1/n → u(0) uniformly on D as n → ∞. Now we are ready to state and prove: Lemma 4.13. Let A be a Banach space which satisfies the conditions (C1), (C2) and (C3). If ϕ(z) = μz with μ = e2πθ i and θ is irrational, u ∈ A(D) and uCϕ : A → A is invertible, then r(uCϕ ) = |u(0)|.  m Proof. For n  1 we have (uCϕ )n = u(n) Cϕn , where u(n) (z) = n−1 m=0 u(μ z) belongs to A(D). Since ϕn is an elliptic automorphism, we obtain by (C2) that there exists a constant α > 0 such that Cϕn  α for all n. Hence, by condition (C3), 1/n

r(uCϕ ) = lim u(n) Cϕn 1/n  lim u(n) ∞ . n→∞

n→∞

It follows again from the proof of Corollary 3.7 that u ∈ A(D) is bounded away from zero. Therefore, we can apply Lemma 4.12 to conclude that |u(n) (z)|1/n converges uniformly to |u(0)| on the unit disc D. Thus we find a subsequence (u(nm ) )m satisfying 1/nm

u(nm ) ∞



  1  u(0) 1 + m

for all m,

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25

and so r(uCϕ )  |u(0)|. Since r(uCϕ )  |u(0)| by Lemma 4.1, the desired statement follows. 2 Suppose that ϕ : D → D is an elliptic automorphism with fixed point a ∈ D and let ϕa (z) = (a − z)/(1 − az). Then ψ := ϕa ◦ ϕ ◦ ϕa has zero as fixed point and takes the form ψ(z) = ϕ  (a)z with |ϕ  (a)| = 1. Moreover, if v := u ◦ ϕa , then uCϕ and vCψ are similar and have the same spectrum. Theorem 4.14. Suppose A is a Banach space which satisfies the conditions (C1), (C2) and (C3). Let uCϕ be invertible on A, ϕ an elliptic automorphism with a fixed point a ∈ D such that ϕn = id for all positive integers n and u ∈ A(D). Then σ (uCϕ ) = {λ; |λ| = |u(a)|}. Proof. By assumption, ψ(z) = e2πθi z, where θ is irrational, so Lemma 4.13 gives that        r(uCϕ ) = r(vCψ ) = v(0) = u ϕa (0)  = u(a). Since ϕ −1 is also an elliptic automorphism with a fixed point a ∈ D such that ϕ−n = id for all positive integers n and (uCϕ )−1 = u◦ϕ1 −1 Cϕ −1 , we obtain that r((uCϕ )−1 ) = |u(a)|−1 . This implies that σ (uCϕ ) ⊂ {λ; |λ| = |u(a)|}. Since zn ∈ A for every integer n  0, the argument used in the proof of [19, Lemma 2.3] shows that v(0)ψ  (0)n ∈ σ (vCψ ) for every integer n  0. The set {v(0)e2πθni ; n  0} is dense in {λ; |λ| = |v(0)|}, so we conclude that 

     λ; |λ| = u(a) = λ; |λ| = v(0) ⊂ σ (vCψ ) = σ (uCϕ ),

which gives the statement.

2

5. Conclusions and examples A summary of the main results of our investigation is given in the following corollary. Although we have proven these results in a more general setting (with the exception of item (ii)), here we formulate them only for the most important A-type spaces: standard weighted Bergman p (D) with standard spaces Aα (D), Hardy spaces H p (D), and weighted Banach spaces Hv∞ p weights vp (z) = (1 − |z|2 )p . Corollary 5.1. Let A be any of the spaces p

(a) Aα (D), p  1, α > −1, and s =

α+2 p ;

(b) H p (D), p  1, and s = p1 ; or (c) Hv∞ (D), 0 < p < ∞, and s = p. p Then the following hold. (i) Suppose u ∈ A(D) is bounded away from zero on D and let ϕ be a parabolic automorphism of D whose Denjoy–Wolff point is a ∈ ∂D. Then    σA (uCϕ ) = λ ∈ C; |λ| = u(a) .

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(ii) Let ϕ be a hyperbolic automorphism of D with the attractive fixed point a and the repulsive fixed point b. If u ∈ A(D) is bounded away from zero and |u(b)/ϕ  (b)s |  |u(a)/ϕ  (a)s |, then   |u(b)| |u(a)| σA (uCϕ ) = λ ∈ C;  s  |λ|   s . ϕ (b) ϕ (a) (iii) Suppose that u ∈ A(D) and ϕ is an automorphism of D such that there is a positive integer j with ϕj (z) = z for all z ∈ D. If n is the smallest such integer, then   σA (uCϕ ) = λ ∈ C; λn = u(n) (z), z ∈ D . (iv) Suppose u ∈ A(D) is bounded away from zero on D and let ϕ be an elliptic automorphism such that ϕn = id for all positive integers n. If a ∈ D is the unique fixed point of ϕ, then σA (uCϕ ) = {λ; |λ| = |u(a)|}. Proof. These items are merely summarizing our previous results, and thus need no further justification; see Theorems 4.3, 4.9, 4.11, and 4.14. 2 Examples. Let us now give two examples in which we determine the spectrum of a weighted composition operator uCϕ . The first example is specific to Hilbert spaces, and it is closely related to [4, Theorem 7]. The second example is a non-Hilbert one. (1) Let uCϕ be a unitary weighted composition operator on H 2 (D), where ϕ is an automorphism. Then s = 1/2. Since uCϕ is unitary, it follows from [4, Theorem 6] that " u(z) = c

1 − |z0 |2 , 1 − z0 z

where ϕ(z0 ) = 0 and |c| = 1. Moreover, r(uCϕ ) = 1 and σ (uCϕ ) ⊆ ∂D. (a) Suppose that ϕ(z) = μz, where |μ| = 1. Then u(z) = c. If μj = 1 for some positive integer j , then if n is the smallest such integer, Theorem 4.11 gives     σ (uCϕ ) = λ; λn = cn = μk c; k = 0, 1, . . . , n − 1 . If there is no such integer, we get σ (uCϕ ) = ∂D by Theorem 4.14. (b) Let ϕ(z) = ((1 + i)z − 1)/(z + i − 1), that is, ϕ is a parabolic automorphism of D. Then z0 = (1 + i)−1 . By Lemma 4.2, r(uCϕ ) = |u(1)| = 1. Hence we can use Theorem 4.3 to obtain σ (uCϕ ) = ∂D. (c) Consider the hyperbolic automorphism ϕ(z) = (z + r)/(1 + rz) where 0 < r < 1. Then z0 = −r. In this case |u(−1)| > |u(1)|, and |u(1)| |u(−1)| =  = 1 = r(uCϕ ).  1/2 ϕ (1) ϕ (−1)1/2 So Theorem 4.9 gives that the spectrum of uCϕ is the unit circle.

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(2) Let us consider the space (z + r)/(1 + rz) with 0 < Denjoy–Wolff point a = 1 (1 − r 2 )p /(1 + rz)2p . Then ible on Hv∞ (D). Now p

27

Hv∞ (D), which is non-Hilbert. Now s = p. Let ϕ(z) = p r < 1, that is, ϕ is a hyperbolic automorphism of D with and the other fixed point b = −1. Put u(z) = (ϕ  (z))p = u ∈ A(D) is bounded away from zero, and so uCϕ is invert-

|u(−1)| |u(1)| =  p = 1,  p ϕ (−1) ϕ (1) and so Theorem 4.9 gives that the spectrum of uCϕ is the unit circle. Let us also observe that from the proof of Theorem 4.6 we know that uCϕ is an isometry. See also [2, Theorem 2]. For u ∈ H (D) and ϕ being an analytic selfmap of D, Li and Stevi´c [21] introduced the generalized composition operator Cϕu given by 

Cϕu f

 (z) =

z

  f  ϕ(ξ ) u(ξ ) dξ.

0 ϕ

ϕ

If we put u = ϕ  , then for any z ∈ D we have (Cϕ f )(z) = (Cϕ f )(z) − (Cϕ f )(0), and so Cϕ = Cϕ − δϕ(0) is almost a composition operator. Now we consider the following linear maps between analytic function spaces S : B(D) → Hv∞ (D) resp. S : Dα2 (D) → A2α (D), 1 T

(D) → B(D) : Hv∞ 1

Sh = h ; z

resp. T

: A2α (D) → Dα2 (D),

(T h)(z) =

h(ξ ) dξ. 0

The operators S and T are bounded and (S ◦ T )f (z) = f (z), and one has T ◦ uCϕ ◦ S = Cϕu

and

S ◦ Cϕu ◦ T = uCϕ .

Hence uCϕ ◦ S = S ◦ Cϕu −: A. Then uCϕ = A ◦ T and Cϕu = T ◦ A, so uCϕ and Cϕu are related operators. By [25, Proposition 27.3.2], this gives that σB (Cϕu ) = σHv∞ (uCϕ ) and σDα2 (Cϕu ) = 1 σA2α (uCϕ ). ϕ

Next we use Corollary 5.1 to describe the spectrum of Cϕ on the Bloch space B(D) as well as on Dirichlet spaces Dα2 (D) when ϕ is either an elliptic or a parabolic automorphism. The cases (i) and (ii) for the Dirichlet space D02 (D) in our next corollary have appeared in the works of Gallardo-Gutiérrez and Montes-Rodríguez [12], and Higdon [17]. Here we get the same results with different techniques. We are not aware of any earlier results of this type for the Bloch space B(D). Corollary 5.2. Let A be any of the spaces B(D) or Dα2 (D) with α  0. Then for u = ϕ  we have the following.

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(i) Suppose that ϕ is an automorphism of D such that there is a positive integer j with ϕj (z) = z ϕ

for all z ∈ D with fixed point a ∈ D and consider the bounded operator Cϕ on A. If n is the smallest such integer, then     σA Cϕϕ = ϕ  (a)k ; k = 0, 1, . . . , n − 1 . (ii) Let ϕ be an elliptic automorphism with fixed point a ∈ D such that ϕn = id for all positive ϕ integers n. Then σA (Cϕ ) = ∂D. (iii) Suppose ϕ is a parabolic automorphism of D, and let a ∈ ∂D be the unique fixed point of ϕ. ϕ Then σA (Cϕ ) = ∂D. ϕ

Proof. For (i), it is clear that ϕ  ∈ A(D) is bounded away from zero. Since σB (Cϕ ) = ϕ

σHv∞ (ϕ  Cϕ ) and σDα2 (Cϕ ) = σA2α (ϕ  Cϕ ), and Theorem 4.11 applies to both Hv∞ (D) and A2α (D) 1 1 we obtain the claim. ϕ For (ii), a similar application of Theorem 4.14 gives that σA (Cϕ ) = ∂D. Finally, in case (iii) we notice that also ϕ  ∈ A(D) has no zeros in D for the parabolic auϕ tomorphism ϕ. Again, as above, we apply Theorem 4.3 to obtain that σA (Cϕ ) = ∂D since ϕ  (a) = 1. 2 In addition to the claims in the previous corollary, Higdon [17] discussed also the hyperbolic case. Here we get the corresponding result on B(D) and D02 (D) with a different technique. Corollary 5.3. Let A be either the Bloch space B(D) or the Dirichlet space D02 (D). Let ϕ be a hyperbolic automorphism of D with the Denjoy–Wolff point a and the other fixed point b. Then ϕ σA (Cϕ ) = ∂D. ϕ

ϕ

Proof. Since σB (Cϕ ) = σHv∞ (ϕ  Cϕ ) and σD2 (Cϕ ) = σA2 (ϕ  Cϕ ), we can apply Theorem 4.9 for 1 0 0 the weighted composition operator ϕ  Cϕ on the spaces Hv∞ (D) and A20 (D), respectively. Indeed, 1  since ϕ ∈ A(D) is bounded away from zero, it follows from Corollary 3.7 that ϕ  Cϕ is invertible on both spaces. The claim follows from Theorem 4.9 as soon as one observes that s = 1 for both of the spaces Hv∞ (D) and A20 (D). 2 1 Acknowledgments The authors are grateful to the referee for helpful comments and for pointing out the reference [3]. References [1] R. Aron, M. Lindström, Spectra of weighted composition operators on weighted Banach spaces of analytic functions, Israel J. Math. 141 (2004) 263–276. [2] J. Bonet, M. Lindström, E. Wolf, Isometric weighted composition operators on weighted Banach spaces of type H ∞ , Proc. Amer. Math. Soc. 136 (2008) 4267–4273. [3] P. Bourdon, Invertible weighted composition operators, Proc. Amer. Math. Soc. (2013), in press, arXiv:1211.4190 [math.FA].

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