COMPOSITION OPERATORS with a MINIMAL COMMUTANT Contents

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COMPOSITION OPERATORS WITH A MINIMAL COMMUTANT

MIGUEL LACRUZ, FERNANDO LEON-SAAVEDRA,´ SRDJAN PETROVIC, AND LUIS RODR´IGUEZ-PIAZZA

2 Abstract. Let Cϕ be a composition operator on the Hardy space H , induced by a linear fractional self-map ϕ of the unit disk. We consider the question whether the commutant of Cϕ is minimal, in the sense that it reduces to the weak closure of the unital algebra generated by Cϕ. We show that this happens in exactly three cases: when ϕ is either a non-periodic elliptic automorphism, or a parabolic non-automorphism, or a loxodromic self-map of the unit disk. Also, we consider the case of a composition operator induced by a univalent, analytic self-map ϕ of the unit disk that ﬁxes the origin and that is not necessarily a linear fractional map, but in exchange its K¨onigs’sdomain is bounded and stricly starlike with respect to the origin, and we show that the operator Cϕ has a minimal commutant. Furthermore, we provide two examples of univalent, analytic self-maps ϕ of the unit disk such that Cϕ is compact but it fails to have a minimal commutant.

Contents Introduction 1 1. Some general results about operators with a minimal commutant 4 2. Composition operators induced by parabolic and hyperbolic automorphisms 5 3. The commutant of a composition operator induced by an elliptic automorphism 7 4. The commutant of a composition operator induced by a parabolic non-automorphism 8 5. The commutant of a composition operator induced by a hyperbolic non-automorphism 11 6. The commutant of a composition operator induced by a loxodromic mapping 12 7. Compact composition operators with univalent symbol and non-minimal commutant 18 Appendix: The Sobolev space as a Banach algebra with an approximate identity 24 References 26

Introduction Let B(H) stand for the algebra of all bounded linear operators on a Hilbert space H and let A ∈ B(H). Recall that the commutant of A is deﬁned as the family of all operators that commute with A, that is, {A}0 : = {X ∈ B(H): AX = XA}. It is a standard fact that {A}0 is a subalgebra of B(H) that is closed in the weak operator topology σ. We shall denote by alg (A) the unital algebra generated in B(H) by the operator A, that is, alg (A): = {p(A): p is a polynomial}.

Date: January 25, 2018. Key words and phrases. Composition operator; Hardy space; Linear fractional map; Operator with minimal commutant. The first and fourth authors were partially supported by Ministerio de Econom´ıay Competitividad, Reino de España, and Fondo Europeo de Desarrollo Regional under Proyecto MTM 2015-63699-P. The second author was partially supported by Vicerrectorado de Investigaciónde la Universidad de Cádiz.This research was in progress when the third author visited Universidad de Sevilla in the spring of 2016. 1 2 MIGUEL LACRUZ, FERNANDO LEON-SAAVEDRA,´ SRDJAN PETROVIC, AND LUIS RODRÍGUEZ-PIAZZA

σ σ It is clear that alg (A) is a commutative algebra with the property that alg (A) ⊆ {A}0. We say that an operator A has a minimal commutant provided that σ alg (A) = {A}0. It is the purpose of this paper to investigate the last equality in the case when A is a composition op- 2 erator Cϕ, acting on H (D), the Hardy space of the unit disk, and ϕ is a linear fractional transformation. We will show that the equality holds precisely when the symbol is either a non-periodic elliptic automorphism, or a parabolic non-automorphism, or a loxodromic self-map of the unit disk (see definitions below). Recall that the Hardy space H2(D) is the Hilbert space of all analytic functions f on the unit disk D that have a Taylor series expansion around the origin of the form ∞ ∞ X n X 2 f(z) = anz , with |an| < ∞, n=0 n=0 endowed with the norm ∞ !1/2 X 2 kfk2 := |an| . n=0 It is a well-known fact that if f ∈ H2(D) then the radial limits f(eiθ) := lim f(reiθ) r→1− exist almost everywhere, and moreover, Z 2π 1/2 1 iθ 2 kfk2 = |f(e )| dθ . 2π 0 Every analytic mapping ϕ: D → D defines a linear composition operator Cϕ on the space of all analytic functions on the unit disk by the formula Cϕf = f ◦ ϕ. The analytic map ϕ is usualy called the symbol of Cϕ. It is a consequence of Littlewood’s subordination principle [30] that every composition operator Cϕ restricts to a bounded linear operator on H2(D). Recall that a linear fractional map ϕ on the Riemann sphere Cb := C ∪ {∞} is any map of the form az + b ϕ(z) = , cz + d where the coefficients a, b, c, d ∈ C satisfy the condition ad − bc 6= 0, just to ensure that ϕ is not constant, and where the standard conventions are applied to the point at infinity. We are interested here in those linear fractional maps that take the unit disk into itself. Every such map has either one or two fixed points. Those with one fixed point are called parabolic. It can be shown that such a fixed point must be on the unit circle. In the remaining cases, the linear fractional map ϕ has two fixed points, one of which must be in the closed unit disk. If it is on ∂D then ϕ is said to be hyperbolic. If it is in the open unit disk, the other one has to lie is in the complement in the Riemann sphere of the closed unit disk. Here we distinguish between two different situations. If ϕ is an automorphism, then ϕ is said to be elliptic. Otherwise, ϕ is said to be loxodromic. We refer the reader to the more detailed presentation in the article by Shapiro [31]. Throughout the paper we will use the facts from [31] about the canonical models for linear fractional maps and the spectral properties of the associated composition operators. The problem of determining whether an operator A has a minimal commutant is not new. Sarason [29] showed that the Volterra operator has a minimal commutant. Erdos [10] obtained a simple, direct proof of this result. Shields and Wallen [34] showed that if A is an injective unilateral shift or if A is the discrete Cesàrooperator then A has a minimal commutant. Shields [33] also proved that if A ∈ B(H) is a non invertible, injective bilateral weighted shift then A has a minimal commutant. COMPOSITION OPERATORS WITH A MINIMAL COMMUTANT 3

Cowen and MacCluer [6] raised the question: when is the commutant of Cϕ equal to the weakly closed algebra generated by those multiplications and composition operators that commute with Cϕ. Cload showed in [4,5] that this is the case when ϕ is an elliptic automorphism of the unit disk and ϕ(0) = 0. He also proved in [5] that, in this situation, the only bounded analytic functions that induce multiplication operators in the commutant of Cϕ are the constant functions. Hence, it follows from his work that when ϕ is a non periodic elliptic automorphism of the unit disk, the corresponding composition operator Cϕ has a minimal commutant. We present in Section 3 a direct proof of this result. Worner [35] showed that, if ϕ is a hyperbolic non-automorphism, the composition operator Cϕ fails to have a minimal commutant. She exhibited two classes of operators that commute with Cϕ but do not belong to the weak closure of the unital algebra generated by Cϕ. First, multiplication operators induced by non-constant, bounded analytic functions, and second, composition operators Cψ, where ψ is a symbol such that ϕ = ψ ◦ ψ. We present a simplified proof of her result in Section 5. She also described a procedure for constructing composition operators that fail to have a minimal commutant, and she pointed out that one of the reasons for this is that the symbol fails to be univalent. She finally asked the question whether a compact composition operator with univalent symbol must have a minimal commutant. We provide in Section 7 a negative answer to this question. Linchuck [20, 21] provided a description of the commutant of a composition operator Cϕ in the cases that ϕ is a hyperbolic or a parabolic automorphism. Although his characterization of the commutants of such composition operators is quite involved, it could perhaps be used to construct examples of operators in the commutant of Cϕ but not in the weak closure of the unital algebra generated by Cϕ. The paper is organized as follows. In Section 1 we provide four general results. The first one is a procedure for constructing multiplication 0 σ operators that belong to {Cϕ} but not to alg (Cϕ) . This procedure was described earlier by Worner [35, Theorem 9], under the more restrictive hypotheses that ϕ is not an automorphism and all its iteration sequences are interpolating. The other three results will establish that the property of having a minimal commutant is closed under the restrictions to reducing subspaces, under similarities and the operation of taking adjoints. In Section 2 we consider the case when ϕ is a parabolic or a hyperbolic automorphism. We will show 0 that in either case the commutant {Cϕ} is not minimal. We present two different proofs of these results 0 in order to shed more light on the structure of {Cϕ} . In Section 3 we consider the case of an elliptic self-map ϕ of the unit disk and we prove first, that 0 σ 0 in the periodic case, {Cϕ} fails to be commutative, so that alg (Cϕ) 6= {Cϕ} , and second, that in the σ 0 non-periodic case, we have alg (Cϕ) = {Cϕ} . Our proof of the latter result relies on a classical extension theorem of Rudin and Carleson. In Section 4 we consider the case of a composition operator induced by a parabolic non-automorphism ϕ of the unit disc, and we prove that its commutant is minimal. The proof is based on a remarkable result ∗ of Montes-Rodr´ıguez,Ponce-Escudero and Shkarin [24], that the adjoint Cϕ is similar to a multiplication operator by a cyclic element on the Sobolev space W 1,2[0, +∞). Our contribution is the proof that the operator of multiplication by a cyclic element in an abstract Banach algebra with an approximate identity has a minimal commutant. The fact that the Sobolev space is a non-unital Banach algebra with an approximate identity is well-known. However, we are not aware of a simple proof, so we offer one in the Appendix to this paper. In Section 5 we consider a composition operator Cϕ induced by a hyperbolic non-automorphism ϕ of the unit disk and we show that its commutant is not minimal. Worner [35] has obtained the same result by studying the case when the iterates of ϕ form an interpolating sequence. Our proof does not require the notion of an interpolating sequence and it is simpler. In Section 6 we consider the case of a composition operator induced by a univalent, analytic self-map ϕ of the unit disk that fixes the origin and that is not necessarily a linear fractional map, but in exchange 4 MIGUEL LACRUZ, FERNANDO LEON-SAAVEDRA,´ SRDJAN PETROVIC, AND LUIS RODRÍGUEZ-PIAZZA its Königs’sdomain is bounded and stricly starlike with respect to the origin, and we show that the operator Cϕ has a minimal commutant. The loxodromic case becomes a particular instance of this result since the range of the Königsfunction is an open disk that contains the origin. In Section 7 we construct an example of a univalent, analytic self-map ϕ of the unit disk such that Cϕ is compact but fails to have a minimal commutant. This answers a question raised by Worner [35]. Finally, in the Appendix we prove the facts about the Sobolev space W 1,2[0, +∞) that were used in Section 4. They can be found in the book [1], where they are stated in a much more general way. Consequently, their proofs are more complicated and we felt that the proofs of these simpler statements would help the exposition.

1. Some general results about operators with a minimal commutant We present in this section some general results that are interesting for our purposes. The ﬁrst one deals with those multiplication operators that commute with a composition operator induced by an analytic self-map ϕ of the unit disk, not necessarily a linear fractional map. This result had been obtained earlier by Worner [35, Theorem 9] under the additional assumptions that ϕ is not an automorphism and all its iteration sequences are interpolating.

Theorem 1.1. Let ϕ be a non-constant, analytic self-map of the unit disk, let b ∈ H∞(D) such that 2 b ◦ ϕ = b, and consider the multiplication operator Mb ∈ B(H (D)) defined by the expression Mbf = b · f. 0 σ Then Mb ∈ {Cϕ} , and if in addition b is non-constant, then Mb ∈/ alg (Cϕ) . 2 0 Proof. For any f ∈ H (D), we have CϕMbf = (b ◦ ϕ) · (f ◦ ϕ) = b · (f ◦ ϕ) = MbCϕf, so Mb ∈ {Cϕ} . Next, if b is non-constant then b and b2 are linearly independent, so that there exists g ∈ H2(D) such that hb, gi = 0 and hb2, gi = 1. Let us define a linear functional Λ: B(H2(D)) → C by the expression Λ(X) = hXb, gi. It is clear that Λ is continuous in the weak operator topology, and therefore, the subspace ker Λ is closed in the weak operator topology. We have Λ(p(Cϕ)) = hp(Cϕ)b, gi = p(1)hb, gi = 0, for every polynomial p, σ 2 and therefore alg (Cϕ) ⊆ ker Λ. On the other hand, Λ(Mb) = hMbb, gi = hb , gi = 1, so that Mb ∈/ ker Λ, σ and it follows that Mb ∈/ alg (Cϕ) , as we wanted. Remark 1.2. Cload [5, Theorem 3 and Theorem 4] showed that, if ϕ is an analytic self-map of the unit disk that is not a periodic elliptic automorphism and fixes a point in the unit disk, then the only bounded analytic functions b ∈ H∞(D) that satisfy b ◦ ϕ = b are the constant functions. His proof is a nice application of the Denjoy-Wolff theorem. Our next result ensures that if a direct sum of Hilbert space operators has a minimal commutant then both summands inherit that property.

Lemma 1.3. Let us consider a direct sum H = H1 ⊕H2, let Aj ∈ B(Hj) for j = 1, 2, and let A = A1 ⊕A2. If A has a minimal commutant, then so do both A1 and A2. 0 Proof. Let Xj ∈ {Aj} , for j = 1, 2, and let X = X1 ⊕ X2. It is easy to see that AX = XA. Since A has a minimal commutant, there is a net of polynomials (pd)d∈D such that pd(A) → X as d ∈ D in the weak operator topology. Then, it is clear that pd(A) = pd(A1) ⊕ pd(A2), and it follows that pd(Aj) → Xj as d ∈ D in the weak operator topology, for j = 1, 2, as we wanted. The next two results in this section ensure that the property for a Hilbert space operator of having a minimal commutant is preserved under similarities and under the operation of taking adjoints. Lemma 1.4. If A ∈ B(H) has a minimal commutant, S is an invertible operator and B = S−1AS, then B has a minimal commutant, too. COMPOSITION OPERATORS WITH A MINIMAL COMMUTANT 5

Proof. Let Y ∈ {B}0 and let X = SYS−1. Then X ∈ {A}0, and since A has a minimal commutant, there is a net (pd)d∈D of polynomials such that pd(A) → X as d ∈ D in the weak operator topology. It follows −1 that pd(B) = S pd(A)S converges in the weak operator topology to Y, as we wanted. Lemma 1.5. An operator A ∈ B(H) has a minimal commutant if and only if so does its adjoint A∗. Proof. Clearly, it suﬃces to establish the implication that if A has a minimal commutant, then so does A∗. ∗ 0 ∗ 0 ∗ Let X ∈ {A } . We have X ∈ {A} , so there exists a net of polynomials (pd)d∈D such that pd(A) → X as d ∈ D in the weak operator topology. Since the operation of taking adjoints is continuous with respect ∗ to the weak operator topology, it follows that pd(A) → X as d ∈ D in the weak operator topology. Finally, consider the net of polynomials (qd)d∈D deﬁned by the expression

qd(z) = pd( z ). σ ∗ ∗ ∗ We get qd(A ) = pd(A) → X as d ∈ D in the weak operator topology, whence X ∈ alg (A ) , and the proof is complete. We end this section with a folk result about the commutant of an operator with a simple eigenvalue, that is easy to prove and is very handy for the study of operators with a minimal commutant.

Lemma 1.6. Let A ∈ B(H) and α ∈ C such that dim ker(A−α) = 1. If X ∈ {A}0 and f ∈ ker(A−α)\{0} then there exists λ ∈ C such that Xf = λf and |λ| ≤ kXk. Proof. Notice that AXf = XAf = αXf, so Xf ∈ ker(A − α). Since dim ker(A − α) = 1, there exists λ ∈ C such that Xf = λf. Finally, kXfk = |λ| · kfk, and since f 6= 0 it follows that |λ| ≤ kXk.

2. Composition operators induced by parabolic and hyperbolic automorphisms Our goal in this section is to show that, when ϕ is either a parabolic or a hyperbolic automorphism, the commutant of Cϕ is not minimal. We present two different proofs. The first one is based on the idea of a strongly compact algebra, and it is an immediate consequence of a theorem by Shapiro [31], while the second one is an application of Theorem 1.1. We say that a subalgebra A ⊆ B(H) is strongly compact provided that its unit ball {A ∈ A: kAk ≤ 1} is precompact in the strong operator topology. This notion was introduced by Lomonosov [22] in connection with the invariant subspace problem for essentially normal operators. An example of a strongly compact algebra is the commutant of a compact operator with dense range. Another example is the commutant of the adjoint of the shift operator on the Hardy space. Marsalli [25] provided a classification of operator algebras and characterized those self-adjoint subal- gebras of operators on Hilbert space that are strongly compact. Lomonosov and the first and fourth authors [16] established a number of results and constructed many examples and counter-examples concerning strongly compact algebras. The first and fourth authors [17] studied the class of strongly compact, normal operators, obtaining results closely related to those of Froelich and Marsalli [12], and they provided further examples and counter-examples of strongly compact algebras. They also considered the issue of strong compactness for algebras of analytic functions [18] and they strentghened earlier results of Froelich and Marsalli [12]. Romero de la Rosa and the first author [19] provided a local spectral condition for strong compactness and they applied it to the class of bilateral weighted shifts. 0 Shapiro [31] studied the strong compactness of the algebras alg (Cϕ) and {Cϕ} , where Cϕ is a composition operator induced on H2(D) by a linear fractional self-map ϕ of the unit disk, completing earlier work of Fernández-Valles and the first author [11]. Marsalli [25] provided a sufficient condition for strong compactness that is interesting for our purposes and that can be stated as follows. 6 MIGUEL LACRUZ, FERNANDO LEON-SAAVEDRA,´ SRDJAN PETROVIC, AND LUIS RODRÍGUEZ-PIAZZA

Theorem 2.1. If a subalgebra A ⊆ B(H) admits a collection of finite-dimensional subspaces invariant under A, whose union is a total subset of H, then the algebra A is strongly compact. Shapiro [31, p. 854] observed that if a subalgebra A ⊆ B(H) meets the hypotheses of Theorem 2.1 σ then so does its closure in the weak operator topology, and therefore the closure A is strongly compact. He also pointed out the following immediate consequence of this observation. σ Theorem 2.2. If an operator A ∈ B(H) has a total set of eigenvectors then alg (A) is strongly compact. This result applies to most composition operators induced by linear fractional self-maps of the unit disk, with the only exception when the symbol is a hyperbolic non-automorphism with a fixed point on ∂D and the other one in D. On the other hand, Shapiro [31, Theorem 4.1.1] proved the following result. 0 Theorem 2.3. If ϕ is a parabolic or hyperbolic automorphism of the unit disk, then the algebra {Cϕ} fails to be strongly compact. The main result in this section is an immediate consequence of Theorem 2.2 and Theorem 2.3. Theorem 2.4. If ϕ is either a parabolic or a hyperbolic automorphism of the unit disk then the operator Cϕ fails to have a minimal commutant. There is a simple, direct proof of Theorem 2.4 that does not require the notion of strong compactness. Instead, it is based on the canonical models for parabolic and hyperbolic automorphisms. These canonical models are explained in detail in the paper [31] of Shapiro. Recall that a linear fractional map is called parabolic if it has precisely one fixed point, which is necessarily on the unit circle. Using conjugation by disk automorphisms, we may assume without loss of generality that the fixed point is 1. The map ϕ then has the canonical form given by the expression (2 − a)z + a ϕ(z) = , −az + 2 + a where a ∈ C is a constant with Re(a) ≥ 0. Further, ϕ is an automorphism if and only if Re(a) = 0. Also, for each σ ≥ 0, the bounded analytic function 1 + z e (z) = exp −σ σ 1 − z −σa −σa −σa satisfies the equation Cϕeσ = e eσ, and hence eσ ◦ ϕ = e eσ. When σ = 2π/|a|, we have e = 1, so the desired result follows from Theorem 1.1. Next, recall that a fractional linear map ϕ is a hyperbolic automorphism if it has two fixed points and both are on ∂D. Without loss of generality, we may assume that these points are 1 and −1, in which case ϕ has the canonical form z + r ϕ(z) = , for some 0 < r < 1. 1 + rz For every t ∈ R, the function 1 + z it e (z) = it 1 − z 2 belongs to H (D), it is bounded and it is an eigenfunction for Cϕ corresponding to the eigenvalue 1 + r it 1 + r λ(it) = = exp it log . 1 − r 1 − r Now, choose t ∈ R such that 1 + r t log = 2π, 1 − r and let b = eit. Then, λ(it) = 1, so that b ◦ ϕ = b, and since b is not constant, the desired result follows once again from Theorem 1.1. COMPOSITION OPERATORS WITH A MINIMAL COMMUTANT 7

3. The commutant of a composition operator induced by an elliptic automorphism In this section we consider composition operators induced by elliptic automorphisms of the unit disk. Recall that a fractional linear self-map of the unit disk ϕ is said to be elliptic if it has one fixed point in D and the other is outside of D. Since ϕ is an automorphism, it maps the circle to itself. Without loss of generality we may assume that ϕ(0) = 0, so by reflection ϕ(∞) = ∞. Therefore, ϕ is a rotation and it has the canonical form ϕ(z) = ωz, for some ω ∈ ∂D\{1}. We will show that the commutant of Cϕ is minimal if and only if ω is not a root of unity. n We know that the point spectrum of Cϕ is the set {ω : n ∈ N0} with the corresponding eigenfunctions the monomials zn. If ω is a root of unity, that is, if there is an m ≥ 1 such that ωm = 1, then there are finitely many eigenvalues for Cϕ of infinite multiplicity, namely, the m-th roots of unity, and the n corresponding eigenspaces are given by Hk = span{z : n ≡ k (mod m)}, 1 ≤ k ≤ m. If ω is not a root of unity then ωn 6= ωm for all n 6= m, so that there is a countable family of eigenvalues that is dense in the unit circle. Also, the eigenvalues are all simple. Let us denote by A(D) the disk algebra. It is a standard fact that A(D) coincides with the uniform closure of the polynomials; see for instance the book of Hoffman [14]. We shall use the following classical extension theorem of Rudin [28] and Carleson [3]. Theorem 3.1 (Rudin-Carleson). Let E ⊆ ∂D be a closed set of Lebesgue measure zero and let f : E → C be a continuous function. Then there exists g ∈ A(D) such that g|E = f and kgkD ≤ kfkE. The next proposition establishes a result that is slightly more general than our target in this section.

Proposition 3.2. Let H be complex, separable, infinite-dimensional Hilbert space and let {en : n ∈ N} be an orthonormal basis of H. Also, let (dn) be a sequence in ∂D with dn 6= dm for all n 6= m, and let σ 0 D ∈ B(H) be the diagonal operator defined by Den = dnen for all n ∈ N. Then alg (D) = {D} . 0 Proof. It is a folklore result that {D} is the family of all diagonal operators with respect to {en : n ∈ N}. 0 Indeed, if X ∈ {D} , it follows from Lemma 1.6 that Xen = λnen for some bounded sequence (λn) of complex scalars. Let n ∈ N be fixed. Consider the finite set En = {d1, d1, . . . , dn}, and the continuous function fn : En → C, defined by the expression fn(dk) = λk, for every k = 1, 2, . . . , n. It follows from Theorem 3.1 that there exists a function gn ∈ A(D) such that gn(dk) = λk, for every k = 1, 2, . . . , n, and such that kgnkD ≤ k(λk)k∞. Next, there is a polynomial pn such that kpn − gnkD < 1/n. We claim that the sequence (pn(D)) converges in the weak operator topology to X. According to the book of Dunford and Schwartz [9], it is enough to check that (pn(D)) is bounded and it converges pointwise in norm to X on {ek : k ∈ N}. To that end, for every k ∈ N we have pn(D)ek = pn(dk)ek, so that pn(D) is a diagonal operator with diagonal sequence (pn(dk))k∈N. Thus, for every n ∈ N, we have

kpn(D)k = sup |pn(dk)| k∈N

≤ kpnkD

≤ kgnkD + kpn − gnkD ≤ k(λk)k∞ + 1.

This shows that the sequence (pn(D)) is bounded. Moreover, for all k ∈ N and for every n ≥ k we get

kpn(D)ek − Xekk = |pn(dk) − λk|

≤ |pn(dk) − gn(dk)| + |gn(dk) − λk|

= |pn(dk) − gn(dk)| + 0

≤ kpn − gnkD → 0, as n → ∞. 8 MIGUEL LACRUZ, FERNANDO LEON-SAAVEDRA,´ SRDJAN PETROVIC, AND LUIS RODR´IGUEZ-PIAZZA

Thus, we arrived at the conclusion that the sequence (pn(D)) converges in the strong operator topology, and therefore in the weak operator topology, to the operator X, as we wanted.

Theorem 3.3. Let ω ∈ ∂D\{1} and let ϕ(z) = ωz, for all z ∈ D. 0 0 σ (1) If ω is a root of unity then {Cϕ} is not commutative, and in particular, {Cϕ} 6= alg(Cϕ) . 0 σ (2) If ω is not a root of unity then {Cϕ} = alg(Cϕ) . Proof. First we consider the case when the multiplier is a root of unity, that is, there is an m ≥ 1 such m that ω = 1. Therefore, there are finitely many eigenvalues for Cϕ of infinite multiplicity, namely, the m-th roots of unity, and the corresponding infinite-dimensional eigenspaces are given by n Hk = span{z : n ≡ k (mod m)}, 1 ≤ k ≤ m. 2 Hence, there is a decomposition of H (D) as an orthogonal direct sum of reducing subspaces for Cϕ, 2 H (D) = H1 ⊕ · · · ⊕ Hm.

Notice that the restriction of Cϕ to the subspace {0}⊕· · ·⊕{0}⊕Hk ⊕{0}⊕· · ·⊕{0} is a scalar operator. 0 0 It follows easily that {Cϕ} = B(H1) ⊕ · · · ⊕ B(Hm), and in particular, {Cϕ} fails to be commutative. The case when the multiplier ω is not a root of unity follows from Proposition 3.2 applied to Hilbert 2 n space H = H (D), the orthonormal basis en(z) = z , the diagonal operator D = Cϕ and the diagonal n sequence (dn) deﬁned by dn = ω . Remark 3.4. Notice that, in the periodic case, the commutant is even bigger than the algebra generated by Cϕ and the multiplications that commute with it.

4. The commutant of a composition operator induced by a parabolic non-automorphism In this section we will demonstrate that, if ϕ is a parabolic non-automorphism of the unit disk, then 2 Cϕ has a minimal commutant. Our path to this result is based on an identiﬁcation between H (D) and the Sobolev space W 1,2[0, +∞). As we mentioned in Section 2, one may assume that a parabolic fractional linear map is of the form (2 − a)z + a ϕ(z) = , for some a ∈ with Re (a) ≥ 0. (4.1) −az + (2 + a) C

Further, ϕ is not an automorphism if and only if Re (a) > 0. The family of eigenvalues for Cϕ is the −aσ set σp(Cϕ) = {e : σ ≥ 0} and every eigenvalue is simple. An eigenfunction eσ corresponding to an eigenvalue e−σa with σ ≥ 0 is the bounded analytic function 1 + z e (z) = exp −σ . (4.2) σ 1 − z Recall that the Sobolev space W 1,2[0, +∞) consists of all functions f ∈ L2[0, +∞) that are absolutely continuous on each bounded subinterval of the interval [0, +∞) and such that f 0 ∈ L2[0, +∞). It is easy to prove the standard fact that W 1,2[0, +∞) is Hilbert space provided with the inner product Z +∞ h i hf, gi: = f(σ)g(σ) + f 0(σ)g0(σ) dσ. 0

The corresponding norm is denoted by k · k1,2. Alfonso Montes-Rodr´ıguez, Manuel Ponce-Escudero and Stanislav Shkarin [24] established the re- ∗ markable fact that the adjoint Cϕ is similar to a certain multiplication operator on the Sobolev space W 1,2[0, +∞). More precisely, they proved the following theorem. COMPOSITION OPERATORS WITH A MINIMAL COMMUTANT 9

Theorem 4.1. Let ϕ be a parabolic non-automorphism of the unit disk in the canonical form (4.1) and let {eσ : σ ≥ 0} be the family of eigenfunctions for Cϕ given by equation (4.2). The linear map S 2 1,2 deﬁned by the expression (Sf)(σ): = hf, eσi is an isomorphism from H (D) onto W [0, +∞). Moreover, ∗ −1 −aσ 1,2 Cϕ = S MψS, where Mψ denotes the operator of multiplication by ψ(σ) := e on W [0, +∞). n Finally, the function ψ is a cyclic vector for the operator Mψ, that is, the family {ψ : n ∈ N} is a total subset of W 1,2[0, +∞).

Theorem 4.1 shows that, in order to establish the results about the operator Cϕ, one should focus 1,2 on those multiplication operators Mψ on W [0, +∞), for which the function ψ is a cyclic vector. This strategy will require some facts about Sobolev spaces which can be found, in a higher generality, in the book of Adams [1]. Since we will need only some special cases, we will state them throughout the section in that manner. For completeness, we will provide their proofs in the Appendix. The ﬁrst such result is that, under a suitable renorming, the Sobolev space W 1,2[0, +∞) provided with pointwise multiplication becomes a commutative Banach algebra without identity. More precisely, √ 1,2 1,2 Theorem 4.2. If f, g ∈ W [0, +∞) then fg ∈ W [0, +∞), and moreover, kfgk1,2 ≤ 2kfk1,2kgk1,2. 1,2 Hence, the Sobolev space W [0, +∞) is a commutative Banach√ algebra without identity with respect to ∗ pointwise multiplication and the equivalent norm kfk1,2 = 2kfk1.2. We will prove Theorem 4.2 in the Appendix. The proof of a more general result can be found in [1, Theorem 5.23]. The key for the proof of Theorem 4.2 is the so-called Sobolev embedding theorem, that can be stated as follows.

1,2 ∞ Theorem 4.3. If f ∈ W [0, +∞), we have f ∈ L [0, +∞), and moreover, kfk∞ ≤ kfk1,2. Once again, we will provide the proof in the Appendix, and refer the reader to [1, Corollary 5.16] for a proof of a more general result. 1,2 Our goal is to show that the operator Mψ, acting on W [0, +∞), has a minimal commutant, whenever ψ is a cyclic vector. We will, in fact, show that this is true on a large class of Banach algebras. Let A be a Banach algebra, let a ∈ A, and consider the multiplication operator Ma defined on A by the expression Mab = ab. Notice that Ma is a bounded linear operator with kMak ≤ kak. An element n a ∈ A is said to be cyclic when it is a cyclic vector for Ma, that is, when the family {a : n ∈ N} is a total subset of A. Notice that if a Banach algebra A has a cyclic element then A is abelian and separable. Remark 4.4. We stated in the Introduction the definition of an operator on Hilbert space with a minimal commutant, and the same definition makes sense for an operator on a general Banach space, and in particular, for a multiplication operator on a Banach algebra. We start our program by considering the case when A has an identity. Theorem 4.5. If A is a unital Banach algebra and a ∈ A is an element such that {1, a, . . . , am,...} is a total subset of A, then the multiplication operator Ma has a minimal commutant.

0 σ n n Proof. Let X ∈ {Ma} . We will show that X ∈ alg (Ma) . It follows by induction that Ma X = XMa , for all n ∈ N. Now, since the family {1, a, . . . , am,...} is a total subset of A, there exists a sequence of polynomials (pn) such that

lim kpn(a) − X1k = 0. (4.3) n→∞ σ We claim that (pn(Ma)) converges in the strong operator topology to X, and therefore X ∈ alg (Ma) . Now, for the proof of our claim, the sequence of vectors (pn(a)) is bounded because it is convergent, so that there is a constant c > 0 such that kpn(a)k ≤ c, for all n ∈ N. Next, the sequence of operators

(pn(Ma)) is also bounded, because kpn(Ma)k = kMpn(a)k ≤ kpn(a)k ≤ c. Finally, in order to show that 10 MIGUEL LACRUZ, FERNANDO LEON-SAAVEDRA,´ SRDJAN PETROVIC, AND LUIS RODR´IGUEZ-PIAZZA

(pn(Ma)) converges to X in the strong operator topology, it is enough to prove the convergence on the total set {1, a, . . . , am,...}. Using (4.3) we see that

kpn(Ma)1 − X1k = kMpn(a)1 − X1k = kpn(a) − X1k → 0, as n → ∞. Also, for every m ∈ N, we have m m m m m m kpn(Ma)a − Xa k = kpn(a)a − XMa 1k = ka pn(a) − Ma X1k m m m = ka pn(a) − a X1k ≤ ka k · kpn(a) − X1k → 0, as n → ∞, and this completes the proof of our claim. 0 Remark 4.6. Notice that, under the hypotheses of Theorem 4.5, we have {Ma} = {Mb : b ∈ A}. 0 Indeed, it is clear that the subalgebra on the right hand side is contained in {Ma} . On the other hand, 0 if X ∈ {Ma} then we know from Theorem 4.5 that there exists a sequence of polynomials (pn) such that pn(Ma) → X, as n → ∞, in the strong operator topology. Let b = X1. We claim that X = Mb. Indeed, we have kpn(a) − bk = kpn(Ma)1 − X1k → 0, as n → ∞, and it follows that pn(Ma) → Mb in the strong operator topology, whence X = Mb. Theorem 4.5 cannot be applied to W 1,2[0, +∞) because the latter is a non-unital Banach algebra. In order to extend Theorem 4.5 we start with the following lemma. Lemma 4.7. If A is a non-unital Banach algebra and a ∈ A is a cyclic element then σ σ alg (Ma) = {Mb + βI : b ∈ A, β ∈ C} .

Proof. It is clear that the family {Mb + βI : b ∈ A, β ∈ C} is a subalgebra of B(A) that contains Ma and σ σ the identity operator, so that it also contains alg (Ma). Hence, alg (Ma) ⊆ {Mb + βI : b ∈ A, β ∈ C} . Conversely, if b ∈ A and β ∈ C, then there is a sequence of polynomials (pn) such that pn(0) = 0 and kb − pn(a)k → 0, as n → ∞. If we set qn(z) = pn(z) + β, then qn(Ma) = Mpn(a) + βI → Mb + βI, as σ n → ∞, in the strong operator topology. This shows that {Mb + βI : b ∈ A, β ∈ C} ⊆ alg (Ma) , and it σ σ ﬁnally follows that {Mb + βI : b ∈ A, β ∈ C} ⊆ alg (Ma) , as we wanted.

Recall that an approximate identity in a Banach algebra A is any sequence (en) such that kenb−bk → 0, as n → ∞, for every b ∈ A. This means that Men → I, as n → ∞, in the strong operator topology. Theorem 4.8. If A is a commutative Banach algebra with an approximate identity and a ∈ A is a cyclic element, then the multiplication operator Ma has a minimal commutant.

Proof. First, let (en) be an approximate identity. Since a is cyclic, there is a sequence of polynomials (pn) such that pn(0) = 0 and ken − pn(a)k → 0, as n → ∞. We have, for every b ∈ A,

kpn(a)b − bk ≤ kpn(a)b − enbk + kenb − bk

≤ kbk · kpn(a) − enk + kenb − bk → 0, as n → ∞, 0 so that (pn(a)) is also an approximate identity. We claim that, for every X ∈ {Ma} , we have XMa = Mb, n n n n n n where b = Xa. Indeed, for every n ∈ N, we get XMaa = XMa a = Ma Xa = a b = ba = Mba , and since a is cyclic, the claim follows. Next, since pn(0) = 0, we have pn(z) = zqn(z), for some polynomial 0 qn. Then, let X ∈ {Ma} . We get Xpn(Ma) = XMaqn(Ma) = Mbqn(Ma) so that Mbqn(Ma) → X, as σ n → ∞, in the strong operator topology. This shows that X ∈ {Mc + γI : c ∈ A, γ ∈ C} , and the desired result ﬁnally follows from Lemma 4.7. The signiﬁcance of Theorem 4.8 comes from the fact that the Sobolev algebra has an approximate identity. COMPOSITION OPERATORS WITH A MINIMAL COMMUTANT 11

Lemma 4.9. The sequence of functions (en) deﬁned by the expression 1, if 0 ≤ σ ≤ n, e (σ) = n e−(σ−n), if σ ≥ n, is an approximate identity for the Sobolev algebra W 1,2[0, +∞). We will prove Lemma 4.9 in the Appendix. Now, everything is ready for the main result of this section.

2 Theorem 4.10. If Cϕ is a composition operator on H (D), induced by a parabolic non-automorphism ϕ of the unit disk, then Cϕ has a minimal commutant. Proof. We may assume without loss of generality that the map ϕ has the canonical form as in (4.1). ∗ By Lemma 1.5, it suffices to show that the adjoint operator Cϕ has a minimal commutant. We know ∗ from Theorem 4.1 that Cϕ is similar to the multiplication operator Mψ on the Sobolev space, where −aσ ψ(σ) = e . Next, according to Lemma 1.4, it suffices to show that Mψ has a minimal commutant. We 1,2 know from Theorem 4.2 that the Sobolev space W [0, +∞) is a commutative Banach√ algebra without ∗ identity with respect to pointwise multiplication and the equivalent norm kfk1,2 = 2kfk1.2, and that the function ψ is a cyclic element. We also know from Lemma 4.9 that the Banach algebra W 1,2[0, +∞) has an approximate identity. It finally follows from Theorem 4.8 that Mψ has a minimal commutant. 5. The commutant of a composition operator induced by a hyperbolic non-automorphism Let us consider a composition operator induced on the Hardy space by a hyperbolic non-automorphism of the unit disk. Worner [35] showed that Cϕ fails to have a minimal commutant using the notion of an analytic self-map ϕ of the unit disk whose iterates (zn), defined as zn+1 = ϕ(zn), are interpolating. We present in this section a direct approach to this issue that is based on the ideas of Worner but that does not require the notion of an interpolating sequence. Recall that ϕ is a hyperbolic non-automorphism if it has one fixed point on ∂D and the other one does not belong to ∂D. There are two possibilities, depending on whether the second fixed point belongs to to Cb\D or to D. In the former case, the fixed points can be assumed to be 1 and ∞, and the symbol has the form ϕ(z) = rz + (1 − r), for some 0 < r < 1. Deddens [8, Theorem 3 (iv)] showed that, in this case, the point spectrum of Cϕ is the punctured disk −1/2 σp(Cϕ) = {λ ∈ C: 0 < |λ| < r }. s He also noticed that for every s ∈ C with Re(s) > −1/2, the function es defined by es(z) = (1 − z) belongs to the Hardy space H2(D) and it satisfies the functional equation s (Cϕes)(z) = r es(z), s so that es is an eigenfunction of Cϕ corresponding to the eigenvalue r . In particular, if we choose t ∈ R it such that t log r = 2π, we get r = 1, so that b = eit is a non-constant, bounded analytic function such that b ◦ ϕ = b. Applying Theorem 1.1 we get

Theorem 5.1. If ϕ is a linear fractional self-map of the unit disk that fixes a point on ∂D and the other 0 σ one in Cb\D, then {Cϕ} 6= alg (Cϕ) . The other possibility, that the second fixed point is in D, is a bit more delicate, because according to Remark 1.2, all the multiplication operators in the commutant of Cϕ are scalar multiples of the identity, so that the strategy we used in the former case does not work here. In this case the fixed points can be assumed to be 1 and 0, and the symbol has the canonical form rz ϕ(z) = , for some 0 < r < 1. (5.1) 1 − (1 − r)z 12 MIGUEL LACRUZ, FERNANDO LEON-SAAVEDRA,´ SRDJAN PETROVIC, AND LUIS RODRÍGUEZ-PIAZZA

Theorem 5.2. If ϕ is a fractional linear self-map of the unit disk that fixes a point on ∂D and the other 0 σ one in D, then {Cϕ} 6= alg (Cϕ) . Proof. Without loss of generality we will assume that ϕ is given by (5.1). The following construction appears in the paper of Shapiro [31, p. 864] and some ideas go back to his joint paper with Bourdon [2]. 2 2 2 Let H0 (D) := {f ∈ H (D): f(0) = 0} and let h1i be the subspace of H (D) consisting of the constant 2 2 functions. Thus, there is an orthogonal decomposition H (D) = h1i ⊕ H0 (D), where both summands are 2 2 invariant under Cϕ. Now, let ψ(z) = rz +(1−r). Then, the unitary operator U : H0 (D) → H (D) defined by (Uf)(z) = z−1f(z) satisfies (U ∗f)(z) = zf(z) and ∗ ∗ (Cϕ| 2 ) = U (rCψ)U, H0 (D) 2 where Cψ is defined on H (D). Since the restriction of Cϕ to the subspace h1i is the identity operator I1, ∗ 2 it follows that Cϕ is unitarily equivalent to I1 ⊕ rCψ, acting on h1i ⊕ H (D). We know from Theorem 5.1 that Cψ, and therefore rCψ fail to have a minimal commutant. Also, it follows from Lemma 1.3 that I1 ⊕ rCψ fails to have a minimal commutant. Finally, it follows from Lemma 1.4 and Lemma 1.5 that Cϕ does not have a minimal commutant, and the proof is complete. Remark 5.3. As a byproduct of Theorem 5.1, the Euler operator does not have a minimal commutant. 2 Recall that the Euler operator Er, where 0 < r < 1, is defined on Hilbert space ` by n X n (E f)(n): = rk(1 − r)n−kf(k). r k k=0

Deddens [8, p. 862] proved that Er is unitarily equivalent to the adjoint of the composition operator Cϕ where ϕ(z) = rz + (1 − r). Hence, Er fails to have a minimal commutant, too.

6. The commutant of a composition operator induced by a loxodromic mapping

In this section we will demonstrate that, if ϕ is a loxodromic linear fractional map, then Cϕ has a minimal commutant. To accomplish this goal we will establish a more general result. Namely, we will assume that ϕ is a univalent, analytic self-map of the unit disk, with a fixed point in D, but not necessarily a linear fractional map. The additional constraints will be imposed, instead, on the range of the associated Königsfunction σ. We will show that, in that case, Cϕ has a minimal commutant (Theorem 6.1 below). In the special case when ϕ is a loxodromic map, we will verify that all the hypotheses of Theorem 6.1 are satisfied, so it will follow that Cϕ has a minimal commutant (Corollary 6.2 below). Let ϕ be a univalent, analytic self-map ϕ of the unit disk, with a fixed point in D. Without loss of generality, we will assume that ϕ(0) = 0. Let us denote α = ϕ0(0). Notice that α 6= 0 because ϕ is a univalent analytic function that fixes the origin, hence it has a non-vanishing derivative at the origin. We will also assume that ϕ is not an automorphism of the unit disk, so that 0 < |α| < 1. Königs[15] showed that there is a univalent, analytic function σ : D → C such that σ ◦ ϕ = ασ and, therefore, σ(0) = 0. The function σ is unique up to a constant factor and it is known as the Königsfunction of ϕ. Moreover, if λ ∈ C and f : D → C is an analytic function such that f ◦ ϕ = λf, then there exists n ∈ N0 such that n n λ = α and f is a constant mutiple of σ . In the language of operators, all eigenspaces of Cϕ are one dimensional. Notice that G = σ(D) is a simply connected domain such that αG ⊆ G. Since σ(0) = 0, we have 0 ∈ G. Also, notice that σ : D → G is a conformal mapping with the nice property that −1 ϕ(z) = σ (ασ(z)), for all z ∈ D. (6.1) Recall that a set S ⊆ C is said to be starlike with respect to the origin provided that rS ⊆ S, COMPOSITION OPERATORS WITH A MINIMAL COMMUTANT 13 for all 0 ≤ r ≤ 1. Notice that if S ⊆ C is starlike with respect to the origin, then it is connected because, in fact, it is pathwise connected. Also, recall that a set S ⊆ C is said to be strictly starlike with respect to the origin if rS ⊆ S, for all 0 ≤ r < 1. The main result of this section can be stated as follows.

Theorem 6.1. Let Cϕ be a composition operator induced by a univalent, analytic self-map ϕ of the unit disk with ϕ(0) = 0. Let σ be the Königsfunction of ϕ, and suppose that its range σ(D) is bounded and strictly starlike with respect to the origin. Then the operator Cϕ has a minimal commutant. An immediate consequence of Theorem 6.1 is the following corollary. Corollary 6.2. If ϕ is a linear fractional loxodromic self-map of the unit disk, then the corresponding 2 composition operator Cϕ on the Hardy space H (D) has a minimal commutant. Proof. Recall that ϕ is loxodromic if it has one fixed point in D and the other one outside of D and if, in addition, ϕ is not an automorphism. Without loss of generality we may assume that ϕ(0) = 0, in which case the canonical form for ϕ is z ϕ(z) = , az + b where a and b are complex numbers such that |b| > 1 and |a| < |1 − b|. Notice that ϕ is univalent. The corresponding Königsfunction is the linear fractional map given by the expression z σ(z) = . az + b − 1 Then, the Königs’sdomain σ(D) is an open disk that contains the origin, so that σ(D) is bounded and convex, hence stricly starlike with respect to the origin. It follows from Theorem 6.1, that the operator Cϕ has a minimal commutant. The proof of Theorem 6.1 relies on several lemmas that we state and prove before we proceed with the proof of the main result.

Lemma 6.3. Assume (ψn) is a sequence of analytic self-maps of the unit disk that converges pointwise to an analytic self-map ψ of the unit disk, and suppose that the sequence of composition operators (Cψn ) is uniformly bounded. Then Cψn → Cψ, as n → ∞, in the weak operator topology. 2 Proof. We need to show that hCψn f, gi → hCψf, gi, as n → ∞, for every f, g ∈ H (D). Since the sequence

(Cψn ) is uniformly bounded, it suﬃces to verify the convergence above when g belongs to a total set. It is clear that the family of reproducing kernels {Kz : z ∈ D} is a total set. Finally, since f is continuous in D, we get

hCψn f, Kzi = f(ψn(z)) → f(ψ(z)) = hCψf, Kzi, as n → ∞, for every z ∈ D, as we wanted. Let G be a domain in C. In what follows it will be of interest to consider the set Ω := {w ∈ C : wG ⊂ G}. (6.2) Lemma 6.4. Let G ⊆ C be a bounded domain that is strictly starlike with respect to the origin, and let Ω ⊆ C be defined by equation (6.2). Then, Ω is a domain in C that contains the interval [0, 1). Proof. Inclusion [0, 1) ⊆ Ω follows from the definition of a strictly starlike set. Next, we notice that Ω is starlike with respect to the origin, hence connected. Indeed, let 0 < r < 1, w ∈ Ω and z ∈ G. Then 14 MIGUEL LACRUZ, FERNANDO LEON-SAAVEDRA,´ SRDJAN PETROVIC, AND LUIS RODRÍGUEZ-PIAZZA wz ∈ G, so rwz ∈ G, whence rw ∈ Ω. Thus, it remains to show that Ω is open. For every w ∈ C, the function fw defined by fw(z) := wz, z ∈ G, is continuous on G. Next, notice that the mapping F : C → C(G)

w 7→ F (w) = fw is Lipschitz continuous, since kF (w1) − F (w2)k∞ ≤ M|w1 − w2|, where M = sup{|z|: z ∈ G}. We will show that the family U : = {f ∈ C(G): f(G) ⊆ G} is an open set in C(G), whence it will follow that Ω = F −1(U) is an open subset of C. In order to show that U is open, we notice that G is a compact set. Let f ∈ U. Then f(G) is compact c as well, so there exists ε > 0 such that dist(f(G),G ) ≥ ε. Let g ∈ U, and suppose that kg − fk∞ < ε. If z ∈ G, then f(z) ∈ G, and |g(z) − f(z)| < ε, so that g(z) ∈ G, and the proof is complete.

The strategy for the proof of Theorem 6.1 is to consider, for every w ∈ Ω, the analytic self-map ϕw of the unit disk deﬁned by the expression −1 ϕw(z) = σ (wσ(z)). (6.3) n n We are using G = σ(D), so ϕw is well-deﬁned. We have σ ◦ϕw = wσ, and therefore σ ◦ϕw = (σ ◦ϕw) = n n n n n w σ , so that Cϕw σ = w σ , for all n ∈ N0. Also, ϕw(0) = 0, and it follows from Littlewood’s theorem

[30, p. 16] that kCϕw k ≤ 1, for all w ∈ Ω. − Corollary 6.5. Under the assumptions of Theorem 6.1, we have Cϕr → I, as r → 1 , in the weak operator topology.

Proof. Let (rn) be a sequence of scalars with 0 < rn < 1 and rn → 1, as n → ∞. Let ψn(z) = ϕrn (z) and ψ(z) = z, for every z ∈ D. Notice that Cψ = I. We know that kCψn k ≤ 1, for all n ∈ N. Then, since σ−1 is continuous in σ(D), which is starlike with respect to the origin by hypothesis, we have −1 −1 ψn(z) = σ (rnσ(z)) → σ (σ(z)) = ψ(z), as n → ∞, for every z ∈ D. It follows from Lemma 6.3 that C → I, as n → ∞, in the weak operator topology. ϕrn Next, we need a result of Mattner [23] about complex diﬀerentiation of parametric integrals that can be stated as follows.

Lemma 6.6. Let (Θ, Σ, µ) be a measure space, let Ω ⊆ C be an open set, and let f :Ω × Θ → C be a function such that (1) f(w, ·) is Σ-measurable for every w ∈ Ω, (2) f(·, θ) is analytic for every θ ∈ Θ, (3) for every w0 ∈ Ω there is some δ > 0 such that D(w0, δ) ⊆ Ω and such that Z sup |f(w, θ)| dµ(θ) < +∞. |w−w0|≤δ Θ Then, the function F :Ω → C deﬁned by the parametric integral Z F (w) := f(w, θ) dµ(θ) Θ is analytic on Ω.

Lemma 6.7. Let Λ: B(H2(D)) → C be a linear functional that is continuous in the weak operator topology, and let X ∈ B(H2(D)). Suppose that ϕ and the corresponding K¨onigsfunction σ satisfy the hypotheses of Theorem 6.1, and let Ω be associated to G = σ(D) by (6.2). Then, the function F :Ω → C, deﬁned by F (w) := Λ(XCϕw ), is analytic. COMPOSITION OPERATORS WITH A MINIMAL COMMUTANT 15

Proof. It suﬃces to prove the result when Λ has the special form Λ(X) = hXf, gi, for some f, g ∈ H2(D). Indeed, a classical result of Dixmier [9, p. 477] is the assertion that, if H is Hilbert space, a linear functional on B(H) is continuous in the weak operator topology if and only if it is continuous in the strong operator topology. Moreover, every such functional can be expressed as a linear combination of ﬁnitely many linear functionals of the form hXf, gi. Then, we have

F (w) = hXCϕw f, gi ∗ = hCϕw f, X gi 1 Z 2π = f(σ−1(wσ(eiθ)))(X∗g)(eiθ) dθ. 2π 0 We shall apply Lemma 6.6 with Θ = [0, 2π] and dµ = dθ/(2π) the normalized Lebesgue measure. It is clear that the function w 7→ f(σ−1(wσ(eiθ)))(X∗g)(eiθ) is analytic on Ω, for every θ ∈ [0, 2π]. It is also clear that the function θ 7→ f(σ−1(wσ(eiθ)))(X∗g)(eiθ) is Lebesgue measurable, for every w ∈ Ω. Finally, it follows from the Cauchy-Schwarz inequality that, for every w ∈ Ω, we have 1 Z 2π f(σ−1(wσ(eiθ)))(X∗g)(eiθ) dθ ≤ kC fk · kX∗gk ϕw 2 2 2π 0 ∗ ≤ kX k · kfk2 · kgk2, as we wanted. The heart of the proof of Theorem 6.1 is enclosed in the following result. Lemma 6.8. Suppose that the map ϕ and the corresponding K¨onigsfunction σ satisfy the hypotheses of 0 Theorem 6.1. There is a constant δ > 0 such that, for every w ∈ C with |w| < δ and for every X ∈ {Cϕ} , there is a sequence of polynomials (pn) such that kpn(Cϕ) − XCϕw k → 0, as n → ∞.

Proof. Let G = σ(D). Since 0 ∈ G and G is open, there is some ρ > 0 such that ρ D ⊆ G. Also, G is bounded, so there exists M > 0 such that |σ(z)| ≤ M, for all z ∈ D. Let f ∈ H2(D). The function f ◦σ−1 is analytic on G and its Taylor series at the origin is ∞ X (f ◦ σ−1)(n)(0) f(σ−1(ξ)) = ξn, |ξ| < ρ. (6.4) n! n=0 Let Ω be deﬁned by (6.2). By Lemma 6.4, Ω is an open set that contains the origin. Thus, there exists 0 < δ < ρ/M such that w ∈ Ω, for all w ∈ C with |w| < δ. Let |w| < δ. Then |wσ(z)| < δM < ρ, for every z ∈ D, so that

(Cϕw f)(z) = f(ϕw(z)) = f(σ−1(wσ(z))) ∞ X (f ◦ σ−1)(n)(0) = wnσ(z)n. n! n=0 We claim that the series of analytic functions on the right hand side of the identity ∞ X (f ◦ σ−1)(n)(0) C f = wnσn (6.5) ϕw n! n=0 2 converges in the Hardy space H (D). Indeed, it suﬃces to show that, for every n ∈ N0, −1 (n) n (f ◦ σ ) (0) n n δM · |w| · kσ k2 ≤ kfk2. (6.6) n! ρ 16 MIGUEL LACRUZ, FERNANDO LEON-SAAVEDRA,´ SRDJAN PETROVIC, AND LUIS RODR´IGUEZ-PIAZZA

n n n n n n We have kσ k2 ≤ kσ k∞ ≤ M , so that |w| · kσ k2 < (δM) for all n ∈ N0. On the other hand, if γ is the circle |ξ| = ρ oriented counterclockwise, then γ is contained in G, and Cauchy’s integral formula for the derivatives yields

(f ◦ σ−1)(n)(0) 1 I f(σ−1(ξ)) = n+1 dξ n! 2πi γ ξ 1 Z 2π f(σ−1(ρeiθ)) = n inθ dθ. 2πρ 0 e −1 Let us consider the analytic self-map of the unit disk ψ(z) = σ (ρz). Since ψ(0) = 0, we have kCψk ≤ 1. Therefore, −1 (n) Z 2π (f ◦ σ ) (0) 1 −1 iθ ≤ n |f(σ (ρe ))| dθ n! 2πρ 0 Z 2π 1/2 1 1 −1 iθ 2 (6.7) ≤ n |f(σ (ρe ))| dθ ρ 2π 0 kC fk kfk = ψ 2 ≤ 2 . ρn ρn 2 Hence, (6.6) holds for all n ∈ N0, and since δM/ρ < 1, the series in (6.5) converges in the norm of H (D). 0 Next, let X ∈ {Cϕ} . The lemma is obviously true when X = 0, so we assume that X 6= 0. The desired result is an immediate consequence of the following assertion: for every ε > 0 there is a polynomial p and an operator Y ∈ B(H2(D)) such that both of the following inequalities hold: ε ε kp(C ) − Y k < , kY − XC k < . (6.8) ϕ 2 ϕw 2

We start by establishing the second inequality in (6.8). Let n ∈ N0. It follows from Lemma 1.6, applied n n n to the operator Cϕ, and f = σ , that there is a scalar λn ∈ C such that Xσ = λnσ and |λn| ≤ kXk. 2 Since n ∈ N0 is arbitrary, (6.5) implies that, for every f ∈ H (D), ∞ X (f ◦ σ−1)(n)(0) XC f = λ wn σn. ϕw n n! n=0

Now, choose n0 ∈ N large enough, so that ∞ n X δM ε < , (6.9) ρ 2kXk n=n0+1 and let Y ∈ B(H2(D)) be the finite rank operator on H2(D) defined by n X0 (f ◦ σ−1)(n)(0) Y f = λ wn σn, for f ∈ H2( ). n n! D n=0 2 It is clear that Y is a bounded linear operator on H (D). Since |λn| ≤ kXk, for all n ∈ N0, the second inequality in (6.8) follows from (6.6) and (6.9). Thus, it remains to establish the first inequality in (6.8). Let α = ϕ0(0), and recall that 0 < |α| < 1. Then, there exists m ∈ N such that |α|m < δ. Let η > 0 be so small that ∞ n X δM ε η < . (6.10) ρ 2 n=0 COMPOSITION OPERATORS WITH A MINIMAL COMMUTANT 17

n Let K = {α : n ∈ N0} ∪ {0}. It is not hard to see that K is a compact set in the complex plane that has empty interior, and that C\K is connected. Notice that the function h : K → C defined by ( λ wn/αmn, if z = α , 0 ≤ n ≤ n , h(z) = n n 0 0, if z = αn, n > n0, or z = 0, is continuous. It follows from Mergelyan’s theorem that there is a polynomial q such that n n λnw q(α ) − < η, for all 0 ≤ n ≤ n0, and αmn n |q(α )| < η, for all n > n0. Then, consider the polynomial p(z) = zmq(z) and notice that n n n |p(α ) − λnw | < ηδ , for all 0 ≤ n ≤ n0, and n n (6.11) |p(α )| < ηδ , for all n > n0. Now, let f ∈ H2(D). Using (6.1) and (6.4) we have that, for every k ≥ m and every z ∈ D, k −1 k (Cϕf)(z) = f(σ (α σ(z))) ∞ X (f ◦ σ−1)(n)(0) = αnkσ(z)n. n! n=0 The same argument as before shows that the last series is norm convergent in H2(D). Indeed, the difference between the last series and the one in (6.5) is that w is replaced by αk. Since |αk| ≤ |α|m < δ, for all k ≥ m, we get an estimate analogous to (6.6), whence the series converges in the norm of H2(D). Pd k If we write p(z) = k=m akz , it follows that d ∞ X X (f ◦ σ−1)(n)(0) (p(C )f)(z) = a αnkσ(z)n ϕ k n! k=m n=0 ∞ d ! X (f ◦ σ−1)(n)(0) X = a αnk σ(z)n, n! k n=0 k=m so that ∞ X (f ◦ σ−1)(n)(0) p(C )f = p(αn)σn. (6.12) ϕ n! n=0 n n Finally, the definition of Y , equations (6.12), (6.7), (6.11) and the estimate kσ k2 ≤ M imply that n0 −1 (n) ∞ −1 (n) X (f ◦ σ ) (0) n n n X (f ◦ σ ) (0) n n kp(Cϕ)f − Y fk2 ≤ (p(a ) − λnw )σ + p(a )σ n! n! n=0 2 n=n0+1 2 ∞ n X δM ≤ η · kfk . ρ 2 n=0 Using (6.10) we obtain the first inequality in (6.8), and the proof is complete. Now, everything is ready for the proof of the main result of this section.

0 Proof of Theorem 6.1. We need to show that alg (Cϕ) is a dense linear subspace of {Cϕ} in the weak operator topology. To that end, let Λ be a linear functional on H2(D) that is continuous in the weak 0 operator topology and that vanishes on alg (Cϕ). We claim that Λ(X) = 0, for all X ∈ {Cϕ} . 0 Now, for the proof of our claim, let X ∈ {Cϕ} , let σ be the Königsfunction for ϕ, and let Ω be the domain associated to G = σ(D) by (6.2). We will consider the function F :Ω → C defined by 18 MIGUEL LACRUZ, FERNANDO LEON-SAAVEDRA,´ SRDJAN PETROVIC, AND LUIS RODRÍGUEZ-PIAZZA

F (w) := Λ(XCϕw ), where ϕw is deﬁned by (6.3). We know from Lemma 6.7 that F is analytic. We also know from Lemma 6.8 that there exists δ > 0 such that, for every w ∈ C with |w| < δ, there is a sequence of polynomials (pn) such that pn(Cϕ) → XCϕw , as n → ∞, in the operator norm and therefore in the weak operator topology. Now, if |w| < δ then

F (w) = Λ(XCϕ ) = lim Λ(pn(Cϕ)) = 0. w n→∞ Since F is analytic and it vanishes in an open disc, it follows from the principle of analytic continuation that F vanishes identically on the domain Ω. By Lemma 6.4, [0, 1) ⊂ Ω, so F (r) = 0, for all 0 < r < 1. Finally, it follows from Corollary 6.5 that

Λ(X) = lim Λ(XCϕr ) = lim F (r) = 0, r→1− r→1− as we wanted.

7. Compact composition operators with univalent symbol and non-minimal commutant As we mentioned in the Introduction, Worner [35] described a procedure for constructing a composition operator without a minimal commutant, but her technique relied on the fact that the symbol be non- univalent. She also asked the question: whether a compact composition operator with a univalent symbol has a minimal commutant. It is the aim of this section to provide a negative answer to this question, by constructing an example of a univalent, analytic self-map ϕ of the unit disk, such that Cϕ is a compact operator whose commutant fails to be minimal. We take advantage of Tami Worner’s original idea that, for every analytic self-map ψ of the unit disk, and for ϕ = ψ ◦ ψ, the operator Cψ commutes with Cϕ, and is a good candidate to be outside the closure in the weak operator topology of the unital algebra generated by the operator Cϕ. We start with an abstract result about the commutant for the square of a Hilbert space operator, that is not necessarily a composition operator on the Hardy space. σ Theorem 7.1. Let A ∈ B(H) such that AH\A2H 6= ∅. Then A ∈ {A2}0, but A/∈ alg (A2) , hence A2 fails to have a minimal commutant.

Proof. We prove the non trivial part of the statement. First of all, for every polynomial p, we have p(A2)f ∈ A2H + p(0)f, for all f ∈ H. (7.1)

2 Since AH is a linear subspace that contains properly A H, it follows that there exists g0 ∈ AH such that 2 kg0k = 1 and such that g0 ⊥ A H, that is,

2 hA f, g0i = 0, for all f ∈ H. (7.2)

Then, there exists a sequence (fn) of vectors in H such that Afn → g0, as n → ∞. We must show that σ A/∈ alg (A2) . We proceed by contradiction. Let us suppose for a little while that there exists a net of 2 polynomials (pd)d∈D such that pd(A ) → A, as d ∈ D, in the weak operator topology. It follows from equations (7.1) and (7.2) that

2 hpd(A )f, g0i = pd(0)hf, g0i, for all f ∈ H. (7.3)

Now, setting f = g0 in equation (7.3), and taking limits, ﬁrst as d ∈ D, and second as n → ∞, we get

2 2 lim pd(0) = limhpd(A )g0, g0i = hAg0, g0i = lim hA fn, g0i = 0. d∈D d∈D n→∞ COMPOSITION OPERATORS WITH A MINIMAL COMMUTANT 19

Then, setting f = fn in equation (7.3), and taking iterated limits, as d ∈ D, and as n → ∞, we obtain

0 = lim lim pd(0)hfn, g0i n→∞ d∈D 2 = lim limhpd(A )fn, g0i n→∞ d∈D

= lim hAfn, g0i n→∞

= hg0, g0i = 1, and a contradiction has arrived. 2 Our next goal is to construct a univalent, analytic self-map ϕ of the unit disk such that Cϕ is compact but fails to have a minimal commutant. It is convenient to apply a result related to the Schwarz-Christoﬀel formula for a conformal mapping τ from the unit disk onto the interior of a polygon P. This result is stated below as Lemma 7.2. We refer to the book of Gamelin [13, p. 298] for this result in connection with the Schwarz reﬂection principle, and for the fact that τ extends to a homeomorphism from the closed unit disk onto P. The last assertion is a particular instance of Caratheodory’s theorem, that can be stated as follows: if the boundary of the range of a conformal mapping is a Jordan curve, then the conformal mapping extends to a homeomorphism from the closed unit disk onto the closure of its range. We refer the reader to the book of Rudin [27, p. 279] for a proof of Caratheodory’s theorem. Lemma 7.2. Let τ be a conformal mapping from the open unit disk onto the interior of a polygon P with vertices wj = τ(zj), where |zj| = 1, and let παj be the angles of the polygon P at the vertices wj. Then, for all 1 ≤ j ≤ n there is an open neighborhood Vj of th point zj and there is a function ξj that is analytic on Vj, such that ξj(zj) 6= 0, and such that

αj τ(z) = wj + (z − zj) ξj(z), for all z ∈ D ∩ Vj. (7.4) iπ/4 Now, let T be the triangle of vertices w1 = 1, w2 = i and w3 = e , and consider a conformal mapping τ : D → int (T ). Using an automorphism of the unit disk allows us to prescribe the images of iπ/4 the three points z1 = 1, z2 = −1 and z3 = i on the unit circle, say τ(1) = 1, τ(−1) = i, and τ(i) = e . Then, the final map ϕ is defined by 4 ϕ(z) := τ(z) , z ∈ D. (7.5) We know that τ(D) is contained in the first open quadrant, and that the quartic q(w) := w4 is univalent on that quadrant. It follows that ϕ = q ◦τ is univalent on the open unit disk. We know, as a consequence of the polygonal compactness theorem [30], that Cτ is compact, hence Cϕ = Cτ Cq is compact, too. Notice that the angles of T at the vertices 1, i and eiπ/4 are given by π/8, π/8 and 3π/4, repectively. If we apply Lemma 7.2 to our conformal mapping τ then we get the following

Lemma 7.3. There are three functions ξ1, ξ2, ξ3 that are analytic at 1, −1, i, respectively, and such that 1/8 (1) τ(z) = 1 + (z − 1) ξ1(z), 1/8 (2) τ(z) = i + (z + 1) ξ2(z), iπ/4 3/4 (3) τ(z) = e + (z − i) ξ3(z). We continue with a result that contains one of the main ideas for the proof of Theorem 7.5 below.

m Lemma 7.4. Let ϕ be the map given by equation (7.5), let p0(z) = (z − 1) , and consider the function it 0 it 0 it it g0(t) := p0(ϕ(ϕ(e )))ϕ (ϕ(e ))ϕ (e )ie . (7.6)

(1) If m ≥ 64 then lim g0(t) = 0. t→0 (2) If m ≥ 10 then lim g0(t) = 0. t→π/2 20 MIGUEL LACRUZ, FERNANDO LEON-SAAVEDRA,´ SRDJAN PETROVIC, AND LUIS RODR´IGUEZ-PIAZZA

Proof. First of all, we shall use Lemma 7.3 in order to estimate the size of |ϕ0(ϕ(eit))| · |ϕ0(eit)| as t → 0. It will be convenient to use the big O notation. Notice that eit → 1 and ϕ(eit) → 1 as t → 0. It follows from the fact that τ − 1 is a factor of τ 4 − 1, and from equation (1) in Lemma 7.3, that |ϕ(eit) − 1| = |τ(eit)4 − 1| = O |τ(eit) − 1| = O |eit − 1|1/8 = O |t|1/8 , as t → 0.

Next, we know that, on a neighborhood of z1 = 1, we have 1 ϕ0(z) = 4τ(z)3τ 0(z) = 4τ(z)3 (z − 1)−7/8ξ (z) + (z − 1)1/8ξ0 (z) , 8 1 1 and from this identity, we obtain the following estimate |ϕ0(z)| = O |z − 1|−7/8 , as z → 1. (7.7)

Thus, |ϕ0(eit)| = O |eit − 1|−7/8 = O(|t|−7/8), as t → 0, (7.8) and −7/8 |ϕ0(ϕ(eit))| = O |ϕ(eit) − 1|−7/8 = O |t|1/8 , as t → 0, (7.9)

Then, if we combine estimates (7.8) and (7.9), then we get |ϕ0(ϕ(eit))| · |ϕ0(eit)| = O |t|−63/64 , as t → 0. (7.10)

Next, we know that, on a neighborhood of z2 = −1, we have 1 ϕ0(z) = 4τ(z)3τ 0(z) = 4τ(z)3 (z + 1)−7/8ξ (z) + (z + 1)1/8ξ0 (z) , 8 2 2 and from this identity we get the following estimate |ϕ0(z)| = O |z + 1|−7/8 as z → −1. (7.11)

Also, we know that, on a neighborhood of z3 = i, we have 3 ϕ0(z) = 4τ(z)3τ 0(z) = 4τ(z)3 (z − i)−1/4ξ (z) + (z − i)3/4ξ0 (z) , 4 3 3 COMPOSITION OPERATORS WITH A MINIMAL COMMUTANT 21 and from this identity, we get the estimate |ϕ0(z)| = O |z − i|−1/4 , as z → i. (7.12) This yields |ϕ0(eit)| = O |eit − i|−1/4 = O |t − π/2|−1/4 , as t → π/2, Moreover, it follows from the estimate (7.11) that |ϕ0(ϕ(eit))| = O |ϕ(eit) + 1|−7/8 −7/8 = O |t − π/2|3/4 = O |t − π/2|−21/32 , as t → π/2. Thus, we arrived at the conclusion that |ϕ0(ϕ(eit))| · |ϕ0(eit)| = O |t − π/2|−21/32 · |t − π/2|−1/4 = O |t − π/2|−29/32 , as t → π/2.

m Now, we consider the polynomial p0(z) = (z −1) , for some m ∈ N to be chosen later on, and we estimate it the size of |p0(ϕ(ϕ(e ))| as t → 0 and as t → π/2. On one hand, we have it it m |p0(ϕ(ϕ(e )))| = |ϕ(ϕ(e )) − 1| m = O |ϕ(eit) − 1|1/8 m/8 = O |eit − 1|1/8 = O |t|m/64 , as t → 0, and it follows from the previous estimates that

it 0 it 0 it (m−63)/64 |g0(t)| = |p0(ϕ(ϕ(e )))| · |ϕ (ϕ(e ))| · |ϕ (e )| = O |t| , as t → 0, hence it suﬃces to take m ≥ 64 to make sure that g0(t) → 0 as t → 0. On the other hand, we have it it m |p0(ϕ(ϕ(e )))| = |ϕ(ϕ(e )) − 1| = O |τ(ϕ(eit))4 − i4|m = O |τ(ϕ(eit)) − i|m = O |ϕ(eit) + 1|m/8 = O |τ(eit)4 − (eiπ/4)4|m/8 = O |τ(eit) − eiπ/4|m/8 = O |eit − i|3m/32 = O |t − π/2|3m/32 , as t → π/2, 22 MIGUEL LACRUZ, FERNANDO LEON-SAAVEDRA,´ SRDJAN PETROVIC, AND LUIS RODR´IGUEZ-PIAZZA and it follows from the previous estimates that

it 0 it 0 it (3m−29)/32 |g0(t)| = |p0(ϕ(ϕ(e )))| · |ϕ (ϕ(e ))| · |ϕ (e )| = O |t| , as t → 0, hence it suffices to take m ≥ 10 to make sure that g0(t) → 0 as t → π/2. 2 2 2 Theorem 7.5. If ϕ is the map defined by equation (7.5) then CϕH (D) fails to be dense in CϕH (D). 2 2 Proof. It suffices to show that there are two functions g0 ∈ L [0, π/2] and h0 ∈ H (D) such that, for every polynomial p, we have Z π/2 it g0(t)p(ϕ(ϕ(e ))) dt = 0, (7.13) 0 Z π/2 it g0(t)h0(ϕ(e )) dt 6= 0. (7.14) 0 it m Consider the closed simple curve γ(t) = ϕ(ϕ(e )), for 0 ≤ t ≤ π/2, and the polynomial p0(z) = (z − 1) , for some m ≥ 64. It follows from Cauchy’s integral theorem that for every polynomial p, I Z π/2 it 0 it 0 it it it 0 = p0(z)p(z) dz = p0(ϕ(ϕ(e )))ϕ (ϕ(e ))ϕ (e )ie · p(ϕ(ϕ(e ))) dt. γ 0 it 0 it 0 it it Next, consider the function g0 given by equation (7.6), that is, g0(t) = p0(ϕ(ϕ(e )))ϕ (ϕ(e ))ϕ (e )ie . Notice that g0 6= 0 because p0 6= 0 and ϕ is a conformal map. It is clear that g0 satisfies equation (7.13). 2 It follows from Lemma 7.4 that g0 ∈ L [0, π/2] because, in fact, g0 is continuous on [0, π/2]. The rest of the proof is easy. Indeed, since g0 6= 0, there is some ε > 0 such that Z π/2 Z π/2 2 |g0(t)| dt − ε |g0(t)| dt > 0. 0 0 Then, consider the compact arc K := {ϕ(eit): 0 ≤ t ≤ π/2}, and notice that there is a homeomorphism β : [0, π/2] → K given by β(t) = ϕ(eit). Since K has empty interior, and its complement in the complex plane is connected, it follows from Mergelyan’s theorem that there is a polynomial h0 such that

−1 h0(z) − g0(β (z)) < ε, for all z ∈ K. (7.15) Finally, we get

Z π/2 Z π/2 Z π/2 it 2 it g0(t)h0(ϕ(e )) dt ≥ |g0(t)| dt − |g0(t)| · h0(e ) − g0(t) dt 0 0 0 Z π/2 Z π/2 2 ≥ |g0(t)| dt − ε |g0(t)| dt > 0, 0 0 so that the inequality (7.14) is also satisfied, as we wanted. Our next result is a consequence of Theorem 7.1 and Theorem 7.5. Corollary 7.6. Let ϕ be the analytic self-map of the unit disk defined by equation (7.5) and let ψ = ϕ◦ϕ. Then, ψ is a univalent, analytic self-map of the unit disk such that the composition operator Cψ is compact but it fails to have a minimal commutant. Remark 7.7. The range of the Jordan curve γ, defined in the proof of Theorem 7.5, is the compact set C := {ϕ(ϕ(t)): 0 ≤ t ≤ π/2} COMPOSITION OPERATORS WITH A MINIMAL COMMUTANT 23 which is homeomorphic to the unit circle. Since the complement of C in the complex plane is not connected, Mergelyan’s theorem cannot be applied. This provides a heuristic obstruction for the approx- 2 imation of Cϕ by polynomials in Cϕ. This argument is one of the main ideas in the construction of our counter-example.

We end this section with an alternative procedure to construct a composition operator Cϕ induced by a univalent, analytic self-map ϕ of the unit disk so that Cϕ fails to have a minimal commutant. Now, the idea is to work with a symbol that ﬁxes the origin, such that the K¨onigs’sdomain is bounded and starlike with respect to the origin, and such that the range of Cϕ fails to be dense. An example of such a symbol is provided by Corollary 7.10 below.

Theorem 7.8. Let ϕ be a univalent, analytic self-map of the unit disk such that ϕ(0) = 0, and let σ denote the corresponding K¨onigs’sfunction. If the domain G := σ(D) is bounded and starlike with respect to the origin and if the range of Cϕ fails to be dense, then Cϕ fails to have a minimal commutant.

Proof. Let α = ϕ0(0) and recall from Königs’stheorem that ϕ(z) = σ−1(ασ(z)). Since ϕ is univalent and it fixes the origin, and in addition, it fails to be a disk automorphism, we have 0 < |α| < 1. Since G is starlike with respect to the origin, for every 0 ≤ r ≤ 1, there exists an analytic self-map −1 ϕr of the unit disk defined by the expression ϕr(z) := σ (rσ(z)). Notice that ϕr ◦ ϕs = ϕrs, hence

Cϕs Cϕr = Cϕrs for all 0 ≤ r, s ≤ 1. 0 Observe that Cϕr ∈ {Cϕ} . Since ϕr(0) = 0, we have kCϕr k = 1, and it follows from Lemma 6.3 that 2 the map r 7→ Cϕr is continuous from [0, 1] into B(H (D)) endowed with the weak operator topology, and − in particular, Cϕr → I, as r → 1 , in the weak operator topology. 2 ∗ Since the range of Cϕ fails to be dense, there exists a function f0 ∈ H (D)\{0} such that Cϕf0 = 0. ∗ Let e0 ≡ 1 and notice that f0(0) = hf0, e0i = hf0,Cϕe0i = hCϕf0, e0i = 0. Then, consider the closed set defined by the expression F := {r ∈ [0, 1]: C∗ f = 0}. ϕr 0 Observe that 0 ∈ F because C∗ f = f (0)e = 0. Also, notice that 1 ∈/ F, because C∗ f = f 6= 0. ϕ0 0 0 0 ϕ1 0 0 Moreover, F is an interval. Indeed, if 0 < s < r ∈ F then C∗ f = C∗ C∗ f = 0, hence s ∈ F. Next, ϕs 0 ϕs/r ϕr 0 let r = sup F. It follows from the continuity of the function r 7→ C∗ f , f that there is some ε > 0 such 0 ϕr 0 0 that hC∗ f , f i= 6 0, and in particular C∗ f 6= 0, whenever 1 − ε < r < 1. This shows that r < 1. ϕr 0 0 ϕ 0 0 We claim that r0 > 0. Indeed, since G is open and 0 ∈ G, there is some δ > 0 such that δD ⊆ G. Further, since G is bounded, there is some M > 0 such that G ⊆ MD. This allows us to consider, for every w ∈ C −1 with |w| ≤ δ/M, the univalent, analytic self-map ϕw of the unit disk defined by ϕw(z) := σ (wσ(z)). Let r := δ/M. It suffices to show that s := r|α| ∈ F. Indeed, let θ ∈ R with α = |α|eiθ, and let w = re−iθ. Observe that s = αw, so that C = C C , so that C∗ f = C∗ C∗ f = 0, hence, s ∈ F, and this ϕs ϕ ϕw ϕs 0 ϕw ϕ 0 completes the proof of our claim. Now, the rest of the proof is easy. We know that F = [0, r0] for some 0 < r0 < 1. Then, there is some 2 2 0 < r < 1 such that r < r0 < r, so that r ∈ F, but r∈ / F. σ We finally show that Cϕr ∈/ alg (Cϕ) , and the result follows at once. We proceed by contradiction.

Let us suppose for a moment that there exists a net of polynomials (pd)d∈D such that pd(Cϕ) → Cϕr , as d ∈ D, in the weak operator topology. Then, for every f ∈ H2(D),

hCϕ f, f0i = limhpd(Cϕ)f, f0i = lim pd(0)hf, f0i. (7.16) r d∈D d∈D

On one hand, if we set f = C∗ f in equation (7.16), we get ϕr 0

∗ 2 ∗ kCϕ f0k = lim pd(0)hCϕ f0, f0i, r d∈D r 24 MIGUEL LACRUZ, FERNANDO LEON-SAAVEDRA,´ SRDJAN PETROVIC, AND LUIS RODR´IGUEZ-PIAZZA

so that hCϕr f0, f0i= 6 0 and lim pd(0) 6= 0, as d ∈ D, and on the other hand, if we set f = Cϕr f0 in equation (7.16), then we get

lim pd(0)hCϕ f0, f0i = 0, d∈D r and it follows that lim pd(0) = 0, as d ∈ D. We arrived at a contradiction. In order to apply Theorem 7.8, we need a condition for an analytic self-map ϕ of the unit disk that ensures that the range of Cϕ fails to be dense. Recall that an analytic self-map ϕ of the unit disk is said to be univalent almost everywhere on ∂D provided that there is a Lebesgue measurable subset E ⊆ ∂D such that |E| = 0 and the restriction of ϕ to ∂D\E is univalent. Here | · | denotes the normalized arc length measure on ∂D. We refer to the paper of Bourdon and Shapiro [2, proof of Theorem 1.7] for the following result.

Theorem 7.9. Let ϕ be an analytic self-map of the unit disk and suppose that Cϕ has dense range. Then ϕ is univalent on D and univalent almost everywhere on ∂D.

Corollary 7.10. Let 0 < r0 < 1, and consider the slit disk G = D\[r0, 1]. Then, consider a conformal mapping σ : D → G such that σ(0) = 0 and σ(1) = r0. Let 0 < r < r0, and let ϕ be the univalent, analytic −1 self-map of the unit disk given by the expression ϕ(z) := σ (rσ(z)). Then, the composition operator Cϕ is compact but it fails to have a minimal commutant. Proof. It is clear that G is bounded and starlike with respect to the origin, so the map ϕ is well deﬁned. −1 Also, it is clear that the composition operator Cϕ is compact, because rD ⊆ G, hence ϕ(D) = σ (rD) is a compact subset of the unit disk. The desired result would follow as a consequence of Theorem 7.8, if we only could show that the range of Cϕ fails to be dense. The last assertion follows from Theorem 7.9, because ϕ fails to be univalent almost everywhere on ∂D. Indeed, it is easy to see that there is some iθ −1 −iθ iθ ε > 0 such that {e : |θ| < ε} ⊆ ϕ ([r0, 1)), and such that ϕ(e ) = ϕ(e ) whenever |θ| < ε. This is an obstruction for ϕ to be univalent almost everywhere on ∂D. Remark 7.11. It follows from Corollary 7.10 that the hypothesis in Theorem 6.1 that a bounded K¨onigs’s domain G be strictly starlike with respect to the origin cannot be replaced by the weaker assumption that G be starlike with respect to the origin. Remark 7.12. The proof of Theorem 7.8 also applies if the assumption that G be bounded is replaced in the statement by the condition ϕ0(0) > 0. Further, the proof also applies when G is assumed to be unbounded and strictly starlike with respect to the origin, because according to a result of Joel Shapiro, Wayne Smith and David Stegenga [32, Lemma 4.1, p. 49], there is some n ∈ N such that ϕ0(0)n > 0.

Appendix: The Sobolev space as a Banach algebra with an approximate identity The aim of this section is to provide the proofs of Theorem 4.2, Theorem 4.3 and Lemma 4.9, just for the sake of completeness. For convenience, we will repeat the statements of these results.

1,2 ∞ Theorem 4.3. If f ∈ W [0, +∞), we have f ∈ L [0, +∞), and moreover, kfk∞ ≤ kfk1,2.

Proof. Let σ0 ≥ 0 and σ > σ0. By deﬁnition, f is absolutely continuous on the bounded interval [σ0, σ], 2 so the same is true of f . Hence, for every s ∈ [σ0, σ], we have Z s 2 2 2 0 f(σ0) − f(s) = − (f ) (t) dt σ0 Z s = −2 f(t)f 0(t) dt. σ0 COMPOSITION OPERATORS WITH A MINIMAL COMMUTANT 25

It follows from the Cauchy-Schwarz inequality that Z s 2 2 0 |f(σ0) − f(s) | = 2 f(t)f (t) dt σ0 Z s 1/2 Z s 1/2 2 0 2 0 ≤ 2 |f(t)| dt |f (t)| dt ≤ 2kfk2kf k2, σ0 σ0 2 2 2 2 0 2 so that |f(σ0)| ≤ |f(σ0) − f(s) | + |f(s)| ≤ 2kfk2kf k2 + |f(s)| . Taking the integral average on both sides of this inequality over the interval [σ0, σ] leads to Z σ 2 0 1 2 |f(σ0)| ≤ 2kfk2kf k2 + |f(s)| ds σ − σ0 σ0 2 0 kfk2 ≤ 2kfk2kf k2 + , σ − σ0 2 0 and taking limits as σ → +∞ yields |f(σ0)| ≤ 2kfk2kf k2. Since σ0 ≥ 0 is arbitrary, we finally get 2 0 2 0 2 2 kfk∞ ≤ 2kfk2kf k2 ≤ kfk2 + kf k2 = kfk1,2, as we wanted. Notice that the inequality in the embedding Theorem 4.3 is sharp. Indeed, the function f(σ) = e−σ 1,2 belongs to the Sobolev space W [0, +∞), and it is easy to check that kfk∞ = kfk1,2 = 1. √ 1,2 1,2 Theorem 4.2. If f, g ∈ W [0, +∞) then fg ∈ W [0, +∞), and moreover, kfgk1,2 ≤ 2kfk1,2kgk1,2. 1,2 Hence, the Sobolev space W [0, +∞) is a commutative Banach√ algebra without identity with respect to ∗ pointwise multiplication and the equivalent norm kfk1,2 = 2kfk1.2. Proof. First, the function fg is absolutely continuous on the bounded subintervals of [0, +∞) because it is a product of two such functions. By Theorem 4.3, both f and g belong to L∞[0, +∞). On the other hand, both f 0 and g0 belong to L2[0, +∞) by definition. It follows that both fg and (fg)0 = f 0g + fg0 belong to L2[0, +∞), because the product of a function in L2[0, +∞) by a bounded measurable function 2 1,2 remains in L [0, +∞). This shows√ that fg ∈ W [0, +∞). Next, we show that kfgk1,2 ≤ 2kfk1,2kgk1,2. We start with the particular case when f = g. We have 2 2 2 2 2 0 2 kf k1,2 = kf k2 + k(f ) k2 2 0 2 = kf · fk2 + 2kf · f k2 2 0 2 ≤ 2kf · fk2 + 2kf · f k2 2 2 0 2 ≤ 2kfk∞(kfk2 + kf k2) 4 ≤ 2kfk1,2, so that √ 2 2 kf k1,2 ≤ 2kfk1,2, (7.17) as we wanted. In the general case, we may assume without loss of generality that both kfk1,2 ≤ 1 and kgk1,2 ≤ 1. The polarization identity 1 fg = (f + g)2 − (f − g)2 , 4 implies 1 kfgk ≤ k(f + g)2k + k(f − g)2k . 1,2 4 1,2 1,2 26 MIGUEL LACRUZ, FERNANDO LEON-SAAVEDRA,´ SRDJAN PETROVIC, AND LUIS RODRÍGUEZ-PIAZZA

Now, (7.17) yields √ 2 kfgk ≤ (kf + gk2 + kf − gk2 ). 1,2 4 1,2 1,2 Finally, using the parallelogram identity in W 1,2[0, +∞), we get √ 2 √ kfgk ≤ (kfk2 + kgk2 ) ≤ 2, 1,2 2 1,2 1,2 as we wanted. √ ∗ ∗ ∗ 1.2 Now, we have kfgk1,2 = 2kfgk1,2 ≤ 2kfk1,2kgk1,2 = kfk1,2kgk1,2, so that W [0, +∞) becomes a commutative Banach algebra without identity with respect to pointwise multiplication and the equivalent ∗ norm k · k1.2.

Lemma 4.9. The sequence of functions (en) deﬁned by the expression 1, if 0 ≤ σ ≤ n, e (σ) = n e−(σ−n), if σ ≥ n, is an approximate identity for the Sobolev algebra W 1,2[0, +∞). 1,2 Proof. It is not hard to see that, for all n ∈ N0, en ∈ W [0, +∞) and kenk∞ ≤ 1. Further, en(σ) → 1, as 1,2 n → ∞, pointwise on the interval [0, +∞). Let f ∈ W [0, +∞). We must show that ken · f − fk1,2 → 0, as n → ∞. We have 2 2 0 2 ken · f − fk1,2 = ken · f − fk2 + k(en · f − f) k2. (7.18) 2 2 Notice that |en(σ)f(σ)−f(σ)| ≤ 4|f(σ)| , and therefore it follows from the bounded convergence theorem 2 0 2 that ken · f − fk2 → 0, as n → ∞. Moreover, since f ∈ L ([0, +∞)), the same argument shows that 0 0 2 0 0 0 0 0 ken · f − f k2 → 0, as n → ∞. Since (en · f − f) = en · f + en · f − f . It remains to prove that en · f → 0, 0 0 as n → ∞. Notice that, for every σ ∈ [0, +∞), en(σ) → 0, as n → ∞. Finally, kenk∞ ≤ 1 and f is bounded, so another application of the bounded convergence theorem completes the proof. References [1] Robert A. Adams and John J. F. Fournier, Sobolev spaces, 2nd ed., Pure and Applied Mathematics (Amsterdam), vol. 140, Elsevier/Academic Press, Amsterdam, 2003. MR2424078 [2] Paul S. Bourdon and Joel H. Shapiro, Cyclic phenomena for composition operators, Mem. Amer. Math. Soc. 125 (1997), no. 596, x+105, DOI 10.1090/memo/0596. MR1396955 [3] Lennart Carleson, Representations of continuous functions, Math. Z. 66 (1957), 447–451. MR0084035 (18,798a) [4] Bruce A. Cload, Generating the commutant of a composition operator, Studies on composition operators (Laramie, WY, 1996), Contemp. Math., vol. 213, Amer. Math. Soc., Providence, RI, 1998, pp. 11–15, DOI 10.1090/conm/213/02845. MR1601052 [5] Bruce Cload, Toeplitz operators in the commutant of a composition operator, Studia Math. 133 (1999), no. 2, 187–196. MR1686697 (2001c:47036) [6] Carl C. Cowen and Barbara D. MacCluer, Some problems on composition operators, Studies on composition operators (Laramie, WY, 1996), Contemp. Math., vol. 213, Amer. Math. Soc., Providence, RI, 1998, pp. 17–25, DOI 10.1090/conm/213/02846. MR1601056 [7] , Composition operators on spaces of analytic functions, Studies in Advanced Mathematics, CRC Press, Boca Raton, FL, 1995. MR1397026 [8] James A. Deddens, Analytic Toeplitz and composition operators, Canad. J. Math. 24 (1972), 859–865. MR0310691 (46 #9789) [9] Nelson Dunford and Jacob T. Schwartz, Linear operators. Part I, Wiley Classics Library, John Wiley & Sons, Inc., New York, 1988. General theory; With the assistance of William G. Bade and Robert G. Bartle; Reprint of the 1958 original; A Wiley-Interscience Publication. MR1009162 (90g:47001a) [10] J. A. Erdos, The commutant of the Volterra operator, Integral Equations Operator Theory 5 (1982), no. 1, 127–130, DOI 10.1007/BF01694033. MR646883 [11] Aurora Fern´andez-Valles and Miguel Lacruz, A spectral condition for strong compactness, J. Adv. Res. Pure Math. 3 (2011), no. 4, 50–60, DOI 10.5373/jarpm.760.020111. MR2859295 (2012j:47151) COMPOSITION OPERATORS WITH A MINIMAL COMMUTANT 27

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Miguel Lacruz, Departamento de Analisis´ Matematico´ and Instituto de Matematicas,´ Universidad de Sevilla, Avenida Reina Mercedes, 41012 Seville (Spain) E-mail address: [email protected]

Fernando Leon-Saavedra,´ Departamento de Matematicas,´ Universidad de Cadiz,´ Avenida de la Universidad, 11405 Jerez de la Frontera, Cadiz´ (Spain) E-mail address: [email protected]

Srdjan Petrovic, Department of Mathematics, Western Michigan University, Kalamazoo, MI 49008 (USA) E-mail address: [email protected]

Luis Rodr´ıguez-Piazza, Departamento de Analisis´ Matematico´ and Instituto de Matematicas,´ Universidad de Sevilla, Avenida Reina Mercedes, 41012 Seville (Spain) E-mail address: [email protected]