Extended eigenvalues of composition operators

J. Math. Anal. Appl. 504 (2021) 125427 Contents lists available at ScienceDirect Journal of Mathematical Analysis and Applications www.elsevier.com/locate/jmaa Extended eigenvalues of composition operators Miguel Lacruz a,∗, Fernando León-Saavedra b, Srdjan Petrovic c, Luis Rodríguez-Piazza a aDepartamento de Análisis Matemático & Instituto de Matemáticas, Universidad de Sevilla, Facultad de Matemáticas, Avenida Reina Mercedes, 41012 Seville, Spain bRegional Mathematical Center of Southern Federal University & Departamento de Matemáticas, Universidad de Cádiz, Avenida de la Universidad, 11405 Jerez de la Frontera, Cádiz, Spain cDepartment of Mathematics, Western Michigan University, Kalamazoo, MI 49008, USA a r t i c l e i n f o a b s t r a c t Article history: Received 31 October 2020 Available online 11 June 2021 Submitted by J. Bonet Keywords: Extended eigenvalue Composition operator Hardy space Linear fractional transformation A complex scalar λis said to be an extended eigenvalue of a bounded linear operator Aon a complex Hilbert space if there is a nonzero operator Xsuch that AX =λXA. The results in this paper provide a full solution to the problem of computing the extended eigenvalues for those composition operators Cϕinduced on the Hardy space H2(D)by linear fractional transformations ϕof the unit disk. © 2021 Universidad de Sevilla. Published by Elsevier Inc. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/). 1. Introduction The goal of this paper is the calculation of the extended eigenvalues for composition operators Cϕinduced on the Hardy space H2(D)by linear fractional transformations ϕof the unit disk. This problem leads to intriguing questions on the interface between complex analysis and operator theory. This is the first part of a twofold project. The second part will appear in a forthcoming paper and it concerns the structure of the linear manifold of the extended eigenoperators (to be defined below) corresponding to a fixed extended eigenvalue, when that manifold is regarded as a bilateral module over the operator’s commutant. The main results of this paper are summarized in Table 1. The first column shows the acronyms that are used for the symbols ϕ. The second column exhibits the fixed points of the symbols ϕwhen they are expressed in a standard form. The third column collects the explicit expressions of such standard forms. *Corresponding author. E-mail addresses: [email protected] (M. Lacruz), [email protected] (F. León-Saavedra), srdjan.petro[email protected] (S. Petrovic), [email protected] (L. Rodríguez-Piazza). https://doi.org/10.1016/j.jmaa.2021.125427 0022-247X/© 2021 Universidad de Sevilla. Published by Elsevier Inc. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/). 2M. Lacruz et al. / J. Math. Anal. Appl. 504 (2021) 125427 Table 1 Extended eigenvalues of linear fractional composition operators. Symbol Fixed points Standard form of the symbol ϕExt (Cϕ) Thm. EA 0,∞ϕ(z)=ωz,|ω|=1 {ωn:n∈Z}3.1 LOX/HNA III c∈D,∞ϕ(z)=a(z−c)+c, |a|+|1−a|·|c|≤1{ϕ(c)n:n∈Z}3.2 HA 1,−1ϕ(z)= z+r 1+rz ,0<r<1∂D4.4 HNA I 1,∞ϕ(z)=rz +1−r,0<r<1D\{0}5.3 HNA II 1,0ϕ(z)= rz 1−(1 −r)z,0<r<1C\D5.7 PA 1,1ϕ(z)= (2 −a)z+a −az +2+a,Re a=0 ∂D6.2 PNA 1,1ϕ(z)= (2 −a)z+a −az +2+a,Re a>0{e−at :t∈R}6.5 The fourth column presents the results for the extended eigenvalues of the composition operators, and the fifth column points at the places in the paper where those results can be found. 1.1. Extended eigenvalues and extended eigenoperators Our setting is an infinite dimensional, complex separable Hilbert space H. A complex scalar λis called an extended eigenvalue of a bounded linear operator Aprovided that there exists a nonzero operator X such that AX =λXA. (1.1) Such an operator Xis called an extended eigenoperator of the operator A. The family of all the extended eigenvalues of an operator Ais called the extended spectrum of A, and it is denoted by Ext (A). Further, the notation Ext(λ, A) stands for the collection of all the extended eigenoperators corresponding to a given extended eigenvalue λ ∈Ext (A). If we accept the zero operator as an extended eigenoperator, then that collection becomes a weakly closed, linear manifold. These notions have their roots in the simultaneous and independent works of Scott Brown [6], and Kim, Moore and Pearcy [11], about a generalization of the celebrated theorem of Victor Lomonosov [20]. They proved that if an operator Ahas a nonzero, compact extended eigenoperator, then the operator Ahas a non trivial, closed hyperinvariant subspace. A systematic study of the extended eigenvalues and the extended eigenoperators of classical operators started with the work of Biswas, Lambert and the third author [2]for the Volterra operator, and continued soon after with the work of Lambert [17], who also gave sufficient conditions for an operator to have non trivial, closed hyperinvariant subspaces, and who calculated the extended spectra for scalar perturbations of the Volterra operator. This line of research has become a very active field of investigation with many contributions, both in the search for invariant subspaces [1,12,16–18], and in the computation of the extended spectral picture for some special classes of operators [2–4,8,9,13,14,19,24,28]. 1.2. Composition operators on the Hardy space We shall be dealing with composition operators defined on the Hardy space H2(D). An analytic self map ϕof the unit disk Dinduces a linear operator Cϕdefined on the space of analytic functions on the unit disk by the expression Cϕf=f◦ϕ, f :D→Canalytic. M. Lacruz et al. / J. Math. Anal. Appl. 504 (2021) 125427 3 Littlewood’s subordination principle [27] ensures that Cϕcan be restricted to a bounded linear operator on the Hardy space H2(D). Perhaps this is the most natural Hilbert space of analytic functions. Recall that the map that takes each function fto its radial limit f∗, defined almost everywhere by the expression f∗(eiθ) := lim f(reiθ)as r→1−, is an isometry from H2(D)into L2(∂D), that is, f2=⎛ ⎝1 2π 2π 0|f∗(eiθ)|2dθ⎞ ⎠ 1/2 ,f∈H2(D). A whole theory has been developed around composition operators on the Hardy space, that relates the function theoretic properties of ϕwith the operator theoretic properties of Cϕ. A very nice treatise on this subject is the book of Joel Shapiro [27]. 1.3. Classification of linear fractional transformations Recall that a linear fractional map defined on the Riemann sphere  C:= C∪{∞}is any map of the form ϕ(z)=az +b cz +d.(1.2) The condition ad −dc = 0 ensures that ϕis not constant. Applying the usual conventions to the point at infinity, it is well known that ϕbecomes an automorphism of the Riemann sphere, and from a metric point of view, ϕis also an isometry of the Riemann sphere provided with the chordal distance. We are interested in linear fractional transformations, that is, the class LFT of linear fractional maps ϕ with the property that ϕ(D) ⊆D. A necessary and sufficient condition for this property is that |bd −ac|+|ad −bc|≤|b|2−|d|2. This simple condition appears very seldom in the literature. It can be found, for instance, in the work of María J. Martín [21], or in the work of Contreras, Díaz-Madrigal, Martín, and Vukotić [7]. The latter reference provides another condition, that is somehow more involved, but has an easier proof. There is a classification of LFTs according to their fixed point configuration. We refer the reader to the paper of Joel Shapiro [26]for that classification. What is important for our purposes is that there are eight classes: elliptic automorphic (EA), parabolic automorphic (PA), parabolic non automorphic (PNA), hyperbolic automorphic (HA), hyperbolic non automorphic of the first, second or third kind (HNA I), (HNA II), (HNA III), and loxodromic (LOX). After conjugation by suitable linear fractional maps, every LFT can be expressed in a standard form that is collected in Table 1. The class HNA III of hyperbolic, non automorphic LFTs without a fixed point on ∂Dis missing in Shapiro’s classification, but there is an easy fix for this issue, because those maps have the same standard form as the loxodromic ones. 2. Preliminary results In this section we will establish several lemmas that will be used throughout the paper. We start with a folk result that applies to any Hilbert space operator. We leave its proof to the reader as a simple but important exercise. Lemma 2.1. An operator A ∈B(H)is injective if and only if 0 /∈Ext(A). 4M. Lacruz et al. / J. Math. Anal. Appl. 504 (2021) 125427 The following result is well known. It is a consequence of the open mapping theorem and the principle of analytic continuation. Lemma 2.2. Let us suppose that ϕis a nonconstant, analytic self map of the open unit disk. Then Cϕis injective. If we combine Lemma 2.1 with Lemma 2.2, we conclude that 0 is never an extended eigenvalue of a composition operator. Proposition 2.3. If ϕis an non constant analytic self map of the open unit disk, then 0 /∈Ext(Cϕ). Next, we present two basic general results about the behavior of the extended spectrum under taking the adjoint and a scalar multiplication. They can be easily derived from equation (1.1). Lemma 2.4. If A ∈B(H)then Ext(A∗) \{0} ={1/λ:λ ∈Ext(A) \{0}}. Lemma 2.5. If A ∈B(H)and α∈C\{0}then Ext(αA) =Ext(A). The following result is a necessary condition for a complex number to be an extended eigenvalue of an injective operator with a total set of eigenvectors. We will use the notation σp(A)for the point spectrum of an operator A ∈B(H), that is, the set of all eigenvalues of A. Lemma 2.6. Let A ∈B(H)be an injective operator with nonempty point spectrum. If Aadmits a total family of eigenvectors, then Ext (A) ⊆{α/β :α, β∈σp(A)}. Proof. Let Fbe a total subset of Hconsisting of eigenvectors for A. Then, let λ ∈Ext (A)and let X∈Ext(λ, A). Next, there exists f∈Fsuch that Xf =0and there exists β∈σp(A)such that Af =βf. It follows that AXf =λXAf =λβXf. Since Xf =0we conclude that α:= λβ ∈σp(A). Since Ais injective, we have β=0so that λ =α/β, as we wanted.  Multiplication operators are a nice source of extended eigenoperators for composition operators. Recall that every bounded analytic function b ∈H∞(D) induces a multiplication operator Mb∈B(H2(D)) defined by Mbf=b ·f. It is easy to check that Mbis indeed a bounded linear operator with Mb =b∞. Lemma 2.7. Let Cϕbe the composition operator induced by ϕ:D→D. Let λ ∈σp(Cϕ)and let bbe a function in H∞(D)such that Cϕb =λb. Then, λ ∈Ext(Cϕ)and CϕMb=λMbCϕ. Proof. Notice that for every f∈H2(D)we have CϕMbf=Cϕ(b·f)=(b◦ϕ)·(f◦ϕ)=(Cϕb)·(Cϕf)=λb ·Cϕf=λMbCϕf, so that CϕMb=λMbCϕ, as we wanted.  The last two results of this section about the extended spectrum of a direct sum of two operators will be useful to reduce the problem of a hyperbolic, non automorphic composition operator of the second kind (HNA II), to the one of the first kind (HNA I). Lemma 2.8. Consider a direct sum decomposition H=H1⊕H2, let A1∈B(H1)and let A2∈B(H2). Then Ext(A1) ∪Ext(A2) ⊆Ext(A1⊕A2). M. Lacruz et al. / J. Math. Anal. Appl. 504 (2021) 125427 5 Proof. Clearly, it suffices to prove that Ext(A1) ⊆Ext(A1⊕A2). To that end, let λ ∈Ext(A1)and let X∈B(H1)be a nonzero operator such that A1X=λXA1. It is not hard to verify that X⊕0is a nonzero operator in B(H) satisfying (A1⊕A2)(X⊕0) =λ(X⊕0)(A1⊕A2). Therefore, λ ∈Ext(A1⊕A2).  Lemma 2.9. Consider a direct sum decomposition H=H1⊕H2, let I1be the identity operator on H1, and let Abe an injective operator in B(H2). Then Ext(I1⊕A∗)=σp(A∗)∪σp(A)−1∪Ext(A∗). Proof. Let λ ∈Ext(I1⊕A∗)and let X∈Ext(I1⊕A∗, λ). Since X∈B(H), there is a representation of X as a block matrix with respect to the decomposition H=H1⊕H2, say X=X11 X12 X21 X22 . A simple computation shows that the equation (I1⊕A∗)X=λX(I1⊕A∗)is equivalent to the following system of equations: (1 −λ)X11 =0,X 12(I1−λA∗)=0, (A∗−λI1)X21 =0,A ∗X22 −λX22A∗=0. Since X=0, one of the four operators Xjk must be nonzero. If X11 =0, then λ =1, which is always an extended eigenvalue of any operator. If X22 =0then λ ∈Ext(A∗), and if X21 =0, then λ ∈σp(A∗). Finally, if X12 =0then λ =0and (I1−λA)X∗ 12 =0, so that 1/λ ∈σp(A), and the proof of the first inclusion is complete. Conversely, if one operator Xj0k0is nonzero, then we get an extended eigenoperator for I1⊕A∗ by taking the other blocks Xjk equal to zero, and this fact completes the proof of the second inclusion, which yields the desired result.  3. Elliptic and loxodromic maps Recall that if ϕis an elliptic automorphism, then ϕhas two fixed points, one in Dand another in  C\D. Given this fixed point configuration, upon conjugation by disk automorphisms, we can suppose that the fixed points are 0, ∞, which leads to the standard form ϕ(z) =ωz. Let us observe that the conjugation by disk automorphisms induces a similarity at the operator level, and the extended spectra are invariant under similarities. Notice that ωnis an eigenvalue of Cϕ, and a corresponding eigenfunction is the monomial en(z) =zn, for every n ∈N0. Theorem 3.1. Let ϕ(z) =ωz be an elliptic automorphism of the unit disk. Then Ext(Cϕ)={ωn:n∈Z}. Proof. The inclusion Ext(Cϕ) ⊆{ωn:n ∈Z}is a consequence of Lemma 2.6. Indeed, we know from Lemma 2.2 that Cϕis injective. On the other hand, the set {en:n ∈N0}of eigenfunctions for Cϕis a total subset of H2(D). Next, each function enis bounded on D, and it follows from Lemma 2.7 that {ωn:n ∈N0} ⊆Ext(Cϕ). Thus, it is sufficient to show the inclusion {ω−n:n ∈N} ⊆Ext(Cϕ). Indeed, fix k∈Nand let X=M∗ ek. Notice that Xen=en−kfor n ≥k, so that 6M. Lacruz et al. / J. Math. Anal. Appl. 504 (2021) 125427 CϕXen=Cϕen−k=ωn−ken−k=ωn−kXen=ω−kXCϕen.(3.1) Since Xen=0for all n <k, the equation (3.1)is valid for all n ∈N0. Finally, the family {en:n ∈N0} spans a dense linear manifold in H2(D), so that CϕX=ω−kXCϕ, and the proof is complete.  Recall that if ϕis a loxodromic map or a hyperbolic nonautomorphism of the third kind ϕcan be assumed to have one fixed point at infinity, while the other one, say c, belongs to D. In this situation ϕhas the standard form ϕ(z)=a(z−c)+c, (3.2) with |a|+|1 −a| ·|c| ≤1, if c =0, and |a| <1, if c =0. The hyperbolic nonautomorphism of the third kind corresponds to the case a >0. In this situation, the composition operator Cϕhas a countable point spectrum, namely σp(Cϕ) ={ϕ(c)n:n ∈N0}. Further, each eigenvalue is of multiplicity one and a corresponding eigenfunction is given by σ(z)=z−c. (3.3) We also know that Cϕσn=ϕ(c)nσn, and that the span of the eigenfunctions {σn:n ∈N0}is a dense linear manifold in H2(D). Notice that σ−1∈L∞(T), hence the multiplication operator Mσ−1is bounded on L2(T). One knows that H2(D)can be identified with a subspace of L2(T) consisting of functions whose negative Fourier coefficients vanish. Therefore, let us regard Mσ−1as a bounded operator from H2(D)to L2(T). Let Pbe the orthogonal projection from L2(T)onto H2(D), and consider the Toeplitz operator Tdefined by Tf =PMσ−1f, for all f∈H2(D).(3.4) Clearly, Tσm=σm−1, for m ≥1. On the other hand, 1 σ(eiθ)=1 eiθ −c=e−iθ 1 1−ce−iθ =e−iθ ∞  n=0 cne−inθ, which shows that T1 =Pσ−1=0. Theorem 3.2. Let ϕbe a map that is either loxodromic or hyperbolic non automorphic of the third kind. Then, we have Ext(Cϕ)={ϕ(c)n:n∈Z}. Proof. Since σp(Cϕ) ={ϕ(c)n:n ∈N0}, it follows from Lemma 2.6 and Proposition 2.3 that Ext(Cϕ) ⊆ {ϕ(c)n:n ∈Z}. By Lemma 2.7, {ϕ(c)n:n ∈N0} ⊆Ext(Cϕ). Therefore, it remains to establish that {ϕ(c)−n:n ∈N} ⊆Ext(Cϕ). We will show for the operator Tdefined by (3.4), that Tnis an extended eigenoperator of Cϕcorresponding to the extended eigenvalue ϕ(c)−n. Since the set {σm:m ∈N0}spans a dense linear manifold in H2(D), it suffices to check that CϕTnσm=ϕ(c)−nTnCϕσm, for all m ∈N0and all n ∈N. The last identity follows from a straightforward computation.  M. Lacruz et al. / J. Math. Anal. Appl. 504 (2021) 125427 7 4. Hyperbolic automorphic maps If ϕis a hyperbolic automorphism of the unit disk, its fixed points lie on the boundary of the unit circle, and upon conjugation by disk automorphisms, we can suppose that the fixed point of ϕare the points 1 and −1. Moreover, we can suppose that ϕhas the following standard form ϕ(z)= z+r 1+rz,for some 0 <r<1.(4.1) In what follows we will take advantage of some spectral information about Cϕthat can be found in [26]. Let R=(1 +r)/(1 −r). Every point in the open annulus G:= {α∈C:R−1/2<|α|<R 1/2},(4.2) belongs to the point spectrum of Cϕ. Moreover, every eigenvalue of Cϕcan be written as γ(w):=Rw,(4.3) where wbelongs to the open strip Ω:={w∈C:−1/2<Re (w)<1/2}.(4.4) A corresponding eigenfunction is given by ew(z):=1+z 1−zw .(4.5) The notion of an operator with rich point spectrum has been introduced recently as a way to determine the extended eigenvalues for Cesàro operators [14]and bilateral weighted shifts [13,14]. An operator A ∈B(H)is said to have a rich point spectrum provided that int σp(A) =∅and for every open disk D⊆σp(A), the family of the corresponding eigenvectors  z∈D ker(A−z) (4.6) is a total subset of H. The following result [14, Lemma 7.1] is a sufficient condition for an operator Ato have rich point spectrum. Lemma 4.1. Let A ∈B(H)and assume there are an open connected set Ω ⊆C, an analytic mapping h:Ω →Hand a nonconstant analytic function γ:Ω →Csuch that (i) h(w) ∈ker[A −γ(w)] \{0}for all w∈Ω, (ii) {h(w): w∈Ω}is a total subset of H, and (iii) σp(A) ⊆clos γ(Ω). Then Ahas rich point spectrum. It turns out that Lemma 4.1 applies to the composition operators under consideration. Theorem 4.2. If ϕis a hyperbolic automorphism of D, then Cϕhas rich point spectrum. 8M. Lacruz et al. / J. Math. Anal. Appl. 504 (2021) 125427 Proof. The property of having rich point spectrum is invariant under similarity, so we may assume that ϕis in the standard form given by the equation (4.1). It is fairly easy to check that the hypotheses of Lemma 4.1 are satisfied with A =Cϕ, and with Ω, ewand γas in the equations (4.3), (4.4)and (4.5), and the mapping h:Ω →H2(D) defined by h(w) := ew. It is clear that Ωis an open, connected set and that γis an analytic function such that γ(Ω) =G =σp(Cϕ), so that the condition (iii) is satisfied. Finally, the condition (i) means that h(w)is an eigenfunction of Cϕ, and we refer the reader to [23]for the proof of the condition (ii). Thus, it remains to show that his an analytic mapping, that is, for every g∈H2(D), the function f:Ω →Cdefined by f(w) =h(w), gis analytic. This follows easily from a suitable application of [15, Lemma 6.6].  The next result [14, Theorem 3.3] establishes a useful fact about operators with rich point spectrum whose point spectrum is between the interior and the closure of an annulus. Lemma 4.3. If an operator A ∈B(H)has rich point spectrum and there are constants C, c >0so that {z∈C:c <|z| <C} ⊆σp(A) ⊆{z∈C:c ≤|z| ≤C}, then Ext(A) ⊆∂D. Now we are ready to describe the family of all the extended eigenvalues for a hyperbolic automorphic composition operator. Theorem 4.4. If Cϕis the composition operator induced by a hyperbolic automorphism of the unit disk then the set of all extended eigenvalues for Cϕis the unit circle, that is, Ext(Cϕ) =∂D. Proof. First, we show that ∂D⊆Ext (Cϕ). Notice that ∂D={γ(it): t ∈R}, where γis given by (4.3). Further, γ(it)is an eigenvalue of Cϕand a corresponding eigenfunction eit is bounded on the unit disk. It follows from Lemma 2.7 that γ(it) ∈Ext(Cϕ). In the other direction, we know from Theorem 4.2 that Cϕhas rich point spectrum, and moreover, its point spectrum σp(Cϕ)is the open annulus given by equation (4.2). Hence, the inclusion Ext(Cϕ) ⊆∂D follows at once from Lemma 4.3. 5. Hyperbolic non automorphic maps In this section, ϕis a hyperbolic, non automorphic linear fractional transformation of the unit disk. First, we consider the class HNA I, which corresponds to the following fixed point configuration: the first fixed point on ∂Dand the second in  C\D. In this case, we may assume that the fixed points are z=1and z=∞. This leads to the standard form ϕ(z)=rz +(1−r),for some 0 <r<1.(5.1) Deddens [10, Theorem 3 (iv)] proved that the point spectrum of Cϕis the punctured disk σp(Cϕ)={λ∈C:0<|λ|<r −1/2}.(5.2) Moreover, he proved that for every w∈Cwith Re(w) >−1/2, the function ew(z) =(1 −z)wis an eigenfunction of Cϕcorresponding to the eigenvalue λ =rw, that is, (Cϕew)(z)=rwew(z).(5.3) In order to compute the extended eigenvalues of Cϕwe make use of the following result. M. Lacruz et al. / J. Math. Anal. Appl. 504 (2021) 125427 9 Theorem 5.1. Let ϕbe a hyperbolic, non automorphic linear fractional selfmap of the unit disk, with a fixed point in  C\D. Then Cϕhas rich point spectrum. Proof. The strategy of the proof is exactly the same as in the proof Theorem 4.2. Namely, we will apply Lemma 4.1 to the operator A =Cϕ, the open half plane Ω := {w∈C:Re(w) >−1/2}, the mapping h:Ω →H2(D) defined by h(w)(z) =(1 −z)w, and the function γ:Ω →Cdefined by the expression γ(w) = rw. Then, conditions (i) and (iii) of Lemma 4.1 follow from (5.3)and (5.2). Also, regarding condition (ii), it suffices to show that the linear span of {en:n ∈N0}is dense in H2(D). This is obvious because, for every n ∈N0, the polynomials {(1 −z)k:0 ≤k≤n}generate the same linear manifold as the monomials {zk:0 ≤k≤n}. Finally, the fact that his an analytic mapping can be derived in the same way as in the proof of Theorem 4.2. It was shown in [14, Theorem 3.1] that the shape of the point spectrum determines the extended spectral picture in the following sense. Lemma 5.2. If A ∈B(H)has rich point spectrum and λ ∈Ext(A)then λ ·int σp(A) ⊆clos σp(A). This result leads to the description of the set of all extended eigenvalues of Cϕ. Theorem 5.3. Let ϕbe a hyperbolic, nonautomorphic linear fractional selfmap of the unit disk with a fixed point in  C\D. Then, we have Ext(Cϕ) =D\{0}. Proof. Once again, we may assume that ϕis in the standard form given by equation (5.1). Let λ ∈D\{0}. Then, there exists w∈Csuch that Re (w) ≥0and λ =rw. Thus, λis an eigenvalue of Cϕand a corresponding eigenfunction is the bounded analytic function ew(z) =(1 −z)w. It follows from Lemma 2.7 that D\{0} ⊆Ext (Cϕ). Conversely, let λ ∈Ext (Cϕ). We know from Proposition 2.3 that λ =0. Since Theorem 5.1 states that ϕhas rich point spectrum, an application of Lemma 5.2 gives λ ·[D(0, r−1/2) \{0}] ⊆D(0, r−1/2), and this easily yields |λ| ≤1, as we wanted.  Now, let us consider the maps in the class HNA II, that is, the LFTs with one fixed point on ∂Dand the other one in D. We can assume without loss of generality that a fixed point is z=0and the other one is z=1. This yields the standard form ϕ(z)= rz 1−(1 −r)z,for some 0 <r<1.(5.4) The case HNA II can be reduced to the case HNA I by using Lemma 2.8 and Lemma 2.9, in combination with the following result, due to Shapiro [26, p. 864] using ideas from his joint paper with Bourdon [5]. Theorem 5.4. If ϕis given by formula (5.4)and ψ(z) =rz +(1 −r), then Cϕis unitarily equivalent to the operator I1⊕rC∗ ψ, where I1is the identity operator on the subspace of constant functions. Now we immediately get a result about the extended spectrum of Cϕ. Theorem 5.5. If ϕis a hyperbolic, non automorphic linear fractional transformation of the unit disk with a fixed point in D, then Ext (Cϕ) ⊇C\D. Proof. First, we may assume without loss of generality that ϕis given by the formula (5.4). We know from Theorem 5.4 that Cϕis unitarily equivalent to I1⊕rC∗ ψ, so that Ext (Cϕ) =Ext(I1⊕rC∗ ψ). Next, it follows from Lemma 2.5 and Lemma 2.8 that