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Conjugations in L2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} ...

Conjugations in L2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym}... Conjugations in space L of the unit circle commuting with multiplication by z or intertwining multiplications by z and z ¯ are characterized. We also study their behaviour with respect to the Hardy space, subspaces invariant for the unilateral shift and model spaces. Keywords Conjugation · C–symmetric operator · Hardy space · Invariant subspaces for the unilateral shift · Model space · Truncated Toeplitz operator Mathematics Subject Classification Primary 47B35; Secondary 30D20 · 30H10 The work of the first author was partially supported by FCT/Portugal through UID/MAT/04459/2019 and the research of the second and the fourth authors was financed by the Ministry of Science and Higher Education of the Republic of Poland. Kamila Klis–Garlicka ´ rmklis@cyfronet.pl M. Cristina Câmara cristina.camara@tecnico.ulisboa.pt Bartosz Łanucha bartosz.lanucha@poczta.umcs.lublin.pl Marek Ptak rmptak@cyf-kr.edu.pl Center for Mathematical Analysis, Geometry and Dynamical Systems, Mathematics Department, Instituto Superior Técnico, Universidade de Lisboa, Av. Rovisco Pais, 1049–001 Lisboa, Portugal Department of Applied Mathematics, University of Agriculture, ul. Balicka 253c, 30-198 Kraków, Poland Department of Mathematics, Maria Curie-Skłodowska University, Maria Curie-Skłodowska Square 1, 20-031 Lublin, Poland 0123456789().: V,-vol 22 Page 2 of 14 M. C. Câmara et al. 1 Introduction Let H be a complex Hilbert space and denote by B(H) the algebra of all bounded linear operators on H.A conjugation C in H is an antilinear isometric involution, i.e., C = id and Cg, Ch=h, g for g, h ∈ H. (1.1) Conjugations have recently been intensively studied and the roots of this subject comes from physics. An operator A ∈ B(H) is called C–symmetric if CAC = A (or equivalently AC = CA ). A strong motivation to study conjugations comes from the study of complex symmetric operators, i.e., those operators that are C–symmetric with respect to some conjugation C. For references see for instance [2,3,7–10]. Hence obtaining the full description of conjugations with certain properties is of great interest. Let T denote the unit circle, and let m be the normalized Lebesgue measure on T. 2 2 ∞ ∞ 2 Consider the spaces L = L (T,m), L = L (T, m), the classical Hardy space H 2 ∞ on the unit disc D identified with a subspace of L , and the Hardy space H of all analytic and bounded functions in D identified with a subspace of L . Denote by M 2 ∞ the operator defined on L of multiplication by a function ϕ ∈ L . 2 2 The most natural conjugation in L is J defined by Jf = f ,for f ∈ L .This conjugation has two natural properties: the operator M is J –symmetric, i.e., M J = z z JM , and J maps an analytic function into a co-analytic one, i.e., JH = H . z ¯ df 2  # # Another natural conjugation in L is J f = f with f (z) = f (z ¯). The conjugation J has a different behaviour: it commutes with multiplication by z 2 2 (M J = J M ) and leaves analytic functions invariant, J H ⊂ H .The map z z J appears for example in connection with Hankel operators (see [11, pp. 146–147]). Its connection with model spaces was studied in [4] (Lemma 4.4, see also [1, p. 37]). Hence a natural question is to characterize conjugations with respect to these proper- ties. The first step was done in [2] where all conjugations in L with respect to which the operator M is C–symmetric were characterized, see Theorem 2.2. In Sect. 2 we give a characterization of all conjugations which commute with M , Theorem 2.4.In Sect. 3, using the above characterizations we show that there are no conjugations in L leaving H invariant, with respect to which the operator M is C–symmetric. We also show that J is the only conjugation commuting with M and leaving H invariant. 2 2 Beurling’s theorem makes subspaces of H of the type θ H (θ inner function, i.e., θ ∈ H , |θ|= 1a.e.on T) exceptionally interesting, as the only invariant subspaces for the unilateral shift S, Sf (z) = zf (z) for f ∈ H . On the other hand, model spaces 2 2 (subspaces of the type K = H  θ H ), which are invariant for the adjoint of the unilateral shift, are important in model theory, [12]. In [2] all conjugations C with respect to which the operator M is C–symmetric and mapping a model space K into another model space K were characterized, with α θ the assumption that α divides θ (α  θ). Recall that for α and θ inner, α  θ means that θ/α is also an inner function. In what follows we will show that the result holds without the assumption α  θ. 2 Conjugations in L and their invariants Page 3 of 14 22 In Sect. 4 conjugations commuting with M and preserving model spaces are described. Section 5 is devoted to conjugations between S–invariant subspaces (i.e., subspaces of the form θ H with θ an inner function). In the last section we deal with θ θ conjugations commuting with the truncated shift A (A = P M where P is the θ z|K θ z z θ 2 θ orthogonal projection from L onto K ) or conjugations such that A is C–symmetric with respect to them. 2 M and M –commuting conjugations in L z z 2 2 Denote by J the conjugation in L defined as Jf = f ,for f ∈ L . This conjugation has the following obvious properties: Proposition 2.1 (1) M J = JM ; z z ¯ (2) M J = JM for all ϕ ∈ L ; ϕ ϕ ¯ (3) JH = H . Let us consider all conjugations C in L satisfying the condition M C = CM . (2.1) z z ¯ Such conjugations were studied in [2] and are called M –conjugations. The following theorem characterizes all M –conjugations in L . Theorem 2.2 [2] Let C be a conjugation in L . Then the following are equivalent: (1) M C = CM , z z ¯ (2) M C = CM for all ϕ ∈ L , ϕ ϕ ¯ (3) there is ψ ∈ L , with |ψ|= 1, such that C = M J, (4) there is ψ ∈ L , with |ψ |= 1, such that C = JM . Another natural conjugation in L is defined as df # # J f = f with f (z) = f (z ¯). (2.2) The basic properties of J are the following: Proposition 2.3 (1) M J = J M ; z z (2) M J = J M ; z ¯ z ¯ (3) M J = J M for all ϕ ∈ L ; 2 2 (4) J H = H . In the context of the Proposition 2.3 it seems natural to consider all conjugations C in L commuting with M , i.e., M C = CM . (2.3) z z 22 Page 4 of 14 M. C. Câmara et al. Such conjugations will be called M –commuting. In what follows we will often deal with functions f ∈ L such that f (z) = f (z ¯) a.e. on T, which will be called symmetric. Observe that if f is symmetric and f ∈ H , then we also have f ∈ H and so it is a constant function. Theorem 2.4 Let C be a conjugation in L . Then the following are equivalent: (1) M C = CM , z z (2) M C = CM # for all ϕ ∈ L , (3) there is a symmetric unimodular function ψ ∈ L such that C = M J , (4) there is a symmetric unimodular function ψ ∈ L such that C = J M . Proof We will show that (1) ⇒ (3).The other implications are straightforward. Assume that M C = CM . Then M CJ = CM J = CJ M . It follows that the linear z z z z z operator CJ commutes with M .By[13, Theorem 3.2], there is ψ ∈ L such that CJ = M . Hence C = M J = J M # . ψ ψ By (1.1) for any f , g ∈ L we have g fdm(z) =g, f =Cf , Cg # # 2 =ψ f ,ψg = |ψ(z)| f (z ¯)g(z ¯) dm(z) = |ψ(z ¯)| f (z)g(z) dm(z). 2 2 Hence |ψ|= 1a.e.on T. On the other hand, since C = I 2, for all f ∈ L we have 2    # # f = C f = M J M J f = M J (ψ f ) = ψψ f , ψ ψ ψ which implies that ψψ = 1a.e.on T. Therefore ψ is symmetric, i.e., ψ(z) = ψ(z ¯) a.e. on T. 3 Conjugations preserving H In the previous section all M –conjugations C in L , i.e., such that M C = CM z z ¯ or M –commuting conjugations, i.e., such that M C = CM z z were characterized. Let us now consider the question which of them preserve H . 2 2 2 2 2 Clearly, if C is a conjugation in L and C (H ) ⊂ H , then C (H ) = H . Since J 2 2 preserves H , it can be considered as a conjugation in H . The following result shows that J is in that sense unique. 2 Conjugations in L and their invariants Page 5 of 14 22 2 2 2 Corollary 3.1 Let C be an M –commuting conjugation in L .IfC (H ) ⊂ H , then C = λJ for some λ ∈ T. Proof By Theorem 2.4 we have that C = M J for some ψ ∈ L with |ψ|= 1 and ψ(z) = ψ(z) a.e. on T. Since C preserves H ,wehave ∞ 2 ∞ ψ = M J (1) = C (1) ∈ L ∩ H = H . Thus ψ is analytic. Since it is symmetric, it is also co-analytic. Hence ψ must be a constant function, so ψ = λ ∈ C and |λ|=|ψ|= 1. 2 2 Corollary 3.2 There are no M –conjugations in L which preserve H . Proof If C is an M –conjugation in L , then by Theorem 2.2 it follows that C = M J z ψ for some ψ ∈ L with |ψ|= 1. As in the proof of Corollary 3.1 the assumption 2 2 ∞ C (H ) ⊂ H implies that ψ ∈ H , which in turn means that ψ is an inner function. Moreover, for n = 0, 1, 2,... we have n+1 n+1 n 0 =Cz , z=ψz , z=ψ, z = 0. So ψ = 0 which is a contradiction. The following example shows that not all conjugations in L satisfy either (2.1)or (2.3). Example 3.3 There is a set of naturally defined conjugations. For k, l ∈ Z, k < l, 2 2 define C : L → L by k,l n k l n C a z = a z + a z + a z , (3.1) k,l n l k n n∈Z n∈{ / k,l} n 2 where {z } is the standard basis in L . Then (2.1) and (2.3) are not satisfied since k l l+1 k k−1 k−1 M C (z ) = M (z ) = z , C M (z ) = C (z ) = z z k,l z k,l z ¯ k,l and k+1 z if k + 1 = l, k k+1 C M (z ) = C (z ) = k,l z k,l z if k + 1 = l. Note that, on the other hand, C preserves H whenever k  0or l < 0. k,l 22 Page 6 of 14 M. C. Câmara et al. 4 Conjugations preserving model spaces There is another class of conjugations in L which appear naturally in connection with model spaces. For a nonconstant inner function θ, denote by K the so called model 2 2 2 space of the form H  θ H . The conjugation C defined in L by C f = θz ¯ f has the important property that it preserves the model space K , i.e., C K = K . θ θ θ θ Thus C can be considered as a conjugation in K . Such conjugations are important θ θ in connection with truncated Toeplitz operators (see for instance [6]). Here we present several simple properties of such conjugations, which we will use later. Proposition 4.1 Let α, β, γ be nonconstant inner functions. Then (1) C C = M , β α βα ¯ (2) M C M is a conjugation in L , γ α γ¯ (3) C M = M C . β γ γ¯ β Now we will consider relations between M –conjugations and model spaces. The theorem below was proved in [2, Theorem 4.2] with the additional assumption that α  θ. As we prove here, this assumption is not necessary. Theorem 4.2 Let α, γ , θ be inner functions (α, θ nonconstant). Let C be a conjugation in L such that M C = CM . Assume that C (γ K ) ⊂ K . Then there is an inner z z ¯ α θ function β such that C = C , with γα  β  γθ and α  θ. Proof Recall the standard notation for the reproducing kernel function at 0 in K , α α α namely, k = 1−α(0)α and its conjugate k = C k =¯z(α −α(0)). By Theorem 2.2 0 0 0 we know that C = M J for some function ψ ∈ L , |ψ|= 1. Hence α α ˜ ˜ K  C (γ k ) = M J (γ k ) = ψ γ z ¯(α − α(0)) =¯ γ α ¯ zψ(1 − α(0)α). θ ψ 0 0 −1 Thus there is h ∈ K such that h =¯ γ α ¯ zψ(1 − α(0)α). Since (1 − α(0)α) is a bounded analytic function, we have −1 2 γ¯ α ¯ zψ = h(1 − α(0)α) ∈ H . Since γ¯ α ¯ zψ ∈ H and |¯ γ α ¯ zψ|= 1a.e.on T, it has to be an inner function. Moreover β = zψ has to be inner and divisible by γα, i.e., γα  β. On the other hand, we have similarly K  C C (γ k ) = C (ψγ(1 − α(0)α) = θγ z ¯ψ(1 − α(0)α), θ θ θ ¯ ¯ and θγ β = θγ z ¯ψ ∈ H . Hence β divides θγ , i.e., β  γθ. It is clear that C = C . Finally, we have α  θ as a consequence of γα  β  γθ. 2 Conjugations in L and their invariants Page 7 of 14 22 Note that if α  θ and C = C for some inner β with γα  β  γθ, then K ⊂ K ⊂ K , C M = C and α β θ β γ β γ γ C (γ K ) = C M (K ) = C (K ) ⊂ K ⊂ K . β β α β γ α α θ γ γ Hence the implication in Theorem 4.2 is actually an equivalence. The corollary bellow strengthens [2, Proposition 4.5]. Corollary 4.3 Let α, θ be nonconstant inner functions, and let C be a conjugation in L such that M C = CM . Assume that C (K ) ⊂ K . Then α  θ and there is an z z ¯ α θ inner function β such that C = C , with α  β  θ. Let us turn to discussing the relations between M –commuting conjugations and model spaces. The following proposition describes some more properties of J . Proposition 4.4 Let α be an inner function. Then 2 # 2 (1) J (αH ) = α H ; (2) J (K ) = K ; (3) J C = C J . Proof The condition (1) is clear, (2) and (3) were proved in [4, Lemma 4.4]. Hence the conjugation J has a nice behaviour in connection with model spaces, namely J (K ) = K # . Theorem 4.6 below says that the conjugation J is, in some α α sense the only M –commuting conjugation with this property. We start with the following: Proposition 4.5 Let α, γ , θ be inner functions (α, θ nonconstant). Let C be an M – 2 # commuting conjugation in L . Assume that C (γ K ) ⊂ K . Then α  θ and there is α θ an inner function β with γα  β  γθ such that C = J M β . γ¯ γα Proof Observe that since C is an M –commuting conjugation, taking antilinear adjoints and applying [2, Proposition 2.1] we get M C = CM . Since by Propo- z ¯ z ¯ sition 4.1 the antilinear operator M C M is a conjugation, then J CM C M is γ α γ γ α γ also a conjugation. Note also that M C M M = M M C M . Hence γ α γ z z ¯ γ α γ J CM C M M = M J CM C M . (4.1) γ α γ z z ¯ γ α γ On the other hand, J CM C M (γ K ) ⊂ J CM C (K ) γ α γ α γ α α ⊂ J C (γ K ) ⊂ J (K ) ⊂ K # . α θ By Theorem 4.2 there is an inner function β such that J CM C M = C , with γ α γ β # # γα  β  γθ and α  θ . Hence C = J C M C M . Therefore β γ α γ 22 Page 8 of 14 M. C. Câmara et al. C = J M β γ¯ γα As in Theorem 4.2 the implication in Proposition 4.5 can be reversed. Indeed, if #  # α  θ and C = J M β for some inner function β with γα  β  γθ , then γα K ⊂ K ⊂ K # and C (γ K ) = J M (K ) = J C C (K ) ⊂ J (K ) ⊂ J (K ) = K . β β # α α α α β θ γ γ Theorem 4.6 Let α, θ be nonconstant inner functions, and let C be an M –commuting 2 # conjugation in L , i.e., M C = CM . Assume that C (K ) ⊂ K . Then α  θ and z z α θ C = λJ with λ ∈ T. Corollary 4.7 Let C be an M –commuting conjugation in L . Assume that there is some nonconstant inner function θ such that C (K ) ⊂ K #. Then C = λJ with λ ∈ T. Proof of Theorem 4.6 By Proposition 4.5 there is an inner function β with α  β  θ such that C = J M β . The function is inner and by Theorem 2.4 it is symmetric. As observed before it follows that it is constant. Hence C = J up to multiplication by a constant of modulus 1. 5 Conjugations preserving S-invariant subspaces of H Beurling’s theorem says that all invariant subspaces for the unilateral shift S are of the 2 2 form θ H with θ inner. We will now investigate conjugations in L which preserve subspaces of this form. Since C transforms θ H onto zH , the operator C J C = M J M θ θ θ is an example of such a conjugation. Note that (C J C )M = M (C J C ). θ θ z z θ θ 2 2 Let α, θ be two inner functions. Then the operator C J C : L → L is an θ α 2 2 antilinear isometry which maps αH onto θ H and commutes with M . This operator however does not have to be an involution. Lemma 5.1 Let α, θ be two inner functions. The operator C J C is an involution θ α 2 # (and hence a conjugation in L ) if and only if the function θ α is symmetric (or # # equivalently αα = θθ ). 2 Conjugations in L and their invariants Page 9 of 14 22 Proof Note that by Proposition 4.1, (C J C )(C J C ) = C C #C #C = M M # = M . θ α θ α θ α # # # α θ θ α θ α θ α θ α Therefore C J C is an involution if and only if θ α # # # θ α θ α =1a.e.on T, i.e., θθ = αα , which means that # # # # (θ α )(z) = (θ α)(z) = θ(z)α (z ¯) = (θ α )(z) a.e. on T. The theorem bellow characterizes all M –commuting conjugations mapping one S– invariant subspace into another S–invariant subspace. Theorem 5.2 Let θ and α be two inner functions and let C be a conjugation in L such 2 2 # # that C M = M C. Then C (αH ) ⊂ θ H if and only if θθ  αα and C = C J C , z z β α # # where β is an inner function such that θ  β, ββ = αα . Moreover, in that case 2 2 C (αH ) = β H . Let α be a fixed inner function. By Lemma 5.1, for each inner function β with # # 2 2 ββ = αα there exists an M –commuting conjugation C which maps αH onto β H , namely C = C J C . On the other hand, if β is an inner function and there exists β α 2 2 an M –commuting conjugation C which maps αH onto β H , then by Theorem 5.2, # #  # ββ  αα and C = C J C for some inner function γ such that β  γ , γγ = γ α # 2 2 2 αα . In particular, C (αH ) = γ H = β H and so γ is a constant multiple of β, # # ββ = αα . It follows from the above that Lemma 5.3 characterizes all possible spaces of type 2 2 β H such that for a given S–invariant subspace αH there is an M –commuting 2 2 conjugation mapping αH onto β H . Lemma 5.3 Let α be a nonconstant inner function. Then # # {β : β is inner, αα = ββ } ={λ uv : u,v are inner, α = uv, λ ∈ T}. (5.1) For two inner functions α and β denote by α ∧ β the greatest common divisor of α and β. We will write α ∧ β = 1 if the only common divisor of α and β is a constant function. # # # Proof Note that for α = uv and β = λuv we have αα = ββ , hence one inclusion is proved. For the other inclusion let u = α ∧ β and we can write α = uv and β = uv . # # From the condition αα = ββ it follows that # # # # uvu v = uv u v . 1 22 Page 10 of 14 M. C. Câmara et al. # # # # Hence vv = v v . Since v ∧ v = 1, we have that v divides v and v divides v and 1 1 1 1 1 # # vice–versa. Thus we can take v = λv with λ ∈ T, and so β = λuv . 2 2 Proof of Theorem 5.2 Assume firstly that CM = M C and C (αH ) ⊂ θ H .By z z Theorem 2.4, C = M J for some unimodular symmetric function ψ ∈ L .In particular, #  2 ψα = M J (α) = C (α) ∈ θ H , 2 # and there exists u ∈ H such that ψα = θu. Note that u must be inner and so # # # # ψ = βα with β = θu, θ  β. Clearly βα is symmetric, i.e., ββ = αα . Hence # # θθ  αα . # # Assume now that θθ  αα , and let α = α · (α ∧ θ) and θ = θ · (α ∧ θ). Since 1 1 # # # # # # # # # # # (α ∧ θ) = α ∧ θ , we get α = α · (α ∧ θ ) and θ = θ · (α ∧ θ ).Notealso 1 1 # # # that θ θ  α α and θ ∧ α = 1, so θ  α . Thus 1 1 1 1 1 1 1 1 # # αα α α α α 1 1 1 1 # = = = uu , # # # θθ θ θ θ θ 1 1 where u = is an inner function. Now we may take β = θu. Since θ  β and # # ββ = αα , by Lemma 5.1 and by Proposition 4.1, C = M J = C J C is a # β α βα 2 2 2 conjugation which maps αH onto β H ⊂ θ H . Corollary 5.4 Let θ be an inner function and let C be an M –commuting conjugation in L . Then 2 2 (1) C (θ H ) ⊂ θ H if and only if C = λC J C with λ ∈ T; θ θ 2 # 2 (2) C (θ H ) ⊂ θ H if and only if C = λJ with λ ∈ T. 2 2 Proof By Theorem 5.2, C (θ H ) ⊂ θ H if and only if there exists an inner function # # β such that θ  β and ββ = θθ . This is only possible if β is constant multiple of θ and (1) is proved. The proof of (2) is similar. Note that by Theorem 5.2 (Lemma 5.1, actually) if θ α is symmetric, then there 2 2 exists an M –commuting conjugation from αH into θ H . The following example shows that in that case there may be no such conjugation between the corresponding model spaces K and K . α θ Example 5.5 Fix a, b ∈ D such that a = b, a = a and b = b, and put a−z b−z a−z b−z α(z) = and θ(z) = . 1−az 1−az 1−bz 1−bz Then # # a−z b−z a−z b−z α (z) = and θ (z) = 1−az 1−bz 1−az 1−bz 2 Conjugations in L and their invariants Page 11 of 14 22 # # 2 and so αα = θθ . Thus there exists an M –commuting conjugation from αH onto 2 # # θ H . In this case however neither α  θ nor θ  α , so by Theorem 4.6 no M – commuting conjugation between K and K exists. Here also neither α  θ nor α θ θ  α, and so by Theorem 4.2 no M –conjugation between K and K exists. z α θ Finally, consider M –conjugations preserving S–invariant subspaces. Proposition 5.6 Let θ and α be two inner functions. There are no M –conjugations in 2 2 2 L which map αH into θ H . Proof If C was such a conjugation, then by Theorem 2.2, C = M J for some uni- modular function ψ ∈ L and, in particular, C (α) = ψ α = θ g for some g ∈ H . Clearly g must be an inner function and ψ = αθ g. Then, for every h ∈ H , C (αh) = αθ gαh = θ gh ∈ θ H , and so gh ∈ H .Itfollows that g = 0 and C (α) = 0 which is a contradiction. 6 Conjugations and truncated Toeplitz operators 2 θ For ϕ ∈ L define the truncated Toeplitz operator A by θ ∞ A f = P (ϕ f ), for f ∈ H ∩ K , θ θ 2 θ were P : L → K is the orthogonal projection (see [14]). The operator A is closed θ θ and densely defined, and if it is bounded, it admits a unique bounded extension to K . The set of all bounded truncated Toeplitz operators on K is denoted by T (θ ). θ θ θ ∞ Note that A ∈ T (θ ) for ϕ ∈ L . It is known that every operator from T (θ ) is C –symmetric (see [14, Lemma 2.1]). Observe that if k  0, then the conjugation C defined by (3.1) satisfies neither k,l M C = C M nor SC = C S. However, for 0  n < k and θ(z) = z , z k,l k,l z k,l k,l θ θ C (K ) = K and A C = C A k,l θ θ k,l k,l z z (since here C = J and θ = θ). k,l|K |K Theorem below characterizes conjugations intertwining truncated shifts A and A . Theorem 6.1 Let θ be a nonconstant inner function and let C be a conjugation in L such that C (K ) ⊂ K #. Then the following are equivalent: θ ∞ (1) A C = CA on K for all ϕ ∈ H , # θ θ θ (2) A C = CA on K , z z 22 Page 12 of 14 M. C. Câmara et al. θ θ (3) there is a function ψ ∈ H such that C = J A and A is an isometry, |K θ ψ ψ # # ∞ θ  θ (4) there is a function ψ ∈ H such that C = A J and A is an isometry. |K ψ |K ψ θ θ Proof We will only prove that (2) ⇒ (3). Since J (K ) = K and J A = A J θ # ϕ for all ϕ ∈ H (see [4, Lemma 4.5]), we have θ  θ θ J CA = J A C = A J C z z z θ ∞ θ on K and so J C = A for some ψ ∈ H ([5, Theorem 14.38]). Hence A is θ |K θ ψ ψ an isometry and C = J A . |K It is much more restrictive if θ = θ. Proposition 6.2 Let θ be an inner function such that θ = θ and let C be a conjugation in K . Then the following are equivalent: θ θ ∞ (1) A C = CA for all ϕ ∈ H , # ϕ θ θ (2) A C = CA , z z (3) C = λJ with λ ∈ T. |K Proof Implications (1) ⇒ (2) and (3) ⇒ (1) are clear. To prove (2) ⇒ (3) apply Theorem 6.1 to the conjugation C in L defined by ⊥ ⊥ C = C ⊕ C : K ⊕ (K ) → K ⊕ (K ) . θ θ θ θ θ θ ∞ θ It follows that C = C = J A for ψ ∈ H such that A is an isometry. Since |K ψ ψ C (K ) = K and J (K ) = K # = K , we see that A = J C maps K onto K θ θ θ θ θ θ and is in fact unitary. Thus we have θ θ θ θ A A = A A = I . ψ ψ θ ψ ψ On the other hand, C = I so 2  θ  θ θ θ θ θ C = J A J A = A A = A A = I . # # K ψ ψ ψ ψ θ ψ ψ θ θ θ # 2 Hence A = A and A = 0, which gives ψ − ψ ∈ θ H + θ H (see [14]). ψ ψ ψ −ψ # 2 In other words, ψ − ψ = θh + θh for some functions h , h ∈ H . Thus there 1 2 1 2 exists a constant λ such that ψ − θh = ψ + θh = λ. 1 2 2 Conjugations in L and their invariants Page 13 of 14 22 We now have θ θ A = A = λI . ψ θ θh +λ Moreover λ ∈ T, since A is unitary. Hence C = J A = λJ . ψ |K θ θ Now we characterize conjugations intertwining the truncated shifts A and A . z ¯ Theorem 6.3 Let θ be an inner function and let C be a conjugation in K . Then the following are equivalent: θ ∞ (1) A C = CA for all ϕ ∈ H , ϕ ϕ θ θ (2) A C = CA , z ¯ θ θ (3) there is a function ψ ∈ H such that C = A C and A is unitary, ψ ψ ∞ θ θ (4) there is a function ψ ∈ H such that C = C A and A is unitary. Proof Let us start with (2) ⇒ (3). Since A is C –symmetric, θ θ θ A CC = CA C = CC A . θ θ θ z z ¯ z θ ∞ θ Hence, by [1, Proposition 1.21], CC = A for some ψ ∈ H . Clearly, A is unitary ψ ψ and θ θ ∗ θ C = A C = C (A ) = C A . θ θ θ ψ ψ θ θ To prove that (4) ⇒ (1) note that, since A and A commute, we have θ θ θ θ θ θ θ θ θ ∗ A C = A C A = C A A = C A A = CA = C (A ) . θ θ θ ϕ ϕ ϕ ϕ ϕ ϕ ψ ψ ψ All other implications are straightforward. Corollary 6.4 If C is a conjugation in K and every A ∈ T (θ ) is C–symmetric, then θ ∞ θ C = A C for some ψ ∈ H such that A is unitary. ψ ψ For a complete description of unitary operators from T (θ ) see [15, Proposition 6.5]. Compliance with ethical standards Conflict of interest The authors declare that there is no conflict of interest. 22 Page 14 of 14 M. C. Câmara et al. Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/. References 1. Bercovici, H.: Operator Theory and Arithmetic in H , Mathematical Surveys and Monographs, vol. 26. American Mathematical Society, Providence (1988) 2. Câmara, C., Klis-Garlicka, ´ K., Ptak, M.: Asymmetric truncated Toeplitz operators and conjugations. Filomat 33, 3697–3710 (2019) 3. Chevrot, N., Frician, E., Timotin, D.: The characteristic function of a complex symmetric contraction. Proc. Am. Math. Soc. 135, 2877–2886 (2007) 4. Cima, J.A., Garcia, S.R., Ross, W.T., Wogen, W.R.: Truncated Toeplitz operators: spatial isomorphism, unitary equivalence, and similarity. Indiana Univ. Math. J. 59(2), 595–620 (2010) 5. Fricain, E., Mashreghi, J.: The Theory of H(b) Spaces, vol. 1. Combridge University Press, Cambridge (2016) 6. Garcia, S.R., Mashreghi, J., Ross, W.T.: Introduction to Model Spaces and Their Operators. Cambridge University Press, Cambridge (2016) 7. Garcia, S.R., Putinar, M.: Complex symmetric operators and applications. Trans. Am. Math. Soc. 358, 1285–1315 (2006) 8. Garcia, S.R., Prodan, E., Putinar, M.: Mathematical and physical aspects of complex symmetric oper- ators. J. Phys. A Math. Theor. 47, 353001 (2014) 9. Garcia, S.R., Putinar, M.: Complex symmetric operators and applications II. Trans. Am. Math. Soc. 359, 3913–3931 (2007) 10. Ko, E., Lee, J.E.: Remark on complex symmetric operator matrices. Linear Multilinear Algebra 67(6), 1198–1216 (2019). https://doi.org/10.1080/03081087.2018.1450350 11. Martínez-Aveñdano, R.A., Rosenthal, P.: An Introduction to Operators on the Hardy–Hilbert Space. Springer, New York (2007) 12. Sz.-Nagy, B., Foias, C., Bercovici, H., Kérchy, L.: Harmonic Analysis of Operators on a Hilbert Space. Springer, London (2010) 13. Radjavi, H., Rosenthal, P.: Invariant Subspaces. Springer, New York (1973) 14. Sarason, D.: Algebraic properties of truncated Toeplitz operators. Oper. Matrices 1, 491–526 (2007) 15. Sedlock, N.A.: Algebras of truncated Toeplitz operators. Oper. Matrices 5, 309–326 (2011) Publisher’s Note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Analysis and Mathematical Physics Springer Journals

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Conjugations in space L of the unit circle commuting with multiplication by z or intertwining multiplications by z and z ¯ are characterized. We also study their behaviour with respect to the Hardy space, subspaces invariant for the unilateral shift and model spaces. Keywords Conjugation · C–symmetric operator · Hardy space · Invariant subspaces for the unilateral shift · Model space · Truncated Toeplitz operator Mathematics Subject Classification Primary 47B35; Secondary 30D20 · 30H10 The work of the first author was partially supported by FCT/Portugal through UID/MAT/04459/2019 and the research of the second and the fourth authors was financed by the Ministry of Science and Higher Education of the Republic of Poland. Kamila Klis–Garlicka ´ rmklis@cyfronet.pl M. Cristina Câmara cristina.camara@tecnico.ulisboa.pt Bartosz Łanucha bartosz.lanucha@poczta.umcs.lublin.pl Marek Ptak rmptak@cyf-kr.edu.pl Center for Mathematical Analysis, Geometry and Dynamical Systems, Mathematics Department, Instituto Superior Técnico, Universidade de Lisboa, Av. Rovisco Pais, 1049–001 Lisboa, Portugal Department of Applied Mathematics, University of Agriculture, ul. Balicka 253c, 30-198 Kraków, Poland Department of Mathematics, Maria Curie-Skłodowska University, Maria Curie-Skłodowska Square 1, 20-031 Lublin, Poland 0123456789().: V,-vol 22 Page 2 of 14 M. C. Câmara et al. 1 Introduction Let H be a complex Hilbert space and denote by B(H) the algebra of all bounded linear operators on H.A conjugation C in H is an antilinear isometric involution, i.e., C = id and Cg, Ch=h, g for g, h ∈ H. (1.1) Conjugations have recently been intensively studied and the roots of this subject comes from physics. An operator A ∈ B(H) is called C–symmetric if CAC = A (or equivalently AC = CA ). A strong motivation to study conjugations comes from the study of complex symmetric operators, i.e., those operators that are C–symmetric with respect to some conjugation C. For references see for instance [2,3,7–10]. Hence obtaining the full description of conjugations with certain properties is of great interest. Let T denote the unit circle, and let m be the normalized Lebesgue measure on T. 2 2 ∞ ∞ 2 Consider the spaces L = L (T,m), L = L (T, m), the classical Hardy space H 2 ∞ on the unit disc D identified with a subspace of L , and the Hardy space H of all analytic and bounded functions in D identified with a subspace of L . Denote by M 2 ∞ the operator defined on L of multiplication by a function ϕ ∈ L . 2 2 The most natural conjugation in L is J defined by Jf = f ,for f ∈ L .This conjugation has two natural properties: the operator M is J –symmetric, i.e., M J = z z JM , and J maps an analytic function into a co-analytic one, i.e., JH = H . z ¯ df 2  # # Another natural conjugation in L is J f = f with f (z) = f (z ¯). The conjugation J has a different behaviour: it commutes with multiplication by z 2 2 (M J = J M ) and leaves analytic functions invariant, J H ⊂ H .The map z z J appears for example in connection with Hankel operators (see [11, pp. 146–147]). Its connection with model spaces was studied in [4] (Lemma 4.4, see also [1, p. 37]). Hence a natural question is to characterize conjugations with respect to these proper- ties. The first step was done in [2] where all conjugations in L with respect to which the operator M is C–symmetric were characterized, see Theorem 2.2. In Sect. 2 we give a characterization of all conjugations which commute with M , Theorem 2.4.In Sect. 3, using the above characterizations we show that there are no conjugations in L leaving H invariant, with respect to which the operator M is C–symmetric. We also show that J is the only conjugation commuting with M and leaving H invariant. 2 2 Beurling’s theorem makes subspaces of H of the type θ H (θ inner function, i.e., θ ∈ H , |θ|= 1a.e.on T) exceptionally interesting, as the only invariant subspaces for the unilateral shift S, Sf (z) = zf (z) for f ∈ H . On the other hand, model spaces 2 2 (subspaces of the type K = H  θ H ), which are invariant for the adjoint of the unilateral shift, are important in model theory, [12]. In [2] all conjugations C with respect to which the operator M is C–symmetric and mapping a model space K into another model space K were characterized, with α θ the assumption that α divides θ (α  θ). Recall that for α and θ inner, α  θ means that θ/α is also an inner function. In what follows we will show that the result holds without the assumption α  θ. 2 Conjugations in L and their invariants Page 3 of 14 22 In Sect. 4 conjugations commuting with M and preserving model spaces are described. Section 5 is devoted to conjugations between S–invariant subspaces (i.e., subspaces of the form θ H with θ an inner function). In the last section we deal with θ θ conjugations commuting with the truncated shift A (A = P M where P is the θ z|K θ z z θ 2 θ orthogonal projection from L onto K ) or conjugations such that A is C–symmetric with respect to them. 2 M and M –commuting conjugations in L z z 2 2 Denote by J the conjugation in L defined as Jf = f ,for f ∈ L . This conjugation has the following obvious properties: Proposition 2.1 (1) M J = JM ; z z ¯ (2) M J = JM for all ϕ ∈ L ; ϕ ϕ ¯ (3) JH = H . Let us consider all conjugations C in L satisfying the condition M C = CM . (2.1) z z ¯ Such conjugations were studied in [2] and are called M –conjugations. The following theorem characterizes all M –conjugations in L . Theorem 2.2 [2] Let C be a conjugation in L . Then the following are equivalent: (1) M C = CM , z z ¯ (2) M C = CM for all ϕ ∈ L , ϕ ϕ ¯ (3) there is ψ ∈ L , with |ψ|= 1, such that C = M J, (4) there is ψ ∈ L , with |ψ |= 1, such that C = JM . Another natural conjugation in L is defined as df # # J f = f with f (z) = f (z ¯). (2.2) The basic properties of J are the following: Proposition 2.3 (1) M J = J M ; z z (2) M J = J M ; z ¯ z ¯ (3) M J = J M for all ϕ ∈ L ; 2 2 (4) J H = H . In the context of the Proposition 2.3 it seems natural to consider all conjugations C in L commuting with M , i.e., M C = CM . (2.3) z z 22 Page 4 of 14 M. C. Câmara et al. Such conjugations will be called M –commuting. In what follows we will often deal with functions f ∈ L such that f (z) = f (z ¯) a.e. on T, which will be called symmetric. Observe that if f is symmetric and f ∈ H , then we also have f ∈ H and so it is a constant function. Theorem 2.4 Let C be a conjugation in L . Then the following are equivalent: (1) M C = CM , z z (2) M C = CM # for all ϕ ∈ L , (3) there is a symmetric unimodular function ψ ∈ L such that C = M J , (4) there is a symmetric unimodular function ψ ∈ L such that C = J M . Proof We will show that (1) ⇒ (3).The other implications are straightforward. Assume that M C = CM . Then M CJ = CM J = CJ M . It follows that the linear z z z z z operator CJ commutes with M .By[13, Theorem 3.2], there is ψ ∈ L such that CJ = M . Hence C = M J = J M # . ψ ψ By (1.1) for any f , g ∈ L we have g fdm(z) =g, f =Cf , Cg # # 2 =ψ f ,ψg = |ψ(z)| f (z ¯)g(z ¯) dm(z) = |ψ(z ¯)| f (z)g(z) dm(z). 2 2 Hence |ψ|= 1a.e.on T. On the other hand, since C = I 2, for all f ∈ L we have 2    # # f = C f = M J M J f = M J (ψ f ) = ψψ f , ψ ψ ψ which implies that ψψ = 1a.e.on T. Therefore ψ is symmetric, i.e., ψ(z) = ψ(z ¯) a.e. on T. 3 Conjugations preserving H In the previous section all M –conjugations C in L , i.e., such that M C = CM z z ¯ or M –commuting conjugations, i.e., such that M C = CM z z were characterized. Let us now consider the question which of them preserve H . 2 2 2 2 2 Clearly, if C is a conjugation in L and C (H ) ⊂ H , then C (H ) = H . Since J 2 2 preserves H , it can be considered as a conjugation in H . The following result shows that J is in that sense unique. 2 Conjugations in L and their invariants Page 5 of 14 22 2 2 2 Corollary 3.1 Let C be an M –commuting conjugation in L .IfC (H ) ⊂ H , then C = λJ for some λ ∈ T. Proof By Theorem 2.4 we have that C = M J for some ψ ∈ L with |ψ|= 1 and ψ(z) = ψ(z) a.e. on T. Since C preserves H ,wehave ∞ 2 ∞ ψ = M J (1) = C (1) ∈ L ∩ H = H . Thus ψ is analytic. Since it is symmetric, it is also co-analytic. Hence ψ must be a constant function, so ψ = λ ∈ C and |λ|=|ψ|= 1. 2 2 Corollary 3.2 There are no M –conjugations in L which preserve H . Proof If C is an M –conjugation in L , then by Theorem 2.2 it follows that C = M J z ψ for some ψ ∈ L with |ψ|= 1. As in the proof of Corollary 3.1 the assumption 2 2 ∞ C (H ) ⊂ H implies that ψ ∈ H , which in turn means that ψ is an inner function. Moreover, for n = 0, 1, 2,... we have n+1 n+1 n 0 =Cz , z=ψz , z=ψ, z = 0. So ψ = 0 which is a contradiction. The following example shows that not all conjugations in L satisfy either (2.1)or (2.3). Example 3.3 There is a set of naturally defined conjugations. For k, l ∈ Z, k < l, 2 2 define C : L → L by k,l n k l n C a z = a z + a z + a z , (3.1) k,l n l k n n∈Z n∈{ / k,l} n 2 where {z } is the standard basis in L . Then (2.1) and (2.3) are not satisfied since k l l+1 k k−1 k−1 M C (z ) = M (z ) = z , C M (z ) = C (z ) = z z k,l z k,l z ¯ k,l and k+1 z if k + 1 = l, k k+1 C M (z ) = C (z ) = k,l z k,l z if k + 1 = l. Note that, on the other hand, C preserves H whenever k  0or l < 0. k,l 22 Page 6 of 14 M. C. Câmara et al. 4 Conjugations preserving model spaces There is another class of conjugations in L which appear naturally in connection with model spaces. For a nonconstant inner function θ, denote by K the so called model 2 2 2 space of the form H  θ H . The conjugation C defined in L by C f = θz ¯ f has the important property that it preserves the model space K , i.e., C K = K . θ θ θ θ Thus C can be considered as a conjugation in K . Such conjugations are important θ θ in connection with truncated Toeplitz operators (see for instance [6]). Here we present several simple properties of such conjugations, which we will use later. Proposition 4.1 Let α, β, γ be nonconstant inner functions. Then (1) C C = M , β α βα ¯ (2) M C M is a conjugation in L , γ α γ¯ (3) C M = M C . β γ γ¯ β Now we will consider relations between M –conjugations and model spaces. The theorem below was proved in [2, Theorem 4.2] with the additional assumption that α  θ. As we prove here, this assumption is not necessary. Theorem 4.2 Let α, γ , θ be inner functions (α, θ nonconstant). Let C be a conjugation in L such that M C = CM . Assume that C (γ K ) ⊂ K . Then there is an inner z z ¯ α θ function β such that C = C , with γα  β  γθ and α  θ. Proof Recall the standard notation for the reproducing kernel function at 0 in K , α α α namely, k = 1−α(0)α and its conjugate k = C k =¯z(α −α(0)). By Theorem 2.2 0 0 0 we know that C = M J for some function ψ ∈ L , |ψ|= 1. Hence α α ˜ ˜ K  C (γ k ) = M J (γ k ) = ψ γ z ¯(α − α(0)) =¯ γ α ¯ zψ(1 − α(0)α). θ ψ 0 0 −1 Thus there is h ∈ K such that h =¯ γ α ¯ zψ(1 − α(0)α). Since (1 − α(0)α) is a bounded analytic function, we have −1 2 γ¯ α ¯ zψ = h(1 − α(0)α) ∈ H . Since γ¯ α ¯ zψ ∈ H and |¯ γ α ¯ zψ|= 1a.e.on T, it has to be an inner function. Moreover β = zψ has to be inner and divisible by γα, i.e., γα  β. On the other hand, we have similarly K  C C (γ k ) = C (ψγ(1 − α(0)α) = θγ z ¯ψ(1 − α(0)α), θ θ θ ¯ ¯ and θγ β = θγ z ¯ψ ∈ H . Hence β divides θγ , i.e., β  γθ. It is clear that C = C . Finally, we have α  θ as a consequence of γα  β  γθ. 2 Conjugations in L and their invariants Page 7 of 14 22 Note that if α  θ and C = C for some inner β with γα  β  γθ, then K ⊂ K ⊂ K , C M = C and α β θ β γ β γ γ C (γ K ) = C M (K ) = C (K ) ⊂ K ⊂ K . β β α β γ α α θ γ γ Hence the implication in Theorem 4.2 is actually an equivalence. The corollary bellow strengthens [2, Proposition 4.5]. Corollary 4.3 Let α, θ be nonconstant inner functions, and let C be a conjugation in L such that M C = CM . Assume that C (K ) ⊂ K . Then α  θ and there is an z z ¯ α θ inner function β such that C = C , with α  β  θ. Let us turn to discussing the relations between M –commuting conjugations and model spaces. The following proposition describes some more properties of J . Proposition 4.4 Let α be an inner function. Then 2 # 2 (1) J (αH ) = α H ; (2) J (K ) = K ; (3) J C = C J . Proof The condition (1) is clear, (2) and (3) were proved in [4, Lemma 4.4]. Hence the conjugation J has a nice behaviour in connection with model spaces, namely J (K ) = K # . Theorem 4.6 below says that the conjugation J is, in some α α sense the only M –commuting conjugation with this property. We start with the following: Proposition 4.5 Let α, γ , θ be inner functions (α, θ nonconstant). Let C be an M – 2 # commuting conjugation in L . Assume that C (γ K ) ⊂ K . Then α  θ and there is α θ an inner function β with γα  β  γθ such that C = J M β . γ¯ γα Proof Observe that since C is an M –commuting conjugation, taking antilinear adjoints and applying [2, Proposition 2.1] we get M C = CM . Since by Propo- z ¯ z ¯ sition 4.1 the antilinear operator M C M is a conjugation, then J CM C M is γ α γ γ α γ also a conjugation. Note also that M C M M = M M C M . Hence γ α γ z z ¯ γ α γ J CM C M M = M J CM C M . (4.1) γ α γ z z ¯ γ α γ On the other hand, J CM C M (γ K ) ⊂ J CM C (K ) γ α γ α γ α α ⊂ J C (γ K ) ⊂ J (K ) ⊂ K # . α θ By Theorem 4.2 there is an inner function β such that J CM C M = C , with γ α γ β # # γα  β  γθ and α  θ . Hence C = J C M C M . Therefore β γ α γ 22 Page 8 of 14 M. C. Câmara et al. C = J M β γ¯ γα As in Theorem 4.2 the implication in Proposition 4.5 can be reversed. Indeed, if #  # α  θ and C = J M β for some inner function β with γα  β  γθ , then γα K ⊂ K ⊂ K # and C (γ K ) = J M (K ) = J C C (K ) ⊂ J (K ) ⊂ J (K ) = K . β β # α α α α β θ γ γ Theorem 4.6 Let α, θ be nonconstant inner functions, and let C be an M –commuting 2 # conjugation in L , i.e., M C = CM . Assume that C (K ) ⊂ K . Then α  θ and z z α θ C = λJ with λ ∈ T. Corollary 4.7 Let C be an M –commuting conjugation in L . Assume that there is some nonconstant inner function θ such that C (K ) ⊂ K #. Then C = λJ with λ ∈ T. Proof of Theorem 4.6 By Proposition 4.5 there is an inner function β with α  β  θ such that C = J M β . The function is inner and by Theorem 2.4 it is symmetric. As observed before it follows that it is constant. Hence C = J up to multiplication by a constant of modulus 1. 5 Conjugations preserving S-invariant subspaces of H Beurling’s theorem says that all invariant subspaces for the unilateral shift S are of the 2 2 form θ H with θ inner. We will now investigate conjugations in L which preserve subspaces of this form. Since C transforms θ H onto zH , the operator C J C = M J M θ θ θ is an example of such a conjugation. Note that (C J C )M = M (C J C ). θ θ z z θ θ 2 2 Let α, θ be two inner functions. Then the operator C J C : L → L is an θ α 2 2 antilinear isometry which maps αH onto θ H and commutes with M . This operator however does not have to be an involution. Lemma 5.1 Let α, θ be two inner functions. The operator C J C is an involution θ α 2 # (and hence a conjugation in L ) if and only if the function θ α is symmetric (or # # equivalently αα = θθ ). 2 Conjugations in L and their invariants Page 9 of 14 22 Proof Note that by Proposition 4.1, (C J C )(C J C ) = C C #C #C = M M # = M . θ α θ α θ α # # # α θ θ α θ α θ α θ α Therefore C J C is an involution if and only if θ α # # # θ α θ α =1a.e.on T, i.e., θθ = αα , which means that # # # # (θ α )(z) = (θ α)(z) = θ(z)α (z ¯) = (θ α )(z) a.e. on T. The theorem bellow characterizes all M –commuting conjugations mapping one S– invariant subspace into another S–invariant subspace. Theorem 5.2 Let θ and α be two inner functions and let C be a conjugation in L such 2 2 # # that C M = M C. Then C (αH ) ⊂ θ H if and only if θθ  αα and C = C J C , z z β α # # where β is an inner function such that θ  β, ββ = αα . Moreover, in that case 2 2 C (αH ) = β H . Let α be a fixed inner function. By Lemma 5.1, for each inner function β with # # 2 2 ββ = αα there exists an M –commuting conjugation C which maps αH onto β H , namely C = C J C . On the other hand, if β is an inner function and there exists β α 2 2 an M –commuting conjugation C which maps αH onto β H , then by Theorem 5.2, # #  # ββ  αα and C = C J C for some inner function γ such that β  γ , γγ = γ α # 2 2 2 αα . In particular, C (αH ) = γ H = β H and so γ is a constant multiple of β, # # ββ = αα . It follows from the above that Lemma 5.3 characterizes all possible spaces of type 2 2 β H such that for a given S–invariant subspace αH there is an M –commuting 2 2 conjugation mapping αH onto β H . Lemma 5.3 Let α be a nonconstant inner function. Then # # {β : β is inner, αα = ββ } ={λ uv : u,v are inner, α = uv, λ ∈ T}. (5.1) For two inner functions α and β denote by α ∧ β the greatest common divisor of α and β. We will write α ∧ β = 1 if the only common divisor of α and β is a constant function. # # # Proof Note that for α = uv and β = λuv we have αα = ββ , hence one inclusion is proved. For the other inclusion let u = α ∧ β and we can write α = uv and β = uv . # # From the condition αα = ββ it follows that # # # # uvu v = uv u v . 1 22 Page 10 of 14 M. C. Câmara et al. # # # # Hence vv = v v . Since v ∧ v = 1, we have that v divides v and v divides v and 1 1 1 1 1 # # vice–versa. Thus we can take v = λv with λ ∈ T, and so β = λuv . 2 2 Proof of Theorem 5.2 Assume firstly that CM = M C and C (αH ) ⊂ θ H .By z z Theorem 2.4, C = M J for some unimodular symmetric function ψ ∈ L .In particular, #  2 ψα = M J (α) = C (α) ∈ θ H , 2 # and there exists u ∈ H such that ψα = θu. Note that u must be inner and so # # # # ψ = βα with β = θu, θ  β. Clearly βα is symmetric, i.e., ββ = αα . Hence # # θθ  αα . # # Assume now that θθ  αα , and let α = α · (α ∧ θ) and θ = θ · (α ∧ θ). Since 1 1 # # # # # # # # # # # (α ∧ θ) = α ∧ θ , we get α = α · (α ∧ θ ) and θ = θ · (α ∧ θ ).Notealso 1 1 # # # that θ θ  α α and θ ∧ α = 1, so θ  α . Thus 1 1 1 1 1 1 1 1 # # αα α α α α 1 1 1 1 # = = = uu , # # # θθ θ θ θ θ 1 1 where u = is an inner function. Now we may take β = θu. Since θ  β and # # ββ = αα , by Lemma 5.1 and by Proposition 4.1, C = M J = C J C is a # β α βα 2 2 2 conjugation which maps αH onto β H ⊂ θ H . Corollary 5.4 Let θ be an inner function and let C be an M –commuting conjugation in L . Then 2 2 (1) C (θ H ) ⊂ θ H if and only if C = λC J C with λ ∈ T; θ θ 2 # 2 (2) C (θ H ) ⊂ θ H if and only if C = λJ with λ ∈ T. 2 2 Proof By Theorem 5.2, C (θ H ) ⊂ θ H if and only if there exists an inner function # # β such that θ  β and ββ = θθ . This is only possible if β is constant multiple of θ and (1) is proved. The proof of (2) is similar. Note that by Theorem 5.2 (Lemma 5.1, actually) if θ α is symmetric, then there 2 2 exists an M –commuting conjugation from αH into θ H . The following example shows that in that case there may be no such conjugation between the corresponding model spaces K and K . α θ Example 5.5 Fix a, b ∈ D such that a = b, a = a and b = b, and put a−z b−z a−z b−z α(z) = and θ(z) = . 1−az 1−az 1−bz 1−bz Then # # a−z b−z a−z b−z α (z) = and θ (z) = 1−az 1−bz 1−az 1−bz 2 Conjugations in L and their invariants Page 11 of 14 22 # # 2 and so αα = θθ . Thus there exists an M –commuting conjugation from αH onto 2 # # θ H . In this case however neither α  θ nor θ  α , so by Theorem 4.6 no M – commuting conjugation between K and K exists. Here also neither α  θ nor α θ θ  α, and so by Theorem 4.2 no M –conjugation between K and K exists. z α θ Finally, consider M –conjugations preserving S–invariant subspaces. Proposition 5.6 Let θ and α be two inner functions. There are no M –conjugations in 2 2 2 L which map αH into θ H . Proof If C was such a conjugation, then by Theorem 2.2, C = M J for some uni- modular function ψ ∈ L and, in particular, C (α) = ψ α = θ g for some g ∈ H . Clearly g must be an inner function and ψ = αθ g. Then, for every h ∈ H , C (αh) = αθ gαh = θ gh ∈ θ H , and so gh ∈ H .Itfollows that g = 0 and C (α) = 0 which is a contradiction. 6 Conjugations and truncated Toeplitz operators 2 θ For ϕ ∈ L define the truncated Toeplitz operator A by θ ∞ A f = P (ϕ f ), for f ∈ H ∩ K , θ θ 2 θ were P : L → K is the orthogonal projection (see [14]). The operator A is closed θ θ and densely defined, and if it is bounded, it admits a unique bounded extension to K . The set of all bounded truncated Toeplitz operators on K is denoted by T (θ ). θ θ θ ∞ Note that A ∈ T (θ ) for ϕ ∈ L . It is known that every operator from T (θ ) is C –symmetric (see [14, Lemma 2.1]). Observe that if k  0, then the conjugation C defined by (3.1) satisfies neither k,l M C = C M nor SC = C S. However, for 0  n < k and θ(z) = z , z k,l k,l z k,l k,l θ θ C (K ) = K and A C = C A k,l θ θ k,l k,l z z (since here C = J and θ = θ). k,l|K |K Theorem below characterizes conjugations intertwining truncated shifts A and A . Theorem 6.1 Let θ be a nonconstant inner function and let C be a conjugation in L such that C (K ) ⊂ K #. Then the following are equivalent: θ ∞ (1) A C = CA on K for all ϕ ∈ H , # θ θ θ (2) A C = CA on K , z z 22 Page 12 of 14 M. C. Câmara et al. θ θ (3) there is a function ψ ∈ H such that C = J A and A is an isometry, |K θ ψ ψ # # ∞ θ  θ (4) there is a function ψ ∈ H such that C = A J and A is an isometry. |K ψ |K ψ θ θ Proof We will only prove that (2) ⇒ (3). Since J (K ) = K and J A = A J θ # ϕ for all ϕ ∈ H (see [4, Lemma 4.5]), we have θ  θ θ J CA = J A C = A J C z z z θ ∞ θ on K and so J C = A for some ψ ∈ H ([5, Theorem 14.38]). Hence A is θ |K θ ψ ψ an isometry and C = J A . |K It is much more restrictive if θ = θ. Proposition 6.2 Let θ be an inner function such that θ = θ and let C be a conjugation in K . Then the following are equivalent: θ θ ∞ (1) A C = CA for all ϕ ∈ H , # ϕ θ θ (2) A C = CA , z z (3) C = λJ with λ ∈ T. |K Proof Implications (1) ⇒ (2) and (3) ⇒ (1) are clear. To prove (2) ⇒ (3) apply Theorem 6.1 to the conjugation C in L defined by ⊥ ⊥ C = C ⊕ C : K ⊕ (K ) → K ⊕ (K ) . θ θ θ θ θ θ ∞ θ It follows that C = C = J A for ψ ∈ H such that A is an isometry. Since |K ψ ψ C (K ) = K and J (K ) = K # = K , we see that A = J C maps K onto K θ θ θ θ θ θ and is in fact unitary. Thus we have θ θ θ θ A A = A A = I . ψ ψ θ ψ ψ On the other hand, C = I so 2  θ  θ θ θ θ θ C = J A J A = A A = A A = I . # # K ψ ψ ψ ψ θ ψ ψ θ θ θ # 2 Hence A = A and A = 0, which gives ψ − ψ ∈ θ H + θ H (see [14]). ψ ψ ψ −ψ # 2 In other words, ψ − ψ = θh + θh for some functions h , h ∈ H . Thus there 1 2 1 2 exists a constant λ such that ψ − θh = ψ + θh = λ. 1 2 2 Conjugations in L and their invariants Page 13 of 14 22 We now have θ θ A = A = λI . ψ θ θh +λ Moreover λ ∈ T, since A is unitary. Hence C = J A = λJ . ψ |K θ θ Now we characterize conjugations intertwining the truncated shifts A and A . z ¯ Theorem 6.3 Let θ be an inner function and let C be a conjugation in K . Then the following are equivalent: θ ∞ (1) A C = CA for all ϕ ∈ H , ϕ ϕ θ θ (2) A C = CA , z ¯ θ θ (3) there is a function ψ ∈ H such that C = A C and A is unitary, ψ ψ ∞ θ θ (4) there is a function ψ ∈ H such that C = C A and A is unitary. Proof Let us start with (2) ⇒ (3). Since A is C –symmetric, θ θ θ A CC = CA C = CC A . θ θ θ z z ¯ z θ ∞ θ Hence, by [1, Proposition 1.21], CC = A for some ψ ∈ H . Clearly, A is unitary ψ ψ and θ θ ∗ θ C = A C = C (A ) = C A . θ θ θ ψ ψ θ θ To prove that (4) ⇒ (1) note that, since A and A commute, we have θ θ θ θ θ θ θ θ θ ∗ A C = A C A = C A A = C A A = CA = C (A ) . θ θ θ ϕ ϕ ϕ ϕ ϕ ϕ ψ ψ ψ All other implications are straightforward. Corollary 6.4 If C is a conjugation in K and every A ∈ T (θ ) is C–symmetric, then θ ∞ θ C = A C for some ψ ∈ H such that A is unitary. ψ ψ For a complete description of unitary operators from T (θ ) see [15, Proposition 6.5]. Compliance with ethical standards Conflict of interest The authors declare that there is no conflict of interest. 22 Page 14 of 14 M. C. Câmara et al. Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. 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