### Abstract

Comput. Methods Funct. Theory (2018) 18:259–267 https://doi.org/10.1007/s40315-017-0219-x Bartosz Łanucha Received: 17 July 2017 / Revised: 8 August 2017 / Accepted: 15 August 2017 / Published online: 30 October 2017 © The Author(s) 2017. This article is an open access publication Abstract A truncated Toeplitz operator is a compression of the multiplication operator to a backward shift invariant subspace of the Hardy space H .An asymmetric truncated Toeplitz operator is a compression of the multiplication operator that acts between two different backward shift invariant subspaces of H . All rank-one truncated Toeplitz operators have been described by Sarason. Here, we characterize all rank-one asymmetric truncated Toeplitz operators. This completes the description given by Łanucha for asymmetric truncated Toeplitz operators on ﬁnite-dimensional backward shift invariant subspaces. Keywords Model space · Truncated Toeplitz operator · Asymmetric truncated Toeplitz operator · Rank-one operator Mathematics Subject Classiﬁcation 47B32 · 47B35 · 30H10 1 Introduction Denote by H the Hardy space of the open unit disk D ={z:|z| < 1} and let P be 2 2 the orthogonal projection from L (∂ D) onto H . Communicated by Stephan Ruscheweyh. B Bartosz Łanucha bartosz.lanucha@poczta.umcs.lublin.pl Department of Mathematics, Maria Curie-Skłodowska University, Maria Curie-Skłodowska Square 1, 20-031 Lublin, Poland 123 260 B. Łanucha ∞ 2 A classical Toeplitz operator T with symbol ϕ ∈ L (∂ D) is deﬁned on H by T f = P(ϕ f ). It is clear that a Toeplitz operator with symbol from L (∂ D) is bounded. Note that if ϕ ∈ L (∂ D), then the above deﬁnition produces an operator T , densely deﬁned on ∞ 2 ∞ 2 H = H ∩ L (∂ D). It is known that such T extends boundedly to H if and only if ϕ ∈ L (∂ D). The operators S = T and S = T are called the unilateral shift and the backward z z shift, respectively. Obviously, Sf (z) = zf (z) and a simple computation reveals that f (z) − f (0) S f (z) = . A truncated Toeplitz operator is a compression of a classical Toeplitz operator to a model space. Recall, that model spaces are the closed subspaces of H of the form 2 2 K = H α H , where α is an inner function, i.e., α ∈ H and |α|= 1a.e.on ∂ D.Asa consequence of the theorem of Beurling, model spaces are the typical non-trivial ∗ 2 S -invariant subspaces of H . Extensive study of the class of truncated Toeplitz operators began in 2007 with Sarason’s paper [10]. Generalizations of these operators, the so-called asymmetric truncated Toeplitz operators, were recently introduced in [2,4,5]. α,β Let α, β be two inner functions. An asymmetric truncated Toeplitz operator A with symbol ϕ ∈ L (∂ D) is given by α,β A f = P (ϕ f ), f ∈ K , β α where P is the orthogonal projection from L (∂ D) onto K . In particular, a truncated β β α α,α Toeplitz operator is the operator A = A . ϕ ϕ ∞ ∞ ∞ Since P (ϕ f ) ∈ K for f ∈ K = K ∩ H and K is dense in K , the operator β β α α α α α,β A is densely deﬁned. It is known that the boundedness of ϕ is not necessary for α,β A to extend boundedly to K . ϕ α Put α,β 2 α,β T (α, β) ={A : ϕ ∈ L (∂ D) and A is bounded} ϕ ϕ and T (α) = T (α, α). Sarason showed that the truncated Toeplitz operators of rank one are generated (in a sense to be explained below) by functions of the form 1 − α(w)α(z) k (z) = 1 − wz 123 Asymmetric Truncated Toeplitz Operators of Rank One 261 and α(z) − α(w) k (z) = . z − w The function k , called the reproducing kernel, is the kernel function for the point evaluation functional f → f (w), which for w ∈ D is bounded on K . In other words, f (w) = f, k for every f ∈ K and each w ∈ D. It is easy to check that α −1 k = P k , where k (z) = (1 − wz) . α w w α α The function k is the image of the reproducing kernel k under the map w w C f (z) = α(z)zf (z), |z|= 1. One can verify that C is an antilinear and isometric involution on L (∂ D) which α α preserves K .The map C is called a conjugation and, since k = C k , the function α α α w w k is often called the conjugate kernel. It should be noted that the functional f → f (w) is bounded on K also for w ∈ ∂ D such that the function α has an angular derivative in the sense of Carathéodory (an ADC) at w. Recall that α is said to have an ADC at w ∈ ∂ D if there exist complex numbers α(w) and α (w) such that α(z) → α(w) ∈ ∂ D and α (z) → α (w) whenever z from D tends to w non-tangentially (that is, with |z − w|/(1 −|z|) bounded). In that case, every f ∈ K has a non-tangential limit f (w) at w, the functions k and α α k belong to K , and f (w) = f, k (for more details, see for example [6,Thm. w w 7.4.1.]). It was proved in [10] that the only rank-one operators in T (α) are the scalar α α α α multiples of k ⊗ k and k ⊗ k for all w ∈ D and all w ∈ ∂ D such that α has an w w w w ADC at w. In the asymmetric case, the authors in [8] showed that the operators k ⊗ k and k ⊗ k belong to T (α, β) for every w ∈ D and every w ∈ ∂ D such that both α and β have an ADC at w. It is natural to ask: are the scalar multiples of these the only rank-one operators in T (α, β)? This issue was addressed in [9]. The author in [9] proved that if K is one dimensional and K has a ﬁnite dimension larger than α β two (or K is one dimensional and dim K > 2), then the answer is negative: there β α exists a rank-one operator in T (α, β) that is neither a scalar multiple of k ⊗ k nor a scalar multiple of k ⊗ k . Nevertheless, it was also showed in [9] that the answer is positive for all other ﬁnite-dimensional cases (that is, whenever α and β are two ﬁnite Blaschke products): Theorem 1.1 ([9,Thm.4.5]) Let α and β be two ﬁnite Blaschke products of degree m > 0 and n > 0, respectively. The only rank-one operators in T (α, β) are the β β α α non-zero scalar multiples of the operators k ⊗ k and k ⊗ k , w ∈ D, if and only w w w w if either mn ≤ 2,orm > 1 and n > 1. In this paper, we resolve the issue in the general case. In particular, we prove that if both K and K have dimension larger than one (and not necessarily ﬁnite), then α β the only rank-one operators in T (α, β) are the non-zero scalar multiples of k ⊗ k and k ⊗ k . 123 262 B. Łanucha 2 Rank-One Asymmetric Truncated Toeplitz Operators In [10], Sarason gave several characterizations of the operators from T (α).For example, he proved that a bounded linear operator A on K is a truncated Toeplitz operator if and only if A − S AS is a rank-two operator of a special kind, where S = A is the so-called compressed shift. Similar characterizations of asymmetric truncated Toeplitz operators were given in [2,3] and [9] for some particular cases. Here, we use the following two characterizations of asymmetric truncated Toeplitz operators. Theorem 2.1 ([7,Thm.2.1]) Let A be a bounded linear operator from K into K . α β Then, A ∈ T (α, β) if and only if there exist ψ ∈ K and χ ∈ K such that β α ∗ α A − S AS = ψ ⊗ k + k ⊗ χ. 0 0 Corollary 2.2 ([7,Cor.2.7(c)]) Let A be a bounded linear operator from K into K . α β Then, A ∈ T (α, β) if and only if A is shift invariant, that is, AS f, Sg = Af, g for all f ∈ K ,g ∈ K such that S f ∈ K ,Sg ∈ K . α β α β Note here that if f ∈ K , then Sf ∈ K if and only if f is orthogonal to k (see α α [10,p.512]). In his proof of [10, Thm. 5.1], Sarason uses the fact that every operator from T (α) is C -symmetric. Recall that an operator A on K is said to be C -symmetric if α α α ∗ ∗ C AC = A . In particular, C S C = S . In the case of asymmetric truncated α α α α α Toeplitz operators it is known that A ∈ T (α, β) if and only if C AC ∈ T (α, β) β α (see [9, p. 9]). So, here we take the approach from [9], that is, we use the following lemma (compare with [9,Lem.4.4]; seealso[10, pp. 503–504]). Lemma 2.3 Let α and β be two inner functions such that K and K have dimension α β m > 1 and n > 1, respectively. Let f ∈ K ,g ∈ K be two non-zero functions such α β that g ⊗ f belongs to T (α, β) and let w ∈ D. Then, (a) g is a scalar multiple of k if and only if f is a scalar multiple of k , (b) g is a scalar multiple of k if and only if f is a scalar multiple of k . Proof Let f ∈ K , g ∈ K be two non-zero functions such that g ⊗ f belongs to α β T (α, β). We ﬁrst show that if g is a scalar multiple of k for some w ∈ D, then f is a scalar multiple of k . Actually, without loss of generality we can assume that g = k for some w ∈ D. Note here, that for w ∈ ∂ D, we automatically assume that β has an ADC at w (this is implied by the fact that k ∈ K ) and we must prove that α also has an ADC at w. w β We consider two cases. 123 Asymmetric Truncated Toeplitz Operators of Rank One 263 Case 1. β(0) = 0. Since the operator k ⊗ f belongs to T (α, β), it must be shift invariant in the sense of Corollary 2.2. That is, β β (k ⊗ f )Sh , Sh = (k ⊗ f )h , h α β α β w w for all h ∈ K and h ∈ K such that Sh ∈ K and Sh ∈ K . Equivalently, α α β β α α β β β β β β (k ⊗ f )Sh , Sh − (k ⊗ f )h , h = Sh , f k , Sh − h , f k , h α β α β α β α β w w w w ∗ β ∗ β = h , h , S k S f − h , k = 0 α β β β w α w for all h ∈ K and h ∈ K , such that h ⊥ k and h ⊥ k . But it follows from α α β β α β 0 0 ∗ β β S k = wk − β(w)k β w w (see [10, Lem. 2.2(a)] and its Corollary) that for every h ⊥ k , we have ∗ β β h , S k = w h , k β β β w w Thus, the shift invariance of k ⊗ f means that β ∗ k , h h ,wS f − f = 0 (2.1) β α w α for all h ⊥ k and h ⊥ k . α β 0 0 β β Since the dimension of K is greater than one, the functions k and k are linearly β w β β β independent. Indeed, if k and k were linearly dependent, then so would be k and w w β β β k ≡ 1. But, here it is easy to verify that if k = ck = c, then β(z) = cz and K is w β 0 0 one dimensional—a contradiction. Therefore, there exists a function h ∈ K orthogonal to k and such that β β k , h = 0. Putting such h into (2.1), we see that w β β h ,wS f − f = 0 for all h ∈ K , h ⊥ k . Thus, α α α ∗ α (I − wS ) f = ck α 0 or, using C -symmetry of S , α α (I − wS )C f = ck (2.2) K α α α 0 for some complex number c (here I denotes the identity operator on K ). K α 123 264 B. Łanucha If c = 0 and |w| < 1, then, by (2.2) and invertibility of I − wS ,wehave K α C f = 0. This implies that f = 0 which contradicts the assumption that f is non-zero. If c = 0 and |w|= 1, then by (2.2), S C f = wC f, α α α which again is not possible since S has no eigenvalues on ∂ D ([10,Lem.2.5]). Therefore, c = 0 and without loss of generality we can assume that c = 1. Hence, instead of (2.2) we consider (I − wS )C f = k . (2.3) K α α α α As above, if |w| < 1, then I − wS is invertible. Since (I − wS )k = k K α K α α α w (for w = 0 this is obvious, for w = 0see [10, Lem. 2.2(b)]), we have −1 α α α f = C (I − wS ) k = C k = k . α K α α α 0 w w If |w|= 1, then applying C to both sides of (2.3) and using C -symmetry of S α α α we get f (z) − f (0) α(z) − α(0) f (z) − w · = , z ∈ D. z z Hence, α(z) − (α(0) + w f (0)) = f (z) ∈ H z − w and by [6, Thm. 7.4.1.] α has an ADC at w with the non-tangential limit α(w) equal to α(0) + w f (0), k belongs to K and α(z) − α(w) f (z) = = k . z − w This completes the proof in the ﬁrst case. Case 2. β(0) = 0. For any two complex numbers a, b and functions f ∈ K , g ∈ K deﬁne α β 2 2 1 −|a| 1 −|b| J f (z) = · f (z) and J g(z) = · g(z). 1 − aα(z) 1 − bβ(z) It is known that J is a unitary map from K onto K , where α α a a a − α(z) α (z) = , 1 − aα(z) 123 Asymmetric Truncated Toeplitz Operators of Rank One 265 and that J is a unitary map from K onto K , where β β b − β(z) β (z) = 1 − bβ(z) (see [10, pp. 521–523] for more details). It was also proved in [8] that A ∈ T (α, β) α −1 if and only if J A(J ) ∈ T (α ,β ) ([8, Prop. 2.5]). a b Take a = 0 and b = β(0). Clearly, α = α and J = I . Therefore, since k ⊗ f 0 K w 0 α belongs to T (α, β), the rank-one operator β β β α −1 β J (k ⊗ f )(J ) = (J k ) ⊗ f w a w b b belongs to T (α, β ). A simple computation reveals that 1 − bβ(w) β β J k = · k b w w 1 −|b| (this can also be seen using Lemma 2.4 from [8]). Hence, the rank-one operator k ⊗ f belongs to T (α, β ), and, since β (0) = 0, f is a scalar multiple of k by Case 1. b b This completes the proof in the second case and the proof of the ﬁrst implication in (a). To prove the ﬁrst implication in (b) assume that g = k for some w ∈ D. Then the rank-one operator k ⊗ C f = C (g ⊗ f )C α β α belongs to T (α, β) [9, p. 9], and it follows from the part of (a) proved above that C f α α is a scalar multiple of k and f is a scalar multiple of k . w w To complete the proof of (a) assume that f = k . Then, α α ∗ k ⊗ g = (g ⊗ k ) w w belongs to T (β, α) [2, Lem. 3.2], and it follows from the part of (b) proved above that g is a scalar multiple of k . Similarly, if f = k , then α α ∗ k ⊗ g = (g ⊗ k ) w w belongs to T (β, α) and it follows from (a) that g is a scalar multiple of k .This completes the proof of the lemma. Using Lemma 2.3, one can modify the proof of [10, Thm. 5.1(c)] to show the following. 123 266 B. Łanucha Theorem 2.4 Let α and β be two inner functions such that K and K have dimension α β m > 1 and n > 1, respectively. Then the only rank-one operators in T (α, β) are the β β α α non-zero scalar multiples of the operators k ⊗ k and k ⊗ k where w ∈ D or w w w w w ∈ ∂ D and α and β have an ADC at w. Note that neither the above nor [9] includes the case when one of the model spaces is one dimensional and the other is inﬁnite dimensional. Let us now assume that α, β are two inner functions such that dim K = 1 and dim K =+∞. Clearly, every bounded linear operator A from K into K is of rank one. Moreover, α β α ∗ α since K is spanned by k , A − S AS is a rank-one operator of the form ψ ⊗ k for α β 0 0 some ψ ∈ K . So here, by Theorem 2.1, all bounded linear operators from K into β α K are asymmetric truncated Toeplitz operators of rank one. As the ﬁnite-dimensional case suggests, one can expect that here there are more β β α α rank-one operators in T (α, β) than just scalar multiples of k ⊗ k and k ⊗ k . w w w w We now prove that this is indeed the case. We do this in two steps: (1) we show that β β α α if every rank-one operator from T (α, β) is a scalar multiple of k ⊗ k or k ⊗ k w w w w for some w ∈ D, then every non-zero function from K must be a scalar multiple of a reproducing kernel or a conjugate kernel; (2) we prove that since dim K =+∞, there must exist f ∈ K that is neither a scalar multiple of a reproducing kernel nor a scalar multiple of a conjugate kernel. Step 1. Let f ∈ K \{0}. Then, f ⊗ k ∈ T (α, β) and there exist w ∈ D and c = 0 such that α β α α β α f ⊗ k = c(k ⊗ k ) or f ⊗ k = c(k ⊗ k ). w w w w 0 0 α α α Since dim K = 1, we can replace k in the above with k or k (multiplying both 0 w w sides of the assumed equality by a constant if necessary). Then it follows easily that β β f is a scalar multiple of either k or k . w w Step 2. Here, we use [1, Prop. 2.8]. Fix w ,w ∈ D, w = w and let 1 2 1 2 β β β f = k + k ∈ K \{0}. Since dim K =+∞,by[1, Prop. 2.8], the functions k , w w β β w 1 2 1 β β k and k are linearly independent for any ﬁxed w ∈ D and for any ﬁxed w ∈ ∂ D w w such that β has an ADC at w. Thus, f cannot be a scalar multiple of k . A similar reasoning shows that f cannot be a scalar multiple of k for any w ∈ D. We proved that if dim K = 1 and dim K =+∞, then there is a rank-one operator α β from T (α, β) that is neither a scalar multiple of k ⊗ k nor a scalar multiple of k ⊗ k . The case dim K =+∞ and dim K = 1 can be treated analogously. w α β Theorem 2.5 Let α and β be two inner functions such that dim K = m and dim K = n (with m or n possibly inﬁnite). The only rank-one operators in T (α, β) β β α α are the non-zero scalar multiples of the operators k ⊗ k and k ⊗ k , w ∈ D,if w w w w and only if either mn ≤ 2,orm > 1 and n > 1. Proof Assume that the only rank-one operators in T (α, β) are the non-zero scalar β β α α multiples of the operators k ⊗ k and k ⊗ k . If both m and n are ﬁnite, then α w w w w 123 Asymmetric Truncated Toeplitz Operators of Rank One 267 and β are two ﬁnite Blaschke products of degree m and n, respectively, and the claim follows from Theorem 1.1.If m or n is inﬁnite, then m > 1 and n > 1 as already observed before. To complete the proof, assume that either mn ≤ 2, or m > 1 and n > 1. 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Computational Methods and Function Theory
– Springer Journals

**Published: ** Oct 30, 2017