Asymmetric Truncated Hankel Operators: Rank One, Matrix Representation

Asymmetric truncated Hankel operators are the natural generalization of truncated Hankel operators. In this paper, we determine all rank one operators of this class. We explore these operators on finite-dimensional model spaces, in particular, their matrix representation. We also give their matrix representation and the one for asymmetric truncated Toeplitz operators in the case of model spaces associated to interpolating Blaschke products.


Introduction
Let H 2 be the standard Hardy space of the unit disc D identified with the subspace of the boundary functions of its functions in L 2 ðT Þ.
A function in H ∞ ðDÞ is inner if it is unimodular on the unit circle T . To each inner function α we associate a model space, K α = H 2 ⊖ αH 2 . The model space is a reproducing kernel Hilbert space with reproducing kernel, k α λ ðzÞ = ð1 − αðλÞαðzÞÞ/ð1 − λzÞ, for z, λ ∈ D. The inner function α has an angular derivative in the sense of Carathéodory (ADC) at a point η ∈ T if and only if every f in K α has nontangential limit at η. In particular, k α η ∈ K α . If α is an inner function, then α # = αð zÞ is also an inner function and the associated model space is noted K α # .
To each model space, we associate a natural conjugation C α such that C α K α = K α , given by C α f ðzÞ = αzf ðzÞ, for z ∈ T . The image of the kernel function for C α , called conjugate kernel functionk α λ , is given byk α λ ðzÞ = ðαðzÞ − αðλÞÞ/ðz − λÞ, z,λ ∈ D. If α has an ADC at η ∈ T, The model spaces can also be defined as the only invariant subspaces of the backward shift operator S * on H 2 . Denote by S α (S * α ) the compression of the shift operator (resp., backward shift) to the model space K α .
The only finite-dimensional model spaces are the one associated to finite Blaschke products, and its dimension is the same as the multiplicity of the associated Blaschke product. Since a finite Blaschke product is analytic on a neighborhood of the unit disc, it has an ADC everywhere on T and k α η ∈ K α , for every η ∈ T . In this case, let m denote the multiplicity of a finite Blaschke product α. If we arbitrarily choose a collection fλ i , i = 1, ⋯, mg of distinct points in D, fk α λ i , i = 1, 2, ::, mg forms a basis for K α . In infinite-dimensional case the kernel functions ðk α λ i Þ i≥1 form a Riesz basis if and only if α is an interpolating Blaschke product or equivalently, ðλ i Þ i≥1 is a uniformly separated Blaschke sequence that satisfies inf k≥1 Q k≥,k≠n jðλ k − λ n Þ/ð1 − λ k λ n Þj > 0.
Asymmetric truncated Toeplitz and Hankel operators were first introduced in [1,2], respectively. For φ ∈ L 2 ðT Þ, let α and β be two inner functions, the asymmetric truncated Toeplitz operator A α,β φ and the asymmetric truncated Hankel operator B where P and P β denote the orthogonal projection on H 2 and K β , respectively, and J is the flip operator from H 2 onto H 2 0 = zH 2 defined on T by J f ðzÞ = zf ð zÞ. We denote the set of bounded asymmetric truncated Toeplitz operators (ATTO) (bounded asymmetric truncated Hankel operators (ATHO)) by Tðα, βÞ (Hðα, βÞ).
In [3], Sarason introduced truncated Toeplitz operators and showed that all rank one TTOs are of the form ck α λ ⊗ k α λ or ck α λ ⊗ k α λ , where c in ℂ and λ in D or in T such that α has an ADC at λ. Similar results were obtained in [4] for truncated Hankel operators ðk α λ ⊗k α λ , k α λ ⊗ k α λ Þ. Surprisingly, this is not always true in the case of asymmetric truncated Toeplitz operators.
(1) Let ω in D or ω in T such that α and β has an ADC at ω (2) The only rank one asymmetric truncated Toeplitz operators in Tðα, βÞ are nonzero scalar multiple of or ω ∈ T such that α and β have an ADC at ω if and only if fmn ≤ 2g or fm > 1 and n > 1g, where m, n ∈ ℕ ∪ f+∞g are the dimensions of K α and K β , respectively In [7], Cima et al. raised the question of which linear transformations on finite-dimensional model spaces are truncated Toeplitz operators, and they proved that the matrix representation of TTOs with respect to kernel basis (conjugate kernel basis, Clark and modified Clark bases) is entirely determined by the entries of the main diagonal and first row. In [8,9], Lanucha obtained similar results for TTOs acting on model spaces associated to interpolating Blaschke products and for THOs. In [6], Jurasik and Lanucha generalized these results to asymmetric truncated Toeplitz operators on finite-dimensional model spaces.
This paper determines all rank one asymmetric truncated Hankel operators and generalizes the results about matrix representation to ATHOs and ATTOs on special model spaces. In Section 2, we cite some results from [3,[10][11][12]. We precise all rank one ATHOs in Section 3. In Section 4, we explore the ATHOs on finite-dimensional spaces, we calculate the dimension of Hðα, βÞ and exhibit a basis for it, and we give the matrix representation of ATHOs in finite dimensional model spaces associated to Blaschke products each with distinct zeros. Section 5 is dedicated to the matrix representation of ATHOs and ATTOS on special infinitedimensional model spaces, in particular, we study the action where τ ξ,c ðzÞ is disc automorphism.

Preliminaries
Like for truncated Toeplitz and Hankel operators, the symbol of an ATTO and an ATHO is not unique, in fact, we have the following theorem.
Theorem 2 (see [10,12] We will use the following technical lemma. Lemma 3 (see [3]). Let α be inner, for λ ∈ D, we have Clark, in [11], proved that the only unitary rank one perturbations of the compressed shift operator are and that the point spectrum of U α λ is the set of the solutions of the equation αðηÞ = αð0Þ + λ/1 + αð0Þλ at which α has an ADC, denote them by η m . The corresponding eigenvectors are the normalized boundary kernels v α η m ≔ kk α η m k −1 k α η m corresponding to the points η m . Whenever the point spectrum of U α λ is pure, the family of eigenvectors forms a basis for K α called the Clark basis.
The modified Clark basis ðe α η m Þ m satisfies C α e α η m = e α η m and is given by One of the most important results about TTOs and THOs is their characterization in terms of compressed shift operator and operators of rank at most 2. Recently, the authors in [10] proved similar characterizations for both ATTO and ATHO.
Theorem 4 (see [10]). Let A be a bounded linear operator from K α to K β . Then for some χ ∈ K α , ψ ∈ K β if and only if for some χ ∈ K α , ψ ∈ K β and some λ α , λ β in T .
In the same paper [10], we also have the following theorem.

Asymmetric Truncated Hankel Operators of Rank One
In this section, we describe all rank one asymmetric truncated Hankel operators through the results for asymmetric truncated Toeplitz operators. In what follows, let α and β denote two arbitrary inner functions.
The following proposition gives some rank one asymmetric truncated Hankel operators. or Choose an arbitrary λ ∈ D or λ ∈ T such that α # and β have an ADC at λ, by Theorem 1, we have For λ ∈ D, the operatorsk β λ ⊗k α λ and k β λ ⊗ k α λ belong to Hðα, βÞ. This is also true for λ ∈ T , since α # has an ADC at λ, if and only if α has an ADC at λ. ?
Now, we give the main theorem of this section.
Theorem 7. All rank one operators in Hðα, βÞ are the nonzero scalar multiples of k β λ ⊗ k α λ andk β λ ⊗k α λ , where λ ∈ D or λ ∈ T such that α and β have an ADC at λ and λ, respectively, if and only if fmn ≤ 2g or fm > 1 and n > 1g, where m, n ∈ ℕ ∪ f+∞g are the dimensions of K α and K β , respectively.
Proof. Every rank one operator in Hðα, βÞ is of the form f ⊗ g for f ∈ K β and g ∈ K α . Since Hðα, if and only if fmn ≤ 2g or fm > 1 and n > 1g. ☐

Asymmetric Truncated Hankel Operators in Finite-Dimensional Model Spaces
In this section, we suppose that both α and β are finite Blaschke products of respective multiplicities m and n.

Dimension and Basis of Hðα, βÞ
Theorem 8. Let K α and K β have dimensions m and n, respectively, then the dimension of Hðα, βÞ equals m + n − 1.

Theorem 10.
Let α, β be two finite Blaschke products of respective multiplicities m and n. Then for any m + n − 1 distinct points from D, denoted by is also a basis of Hðα, βÞ Proof. We only prove (1), the other case follows by application of the conjugations. From [6], a basis of Tðα # , βÞ is fk Using the proof of Propo- We can prove the result directly as in [6] for ATTOs.

Matrix Representation of ATHO on Finite-Dimensional
Spaces. In this subsection, we give the matrix representation of an ATHO acting on two finite dimensional model spaces, with respect to kernel and conjugate kernel bases, Clark and modified Clark bases. In all this section, suppose that the inner function α has distinct zeros ða i Þ m i=1 and β has distinct zeros ðb j Þ n j=1 .

Matrix Representation with respect to the Kernel Bases and Conjugate Kernel Bases. Choose
, ::, m + n − 1g is a basis for Hðα, βÞ (Theorem 10). Since the zeros of the considered inner functions are distinct, fk α a j , j = 1, ⋯, mg and fk β b i , i = 1, ⋯, ng are bases of K α and K β (same for the conjugate kernel functions). We denote the matrix representation of a bounded operator B with respect to the abovementioned kernel bases (conjugate kernel bases) by ðs i,j Þððp i,j ÞÞ, where 1 ≤ i ≤ n and 1 ≤ j ≤ m. For the asymmetric case, we also have (see [9]), Proof. Suppose that B = B α,β φ ∈ Hðα, βÞ for some φ ∈ L 2 . We show that its matrix representation with respect to the kernel basis is of the form (21). From Theorem 10, B Now replace in (20), we have We can write By adding and Journal of Function Spaces Conversely, suppose that the matrix representation of the bounded transformation B satisfies (21). From the proof of the first implication, we know that the subspace of the matrices satisfying (21) is a subspace of Hðα, βÞ, and its dimension is obviously m + n − 1. But dim Hðα, βÞ = m + n − 1, we have the equality. ☐ The following theorem establishes the matrix representation with respect to conjugate kernel bases.
Theorem 13. Let B be a bounded linear transformation from K α to K β . B ∈ Hðα, βÞ if and only if for 1 ≤ i ≤ n and 1 ≤ j ≤ m , Proof. We have Therefore where s i,j is the entry in the ith row and jth column of the matrix representing B α,β αβ # φ with respect to the kernel bases, by Theorem 12, it satisfies (21) and we have have exactly m and n distinct solutions, respectively, denoted by ðη j Þ m j=1 and ðζ i Þ n i=1 . The families of normalized eigenvec- bases for K α and K β , respectively. We also have Reorder the sequences ðη j Þ m j=1 and ðζ i Þ n i=1 so that l be the maximal integer such that η i = ζ i , for any i ≤ l. l = 0 when η i =ζ i for every i. Denote by ðt i,j Þ, for 1 ≤ i ≤ n and 1 ≤ j ≤ m , the matrix representation of any bounded transformation from K α into K β with respect to Clark bases, and by ðu i,j Þ, for 1 ≤ i ≤ n and 1 ≤ j ≤ m, the matrix representation with respect to the modified Clark bases, we have For l ≥ 1 with i > l, (2) For l = 0 Proof. We proceed as in the proof of Theorem 12. ☐

Journal of Function Spaces
We deduce the following theorem from Theorem 14 and formula (32).
Theorem 15. A bounded linear transformation B from K α to K β belongs to Hðα, βÞ if and only if for 1 ≤ i ≤ n and 1 ≤ j ≤ m , except that i = j ≤ l.
(1) For l ≥ 1 with i ≤ l and i=j For l ≥ 1 with i > l, (2) For l = 0

Matrix Representation in Infinite-Dimensional Case
In all this section, unless mentioned, we will suppose that α and β are two interpolating Blaschke sequences with respective zeros ða m Þ m≥1 and ðb n Þ n≥1 , then the kernel functions ðk α a i Þ i≥1 and conjugate kernel functions ðk α a i Þ i≥1 form two Riesz bases for K α (same for K β ) (see [8,9]). For all sequence of complex numbers ðf m Þ m≥1 such that ∑ ∞ m=1 jf m j 2 ð1 − ja m j 2 Þ < ∞, the unique solution f in K α of the interpolation problem f ða m Þ = f m is given by We will also keep the previous notations of the matrix representations of an operator with respect to the different bases (s i,j , p i,j , t i,j , and u i,j ).

Matrix Representation of ATHOs.
Denote by ða l k Þ k≥1 = ð b l k Þ k≥1 the subsequence of common elements between ða m Þ m and ð b n Þ n ordered such that a l k = b l k and that 1 = l 1 ∈ ðl k Þ k≥1 . We will prove that the above matrix representations are also true in the case of model spaces associated to interpolating Blaschke products.
Theorem 16. Let B be a bounded linear transformation from K α to K β . Then, B ∈ Hðα, βÞ if and only if for any i, j Proof. Suppose that B = B α,β φ and we will prove that it satisfies (40). As in the proof of Theorem 8, for any φ in Then, there are χ ∈ K α and ψ ∈ K β # such that B By (6), we have So Journal of Function Spaces Replacing in (20), we get When ðl k Þ k is empty, we have When ðl k Þ k is not empty, we havẽ Therefore, we have 4 cases, {j ∈ ðl k Þ k and i ∈ ðl k Þ k }, fj ∈ ðl k Þ k and i∈ðl k Þ k g, fj∈ðl k Þ k and i ∈ ðl k Þ k g, and fj∈ðl k Þ k andi∈ ðl k Þ k g. In all these cases, we decompose, add, and subtract as in the finite-dimensional case. Using the fact that a 1 = b 1 , we obtain Conversely, we proceed as in [9] for truncated Hankel operators but using the generalized characterization in Theorem 4. ☐ As for Theorem 10, we also generalize the matrix representation of the ATHOs with respect to conjugate kernel bases.
Theorem 17. Let B be a bounded linear transformation from K α to K β . We have B ∈ Hðα, βÞ if and only if for any i, j ≥ 1, we have Suppose that α and β are inner functions such that K α and K β have Clark bases ðv α η j Þ j≥1 and ðv , where ðη j Þ j and ðζ i Þ i are sequences of eigenvalues for some U α λ 1 and some U β λ 2 satisfying the equations (30). We have the relations ffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi jα ′ ðη j Þj q = kk α η j k and Reorder these sequences such that η l k = ζ l k , for any k ≥ 1 and that 1 = l 1 ∈ ðl k Þ k . We have the following theorem.
Theorem 18. Let B be a bounded linear transformation from K α to K β . B ∈ Hðα, βÞ if and only if (1) When ðl k Þ k≥1 is not empty, for fi ∈ ðl k Þ k and j∈ðl k Þ k g or fi ∈ ðl k Þ k and j ∈ ðl k Þ k and j ≠ ig For fi∈ðl k Þ k and j ∈ ðl k Þ k g or fi∈ðl k Þ k and j∈ðl k Þ k g, (2) When ðl k Þ k≥1 is empty, for any i, j ≥ 1

Journal of Function Spaces
Proof. The proof is similar to the one of Theorem 14, but we use formula (44). For the converse implication, we proceed as in Theorem 21, except using the following characterization of ATHOs in Theorem 4, there exist χ ∈ K α and ψ ∈ K β , ☐ As for Theorem 15, we deduce the matrix representation with respect to modified Clark bases.
Theorem 19. A bounded linear transformation B from K α to K β belongs to Hðα, βÞ if and only if (1) When ðl k Þ k≥1 is not empty, for fi ∈ ðl k Þ k and j∈ðl k Þ k g or fi ∈ ðl k Þ k and j ∈ ðl k Þ k and j ≠ ig For fi∈ðl k Þ k and j ∈ ðl k Þ k g or fi∈ðl k Þ k and j∈ðl k Þ k g, (2) When ðl k Þ k≥1 is empty
Proposition 20. Let α and β be two inner functions, φ ∈ L 2 and A α,β φ ∈ Tðα, βÞ. Then for some c ∈ D and ξ ∈ T , Proof. For f ∈ K α , φ ∈ L 2 , ξ ∈ T and a ∈ D, we have Let χ = τ ξ,c ðωÞ. Since τ′ ξ,c ðzÞ = ξðjcj 2 − 1Þ/ðð1 − czÞ 2 Þ, we have Since we have Journal of Function Spaces In conclusion, Applying V −1 ξ,c = V * ξ,c , we get the result. ☐ As before, the formula (20) is also true for ATTOs. We will prove that the matrix characterizations of ATTOs on finite-dimensional model space obtained in [6] are also true in the infinite case. Denote by ða l k Þ k≥1 the subsequence of common zeros between α and β ordered such that a l k = b l k , for any k ≥ 1 and that 1 = l 1 ∈ ðl k Þ k when ðl k Þ k is not empty.
Theorem 21. A bounded linear transformation A from K α to K β belongs to Tðα, βÞ if and only if its matrix representation with respect to the kernel bases satisfies (1) When ðl k Þ k≥1 is not empty, for fi ∈ ðl k Þ k and j∈ðl k Þ k g or fi ∈ ðl k Þ k and j ∈ ðl k Þ k and j ≠ ig For fi ∈ ðl k Þ k and j ∈ ðl k Þ k g or fi ∈ ðl k Þ k and j ∈ ðl k Þ k g, (2) When ðl k Þ k≥1 is empty Proof. The proof of necessity is the same as in [6] for ATTOs acting on finite-dimensional model spaces. Conversely, consider any bounded linear transformation from K α into K β whose matrix representation satisfies Theorem 21. To show that A ∈ Tðα, βÞ, or equivalently to show that A satisfies the characterization in Theorem 4, we will find a χ ∈ K α and a ψ ∈ K β such that This is equivalent to for any i, j ≥ 1. Using the relations in Lemma 3, when b i ≠ 0, for any i ≥ 1, we have for every i, j ≥ 1 for any i, j ≥ 1.
When ðl k Þ k is not empty, suppose that βð0Þ ≠ 0. Since the matrix representation of A satisfies the formulas in Theorem 21, the above system is equivalent to 9 Journal of Function Spaces Set an arbitraryψðb 1 Þ, then the solution of the system is To show that the solutions of the system χ and ψ are in K α and K β , respectively, it suffices to prove that χ and ψ are the unique solutions in K α and K β of the interpolation problems corresponding to ða m Þ m≥1 and ðb n Þ n≥1 (39). In fact, 10

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If ðl k Þ k is empty, then the equation (70) becomes with the assumption βð0Þ ≠ 0, If we set an arbitraryψðb 1 Þ, then, the solutions are It remains to check that χ ∈ K α and ψ ∈ K β by showing that they are solutions of the corresponding interpolation problems. The case βð0Þ = 0 can be treated in the same way as in the previous case.
and since C β BJ # and J # BC α are ATTOs by Theorem 4 ([10]), then we can deduce the matrix representation of an ATTO with respect to kernel and conjugate kernel bases, and to conjugate kernel and kernel bases in both finite and infinite dimensional cases, which matches with the work done in [14].
Similarly, we also can obtain the matrix representation of an ATHO with respect to kernel and conjugate kernel bases, and to conjugate kernel and kernel bases via the passage formulas.

Data Availability
There is no underlying data in this paper, and all of its research is the derivation of basic theory.

Conflicts of Interest
The authors declare that they have no conflicts of interest.