# Asymmetric Truncated Hankel Operators: Rank One, Matrix Representation

Authors: Firdaws Rahmani; Yufeng Lu; Ran Li
Journal: Journal of Function Spaces
 (2021)

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## Abstract

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.


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## Body

## 1. Introduction

LetH2 be the standard Hardy space of the unit disc D identified with the subspace of the boundary functions of its functions in L2T.A function inH∞D is inner if it is unimodular on the unit circle T. To each inner function α we associate a model space, Kα=H2⊖αH2. The model space is a reproducing kernel Hilbert space with reproducing kernel, kλαz=1−αλ¯αz/1−λ¯z, for z, λ∈D.The inner functionα has an angular derivative in the sense of Carathéodory (ADC) at a point η∈T if and only if every f in Kα has nontangential limit at η. In particular, kηα∈Kα.Ifα is an inner function, then α#=αz¯¯ is also an inner function and the associated model space is noted Kα#.To each model space, we associate a natural conjugationCα such that CαKα=Kα, given by Cαfz=αzfz¯, for z∈T. The image of the kernel function for Cα, called conjugate kernel function k~λα, is given by k~λαz=αz−αλ/z−λ, z,λ∈D. If α has an ADC at η∈T,
(1)k~ηαz=αz−αηz−η=αηη¯1−αη¯αz1−η¯z=αηη¯kηαz.The model spaces can also be defined as the only invariant subspaces of the backward shift operatorS∗ on H2. Denote by Sα (Sα∗) the compression of the shift operator (resp., backward shift) to the model space Kα.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 onT and kηα∈Kα, for every η∈T. In this case, let m denote the multiplicity of a finite Blaschke product α. If we arbitrarily choose a collection λi,i=1,⋯,m of distinct points in D, kλiα,i=1,2,..,m forms a basis for Kα.In infinite-dimensional case the kernel functionskλiαi≥1 form a Riesz basis if and only if α is an interpolating Blaschke product or equivalently, λii≥1 is a uniformly separated Blaschke sequence that satisfies infk≥1∏k≥,k≠nλk−λn/1−λk¯λn>0.Asymmetric truncated Toeplitz and Hankel operators were first introduced in [1, 2], respectively. For φ∈L2T, let α and β be two inner functions, the asymmetric truncated Toeplitz operator Aφα,β and the asymmetric truncated Hankel operator Bφα,β are defined on Kα∞=Kα∩H∞ by
(2)Aφα,β:Kα∞⟶Kβ,(3)f↦Aφα,βf=Pβφf,(4)Bφα,β:Kα∞⟶Kβ,(5)f↦Bφα,βf=PβJI−Pφf,where P and Pβ denote the orthogonal projection on H2 and Kβ, respectively, and J is the flip operator from H2 onto H¯02=zH2¯ defined on T by Jfz=z¯fz¯. We denote the set of bounded asymmetric truncated Toeplitz operators (ATTO) (bounded asymmetric truncated Hankel operators (ATHO)) by Tα,β (Hα,β).In [3], Sarason introduced truncated Toeplitz operators and showed that all rank one TTOs are of the form ckλα⊗k~λα or ck~λα⊗kλα, 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 k~λ¯α⊗k~λα,kλ¯α⊗kλα. Surprisingly, this is not always true in the case of asymmetric truncated Toeplitz operators.Theorem 1 (see [5, 6]).
Letα and β be two inner functions.
(1)
Letω in D or ω in T such that αandβ has an ADC at ω(6)kωβ⊗k~ωα=Aα¯z−ω¯α,β∈Tα,βandk~ωβ⊗kωα=Aβz−ωα,β∈Tα,β.(2)
The only rank one asymmetric truncated Toeplitz operators inTα,β are nonzero scalar multiple of kωβ⊗k~ωα and k~ωβ⊗kωα where ω∈D or ω∈T such that α and β have an ADC at ω if and only if mn≤2 or m>1andn>1, where m,n∈ℕ∪+∞ are the dimensions of Kα and Kβ, respectivelyIn [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 Section2, we cite some results from [3, 10–12]. 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 Hα,β 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 infinite-dimensional model spaces, in particular, we study the action of the unitary operator Vξ,c=τξ,c′f∘τξ,c, where τξ,cz is disc automorphism.


## 2. 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]).
Letα and β be inner functions. For φ∈L2T, we have Aφα,β=0 if and only if φ∈αH2¯+βH2 and Bφα,β=0 if and only if φ∈H2+αβ#H2¯. Where β#z=βz¯¯.We will use the following technical lemma.Lemma 3 (see [3]).
Letα be inner, for λ∈D, we have
(7)Sα∗kλα=λ¯kλα−αλ¯k0α~,andSαk~λα=λk~λα−αλk0α.
Forλ∈D\0, we have
(8)Sα∗kλα~=1λ¯kλα~−1λk0α~.
Clark, in [11], proved that the only unitary rank one perturbations of the compressed shift operator are
(9)Uλα=Sα+α0+λ1−α02k0α⊗k0α~,forλ∈T,and that the point spectrum of Uλα is the set of the solutions of the equation αη=α0+λ/1+α0¯λ at which α has an ADC, denote them by ηm. The corresponding eigenvectors are the normalized boundary kernels vηmα≔kηmα−1kηmα corresponding to the points ηm. Whenever the point spectrum of Uλα is pure, the family of eigenvectors forms a basis for Kα called the Clark basis.
The modified Clark basiseηmαm satisfies Cαeηmα=eηmα and is given by eηmα=ωmαvηmα, where ωmα=exp−iargηm−argλ.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]).
LetA be a bounded linear operator from Kα to Kβ. Then
(1)
A∈Tα,β if and only if
(10)A−SβASα∗=ψ⊗k0α+k0β⊗χ,for some χ∈Kα,ψ∈Kβ if and only if
(11)A−UλββAUλαβ∗=ψ⊗k0α+k0β⊗χ,for some χ∈Kα,ψ∈Kβ and some λα,λβ in T.(2)
B∈Hα,β if and only if
(12)B−Sβ∗BSα∗=ψ⊗k0α+k~0β⊗χ,for some χ∈Kα,ψ∈Kβ if and only if
(13)B−Uλββ∗BUλαα∗=ψ⊗k0α+k~0β⊗χ,for some χ∈Kα,ψ∈Kβ and some λα,λβ in T.
WhereUλαα,Uλββ are Clark operators (9).In the same paper [10], we also have the following theorem.Theorem 5 (see [10]).
Letα and β be two inner functions and φ∈L2T. Then
(1)
A∈Tα,β if and only if CβACα∈Tα,β. In addition, CβAφα,βCα=Aα¯βφ¯α,β(2)
B∈Hα,β if and only if CβBCα∈Hα,β. In addition, CβBφα,βCα=Bαβ#φ¯α,β(3)
CβAφα,βJ#=Bβ#¯φ#α#,β, where J# is a conjugation on L2 defined by J#f=fz¯¯


## 3. 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, letα and β denote two arbitrary inner functions.The following proposition gives some rank one asymmetric truncated Hankel operators.Proposition 6.
We have(1)
For everyλ∈D, the operators kλ¯β⊗kλα=B1/z−λα,β and k~λ¯β⊗k~λα=Bαβ#¯/z−λ¯α,β belong to Hα,β(2)
Ifα and β have an ADC at η and η¯, respectively, the operators k~η¯β⊗k~ηα=Bαβ#¯/z−ηα,β=Bk~ηαβ#¯α,β and kη¯β⊗kηα=Bηk~ηαβ#¯α,β belong to Hα,βProof.
From Theorem5, we have
(14)CβAφα,βJ#=Bβ#¯φ#α#,β,or
(15)CβAφα#,βJ#=Bβ#¯φ#α,β.
Choose an arbitraryλ∈D or λ∈T such that α# and β have an ADC at λ, by Theorem 1, we have
(16)Cβkλβ⊗k~λα#J#=CβAα#¯z−λ¯α#,βJ#=Bαβ#¯z¯−λ¯α,β=k~λβ⊗k~λ¯α,(17)Cβk~λβ⊗kλα#J#=CβAβz−λα#,βJ#=B1z−λ¯α,β=kλβ⊗kλ¯α.
Forλ∈D, the operators k~λβ⊗k~λ¯α and kλβ⊗kλ¯α belong to Hα,β. 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 inHα,β are the nonzero scalar multiples of kλ¯β⊗kλα and k~λ¯β⊗k~λα, where λ∈D or λ∈T such that α and β have an ADC at λ and λ¯, respectively, if and only if mn≤2 or m>1andn>1, where m,n∈ℕ∪+∞ are the dimensions of Kα and Kβ, respectively.Proof.
Every rank one operator inHα,β is of the form f⊗g for f∈Kβ and g∈Kα. Since Hα,β=CβTα#,βJ# ([10]), we can find f′∈Kβ and g′∈Kα# such that f⊗g=Cβf′⊗g′J#. For f′⊗g′ is rank one in Tα#,β, by Theorem 1, there exist c∈ℂ and λ∈D or λ∈T (α#,β have an ADC at λ) such that f′⊗g′=ckλβ⊗k~λα# or f′⊗g′=ck~λβ⊗kλα#, if and only if mn≤2 or m>1andn>1. Finally, f⊗g=cCβkλβ⊗k~λα#J#=ck~λβ⊗k~λ¯α or f⊗g=cCβk~λβ⊗kλα#J#=ckλβ⊗kλ¯α if and only if mn≤2 or m>1andn>1.


## 4. Asymmetric Truncated Hankel Operators in Finite-Dimensional Model Spaces

In this section, we suppose that bothα and β are finite Blaschke products of respective multiplicities m and n.


### 4.1. Dimension and Basis ofHα,β

Theorem 8.
LetKα and Kβ have dimensions m and n, respectively, then the dimension of Hα,β equals m+n−1.Proof.
Using Theorem2, we can write
(18)Hα,β=Bχ¯α,β:χ∈Kαβ#,where dimKαβ#=m+n. Since the constants are in Kαβ# and Bcα,β=0forallc∈ℂ,
(19)dimHα,β=dimKαβ#−1=m+n−1.Remark 9.
From Theorem4, CβTα#,βJ#=Hα,β. Since Cβ and J# preserve the dimensions, dimHα,β=dimTα#,β=m+n−1.Theorem 10.
Letα,β be two finite Blaschke products of respective multiplicities m and n. Then for any m+n−1 distinct points from D¯, denoted by λii=1m+n−1,
(1)
kλi¯β⊗kλiα,i=1,..,m+n−1 is a basis of Hα,β(2)
k~λi¯β⊗k~λiα,i=1,..,m+n−1 is also a basis of Hα,βProof.
We only prove (1), the other case follows by application of the conjugations. From [6], a basis of Tα#,β is k~λi¯β⊗kλi¯α#,i=1,⋯,m+n−1. Using the proof of Proposition 6, Cβk~λi¯β⊗kλi¯α#J#=kλi¯β⊗kλiα,i=1,⋯,m+n−1 is a basis of Hα,β.Remark 11.
We can prove the result directly as in [6] for ATTOs.


### 4.2. 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 functionα has distinct zeros aii=1m and β has distinct zeros bjj=1n.


#### 4.2.1. Matrix Representation with respect to the Kernel Bases and Conjugate Kernel Bases

Choosem+n−1 points λii=1m+n−1 in D distinct from bj¯j=1n, then kλi¯β⊗kλiα,i=1,..,m+n−1 is a basis for Hα,β (Theorem 10). Since the zeros of the considered inner functions are distinct, kajα,j=1,⋯,m and kbiβ,i=1,⋯,n are bases of Kα and Kβ (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 si,jpi,j, where 1≤i≤n and 1≤j≤m. For the asymmetric case, we also have (see [9]),
(20)si,j=1β′bi¯Bkajα,k~biβ.Theorem 12.
LetB be a bounded linear transformation from Kα to Kβ.
B∈Hα,β if and only if for 1≤i≤n and 1≤j≤m,
(21)si,j=β′b1¯1−ajb1¯s1,j−β′b1¯1−a1b1¯s1,1+β′bi¯1−a1bi¯si,1β′bi¯1−ajbi¯.Proof.
Suppose thatB=Bφα,β∈Hα,β for some φ∈L2. We show that its matrix representation with respect to the kernel basis is of the form (21). From Theorem 10, Bφα,β=∑q=1m+n−1dqkλq¯β⊗kλqα. We have
(22)Bφα,βkajα=∑q=1m+n−1dqkλq¯β⊗kλqαkajα=∑q=1m+n−1dq11−aj¯λqkλq¯β.
Now replace in (20), we have
(23)si,j=1β′bi¯∑q=1m+n−1dq11−aj¯λqβ#λqλq−bi¯.
We can write(24)11−aj¯λq1λq−bi¯=11−aj¯bi¯1−λq1−bi¯λq−bi¯−1−aj¯1−λqaj¯.
By adding and subtracting1−b1¯/λq−b1¯ and 1−a1¯/1−λqa1¯, we find
(25)si,j=β′b1¯1−ajb1¯s1,j−β′b1¯1−a1b1¯s1,1+β′bi¯1−a1bi¯si,1β′bi¯1−ajbi¯.
Conversely, suppose that the matrix representation of the bounded transformationB satisfies (21). From the proof of the first implication, we know that the subspace of the matrices satisfying (21) is a subspace of Hα,β, and its dimension is obviously m+n−1. But dimHα,β=m+n−1, we have the equality.The following theorem establishes the matrix representation with respect to conjugate kernel bases.Theorem 13.
LetB be a bounded linear transformation from Kα to Kβ. B∈Hα,β if and only if for 1≤i≤n and 1≤j≤m,
(26)pi,j=β′b11−ajb1p1,j−β′b11−a1b1p1,1+β′bi1−a1bipi,1β′bi1−ajbi.Proof.
We have(27)pi,j=1β′biBk~ajα,kbiβ=1β′biBCαkajα,Cβk~biβ=1β′bi¯CβBCαkajα,k~biβ¯=1β′bi¯Bαβ#φ¯α,βkajα,k~biβ¯.
Therefore(28)pi,j=si,j¯,where si,j is the entry in the ith row and jth column of the matrix representing Bαβ#φ¯α,β with respect to the kernel bases, by Theorem 12, it satisfies (21) and we have
(29)pi,j=β′b1¯1−ajb1¯s1,j−β′b1¯1−a1b1¯s1,1+β′bi¯1−a1bi¯si,1β′bi¯1−ajbi¯¯¯=β′b11−ajb1s1,j¯−β′b11−a1b1s1,1¯+β′bi1−a1bisi,1¯β′bi1−ajbi=β′b11−ajb1p1,j−β′b11−a1b1p1,1+β′bi1−a1bipi,1β′bi1−ajbi.


#### 4.2.2. Matrix Representation with respect to Clark and Modified Clark Bases

Letα be m-finite Blaschke product and β be n-finite Blaschke product. Choose arbitrary λ1,λ2∈T, then the following equations
(30)αη=α0+λ11+α0¯λ1≔αλ1andβη=β0+λ21+β0¯λ2≔βλ2,have exactly m and n distinct solutions, respectively, denoted by ηjj=1m and ζii=1n. The families of normalized eigenvectors vηjα≔kηjα−1kηjα and vζiβ≔kζiβ−1kζiβ form orthonormal bases for Kα and Kβ, respectively. We also have
(31)kηjα=α′ηjandkζiβ=β′ζi.Reorder the sequencesηjj=1m and ζii=1n so that l be the maximal integer such that ηi¯=ζi, for any i≤l. l=0 when ηi¯=ζi for every i. Denote by ti,j, for 1≤i≤n and 1≤j≤m, the matrix representation of any bounded transformation from Kα into Kβ with respect to Clark bases, and by ui,j, for 1≤i≤n and 1≤j≤m, the matrix representation with respect to the modified Clark bases, we have
(32)ui,j=Beηjα,eζiβ=ωjαωiβ¯Bvηjα,vζiβ=ωjαωiβ¯ti,j.Theorem 14.
A bounded linear transformationB from Kα to Kβ belongs to Hα,β if and only if
(1)
Forl≥1 with i≤l and i=j(33)ti,j=β′ζ1ηj¯−ζ1β′ζiηj¯−ζit1,j+α′ηiβ′ζ1ηj¯−ζiα′ηjβ′ζiηj¯−ζit1,i.Forl≥1 with i>l,
(34)ti,j=β′ζ1ηj¯−ζ1β′ζiηj¯−ζit1,j+α′η1ηj¯−ζiα′ηjηj¯−ζiti,1.(2)
Forl=0(35)ti,j=β′ζ1ηj¯−ζ1β′ζiηj¯−ζit1,j−α′η1β′ζ1η1¯−ζ1α′ηjβ′ζiηj¯−ζit1,1+α′η1η1¯−ζiα′ηjηj¯−ζiti,1.Proof.
We proceed as in the proof of Theorem12.We deduce the following theorem from Theorem14 and formula (32).Theorem 15.
A bounded linear transformationB from Kα to Kβ belongs to Hα,β if and only if for 1≤i≤n and 1≤j≤m, except that i=j≤l.
(1)
Forl≥1 with i≤l and i=j(36)ui,j=ω1βωiββ′ζ1ηj¯−ζ1β′ζiηj¯−ζiu1,j+ωjαω1αα′ηiβ′ζ1ηj¯−ζiα′ηjβ′ζiηj¯−ζiu1,i.Forl≥1 with i>l,
(37)ui,j=ω1βωiββ′ζ1ηj¯−ζiβ′ζiηj¯−ζiu1,j+ωjαω1αα′η1ηj¯−ζiα′ηjηj¯−ζiui,1.(2)
Forl=0(38)ui,j=ω1βωiββ′ζ1ηj¯−ζiβ′ζiηj¯−ζiu1,j−ωjαωiβω1βω1αα′η1β′ζ1ηj¯−ζiα′ηjβ′ζiηj¯−ζiu1,1+ωjαω1αα′η1ηj¯−ζiβ′ζiηj¯−ζiui,1.


## 4.1. Dimension and Basis ofHα,β

Theorem 8.
LetKα and Kβ have dimensions m and n, respectively, then the dimension of Hα,β equals m+n−1.Proof.
Using Theorem2, we can write
(18)Hα,β=Bχ¯α,β:χ∈Kαβ#,where dimKαβ#=m+n. Since the constants are in Kαβ# and Bcα,β=0forallc∈ℂ,
(19)dimHα,β=dimKαβ#−1=m+n−1.Remark 9.
From Theorem4, CβTα#,βJ#=Hα,β. Since Cβ and J# preserve the dimensions, dimHα,β=dimTα#,β=m+n−1.Theorem 10.
Letα,β be two finite Blaschke products of respective multiplicities m and n. Then for any m+n−1 distinct points from D¯, denoted by λii=1m+n−1,
(1)
kλi¯β⊗kλiα,i=1,..,m+n−1 is a basis of Hα,β(2)
k~λi¯β⊗k~λiα,i=1,..,m+n−1 is also a basis of Hα,βProof.
We only prove (1), the other case follows by application of the conjugations. From [6], a basis of Tα#,β is k~λi¯β⊗kλi¯α#,i=1,⋯,m+n−1. Using the proof of Proposition 6, Cβk~λi¯β⊗kλi¯α#J#=kλi¯β⊗kλiα,i=1,⋯,m+n−1 is a basis of Hα,β.Remark 11.
We can prove the result directly as in [6] for ATTOs.


## 4.2. 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 functionα has distinct zeros aii=1m and β has distinct zeros bjj=1n.


### 4.2.1. Matrix Representation with respect to the Kernel Bases and Conjugate Kernel Bases

Choosem+n−1 points λii=1m+n−1 in D distinct from bj¯j=1n, then kλi¯β⊗kλiα,i=1,..,m+n−1 is a basis for Hα,β (Theorem 10). Since the zeros of the considered inner functions are distinct, kajα,j=1,⋯,m and kbiβ,i=1,⋯,n are bases of Kα and Kβ (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 si,jpi,j, where 1≤i≤n and 1≤j≤m. For the asymmetric case, we also have (see [9]),
(20)si,j=1β′bi¯Bkajα,k~biβ.Theorem 12.
LetB be a bounded linear transformation from Kα to Kβ.
B∈Hα,β if and only if for 1≤i≤n and 1≤j≤m,
(21)si,j=β′b1¯1−ajb1¯s1,j−β′b1¯1−a1b1¯s1,1+β′bi¯1−a1bi¯si,1β′bi¯1−ajbi¯.Proof.
Suppose thatB=Bφα,β∈Hα,β for some φ∈L2. We show that its matrix representation with respect to the kernel basis is of the form (21). From Theorem 10, Bφα,β=∑q=1m+n−1dqkλq¯β⊗kλqα. We have
(22)Bφα,βkajα=∑q=1m+n−1dqkλq¯β⊗kλqαkajα=∑q=1m+n−1dq11−aj¯λqkλq¯β.
Now replace in (20), we have
(23)si,j=1β′bi¯∑q=1m+n−1dq11−aj¯λqβ#λqλq−bi¯.
We can write(24)11−aj¯λq1λq−bi¯=11−aj¯bi¯1−λq1−bi¯λq−bi¯−1−aj¯1−λqaj¯.
By adding and subtracting1−b1¯/λq−b1¯ and 1−a1¯/1−λqa1¯, we find
(25)si,j=β′b1¯1−ajb1¯s1,j−β′b1¯1−a1b1¯s1,1+β′bi¯1−a1bi¯si,1β′bi¯1−ajbi¯.
Conversely, suppose that the matrix representation of the bounded transformationB satisfies (21). From the proof of the first implication, we know that the subspace of the matrices satisfying (21) is a subspace of Hα,β, and its dimension is obviously m+n−1. But dimHα,β=m+n−1, we have the equality.The following theorem establishes the matrix representation with respect to conjugate kernel bases.Theorem 13.
LetB be a bounded linear transformation from Kα to Kβ. B∈Hα,β if and only if for 1≤i≤n and 1≤j≤m,
(26)pi,j=β′b11−ajb1p1,j−β′b11−a1b1p1,1+β′bi1−a1bipi,1β′bi1−ajbi.Proof.
We have(27)pi,j=1β′biBk~ajα,kbiβ=1β′biBCαkajα,Cβk~biβ=1β′bi¯CβBCαkajα,k~biβ¯=1β′bi¯Bαβ#φ¯α,βkajα,k~biβ¯.
Therefore(28)pi,j=si,j¯,where si,j is the entry in the ith row and jth column of the matrix representing Bαβ#φ¯α,β with respect to the kernel bases, by Theorem 12, it satisfies (21) and we have
(29)pi,j=β′b1¯1−ajb1¯s1,j−β′b1¯1−a1b1¯s1,1+β′bi¯1−a1bi¯si,1β′bi¯1−ajbi¯¯¯=β′b11−ajb1s1,j¯−β′b11−a1b1s1,1¯+β′bi1−a1bisi,1¯β′bi1−ajbi=β′b11−ajb1p1,j−β′b11−a1b1p1,1+β′bi1−a1bipi,1β′bi1−ajbi.


### 4.2.2. Matrix Representation with respect to Clark and Modified Clark Bases

Letα be m-finite Blaschke product and β be n-finite Blaschke product. Choose arbitrary λ1,λ2∈T, then the following equations
(30)αη=α0+λ11+α0¯λ1≔αλ1andβη=β0+λ21+β0¯λ2≔βλ2,have exactly m and n distinct solutions, respectively, denoted by ηjj=1m and ζii=1n. The families of normalized eigenvectors vηjα≔kηjα−1kηjα and vζiβ≔kζiβ−1kζiβ form orthonormal bases for Kα and Kβ, respectively. We also have
(31)kηjα=α′ηjandkζiβ=β′ζi.Reorder the sequencesηjj=1m and ζii=1n so that l be the maximal integer such that ηi¯=ζi, for any i≤l. l=0 when ηi¯=ζi for every i. Denote by ti,j, for 1≤i≤n and 1≤j≤m, the matrix representation of any bounded transformation from Kα into Kβ with respect to Clark bases, and by ui,j, for 1≤i≤n and 1≤j≤m, the matrix representation with respect to the modified Clark bases, we have
(32)ui,j=Beηjα,eζiβ=ωjαωiβ¯Bvηjα,vζiβ=ωjαωiβ¯ti,j.Theorem 14.
A bounded linear transformationB from Kα to Kβ belongs to Hα,β if and only if
(1)
Forl≥1 with i≤l and i=j(33)ti,j=β′ζ1ηj¯−ζ1β′ζiηj¯−ζit1,j+α′ηiβ′ζ1ηj¯−ζiα′ηjβ′ζiηj¯−ζit1,i.Forl≥1 with i>l,
(34)ti,j=β′ζ1ηj¯−ζ1β′ζiηj¯−ζit1,j+α′η1ηj¯−ζiα′ηjηj¯−ζiti,1.(2)
Forl=0(35)ti,j=β′ζ1ηj¯−ζ1β′ζiηj¯−ζit1,j−α′η1β′ζ1η1¯−ζ1α′ηjβ′ζiηj¯−ζit1,1+α′η1η1¯−ζiα′ηjηj¯−ζiti,1.Proof.
We proceed as in the proof of Theorem12.We deduce the following theorem from Theorem14 and formula (32).Theorem 15.
A bounded linear transformationB from Kα to Kβ belongs to Hα,β if and only if for 1≤i≤n and 1≤j≤m, except that i=j≤l.
(1)
Forl≥1 with i≤l and i=j(36)ui,j=ω1βωiββ′ζ1ηj¯−ζ1β′ζiηj¯−ζiu1,j+ωjαω1αα′ηiβ′ζ1ηj¯−ζiα′ηjβ′ζiηj¯−ζiu1,i.Forl≥1 with i>l,
(37)ui,j=ω1βωiββ′ζ1ηj¯−ζiβ′ζiηj¯−ζiu1,j+ωjαω1αα′η1ηj¯−ζiα′ηjηj¯−ζiui,1.(2)
Forl=0(38)ui,j=ω1βωiββ′ζ1ηj¯−ζiβ′ζiηj¯−ζiu1,j−ωjαωiβω1βω1αα′η1β′ζ1ηj¯−ζiα′ηjβ′ζiηj¯−ζiu1,1+ωjαω1αα′η1ηj¯−ζiβ′ζiηj¯−ζiui,1.


## 4.2.1. Matrix Representation with respect to the Kernel Bases and Conjugate Kernel Bases

Choosem+n−1 points λii=1m+n−1 in D distinct from bj¯j=1n, then kλi¯β⊗kλiα,i=1,..,m+n−1 is a basis for Hα,β (Theorem 10). Since the zeros of the considered inner functions are distinct, kajα,j=1,⋯,m and kbiβ,i=1,⋯,n are bases of Kα and Kβ (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 si,jpi,j, where 1≤i≤n and 1≤j≤m. For the asymmetric case, we also have (see [9]),
(20)si,j=1β′bi¯Bkajα,k~biβ.Theorem 12.
LetB be a bounded linear transformation from Kα to Kβ.
B∈Hα,β if and only if for 1≤i≤n and 1≤j≤m,
(21)si,j=β′b1¯1−ajb1¯s1,j−β′b1¯1−a1b1¯s1,1+β′bi¯1−a1bi¯si,1β′bi¯1−ajbi¯.Proof.
Suppose thatB=Bφα,β∈Hα,β for some φ∈L2. We show that its matrix representation with respect to the kernel basis is of the form (21). From Theorem 10, Bφα,β=∑q=1m+n−1dqkλq¯β⊗kλqα. We have
(22)Bφα,βkajα=∑q=1m+n−1dqkλq¯β⊗kλqαkajα=∑q=1m+n−1dq11−aj¯λqkλq¯β.
Now replace in (20), we have
(23)si,j=1β′bi¯∑q=1m+n−1dq11−aj¯λqβ#λqλq−bi¯.
We can write(24)11−aj¯λq1λq−bi¯=11−aj¯bi¯1−λq1−bi¯λq−bi¯−1−aj¯1−λqaj¯.
By adding and subtracting1−b1¯/λq−b1¯ and 1−a1¯/1−λqa1¯, we find
(25)si,j=β′b1¯1−ajb1¯s1,j−β′b1¯1−a1b1¯s1,1+β′bi¯1−a1bi¯si,1β′bi¯1−ajbi¯.
Conversely, suppose that the matrix representation of the bounded transformationB satisfies (21). From the proof of the first implication, we know that the subspace of the matrices satisfying (21) is a subspace of Hα,β, and its dimension is obviously m+n−1. But dimHα,β=m+n−1, we have the equality.The following theorem establishes the matrix representation with respect to conjugate kernel bases.Theorem 13.
LetB be a bounded linear transformation from Kα to Kβ. B∈Hα,β if and only if for 1≤i≤n and 1≤j≤m,
(26)pi,j=β′b11−ajb1p1,j−β′b11−a1b1p1,1+β′bi1−a1bipi,1β′bi1−ajbi.Proof.
We have(27)pi,j=1β′biBk~ajα,kbiβ=1β′biBCαkajα,Cβk~biβ=1β′bi¯CβBCαkajα,k~biβ¯=1β′bi¯Bαβ#φ¯α,βkajα,k~biβ¯.
Therefore(28)pi,j=si,j¯,where si,j is the entry in the ith row and jth column of the matrix representing Bαβ#φ¯α,β with respect to the kernel bases, by Theorem 12, it satisfies (21) and we have
(29)pi,j=β′b1¯1−ajb1¯s1,j−β′b1¯1−a1b1¯s1,1+β′bi¯1−a1bi¯si,1β′bi¯1−ajbi¯¯¯=β′b11−ajb1s1,j¯−β′b11−a1b1s1,1¯+β′bi1−a1bisi,1¯β′bi1−ajbi=β′b11−ajb1p1,j−β′b11−a1b1p1,1+β′bi1−a1bipi,1β′bi1−ajbi.


## 4.2.2. Matrix Representation with respect to Clark and Modified Clark Bases

Letα be m-finite Blaschke product and β be n-finite Blaschke product. Choose arbitrary λ1,λ2∈T, then the following equations
(30)αη=α0+λ11+α0¯λ1≔αλ1andβη=β0+λ21+β0¯λ2≔βλ2,have exactly m and n distinct solutions, respectively, denoted by ηjj=1m and ζii=1n. The families of normalized eigenvectors vηjα≔kηjα−1kηjα and vζiβ≔kζiβ−1kζiβ form orthonormal bases for Kα and Kβ, respectively. We also have
(31)kηjα=α′ηjandkζiβ=β′ζi.Reorder the sequencesηjj=1m and ζii=1n so that l be the maximal integer such that ηi¯=ζi, for any i≤l. l=0 when ηi¯=ζi for every i. Denote by ti,j, for 1≤i≤n and 1≤j≤m, the matrix representation of any bounded transformation from Kα into Kβ with respect to Clark bases, and by ui,j, for 1≤i≤n and 1≤j≤m, the matrix representation with respect to the modified Clark bases, we have
(32)ui,j=Beηjα,eζiβ=ωjαωiβ¯Bvηjα,vζiβ=ωjαωiβ¯ti,j.Theorem 14.
A bounded linear transformationB from Kα to Kβ belongs to Hα,β if and only if
(1)
Forl≥1 with i≤l and i=j(33)ti,j=β′ζ1ηj¯−ζ1β′ζiηj¯−ζit1,j+α′ηiβ′ζ1ηj¯−ζiα′ηjβ′ζiηj¯−ζit1,i.Forl≥1 with i>l,
(34)ti,j=β′ζ1ηj¯−ζ1β′ζiηj¯−ζit1,j+α′η1ηj¯−ζiα′ηjηj¯−ζiti,1.(2)
Forl=0(35)ti,j=β′ζ1ηj¯−ζ1β′ζiηj¯−ζit1,j−α′η1β′ζ1η1¯−ζ1α′ηjβ′ζiηj¯−ζit1,1+α′η1η1¯−ζiα′ηjηj¯−ζiti,1.Proof.
We proceed as in the proof of Theorem12.We deduce the following theorem from Theorem14 and formula (32).Theorem 15.
A bounded linear transformationB from Kα to Kβ belongs to Hα,β if and only if for 1≤i≤n and 1≤j≤m, except that i=j≤l.
(1)
Forl≥1 with i≤l and i=j(36)ui,j=ω1βωiββ′ζ1ηj¯−ζ1β′ζiηj¯−ζiu1,j+ωjαω1αα′ηiβ′ζ1ηj¯−ζiα′ηjβ′ζiηj¯−ζiu1,i.Forl≥1 with i>l,
(37)ui,j=ω1βωiββ′ζ1ηj¯−ζiβ′ζiηj¯−ζiu1,j+ωjαω1αα′η1ηj¯−ζiα′ηjηj¯−ζiui,1.(2)
Forl=0(38)ui,j=ω1βωiββ′ζ1ηj¯−ζiβ′ζiηj¯−ζiu1,j−ωjαωiβω1βω1αα′η1β′ζ1ηj¯−ζiα′ηjβ′ζiηj¯−ζiu1,1+ωjαω1αα′η1ηj¯−ζiβ′ζiηj¯−ζiui,1.


## 5. Matrix Representation in Infinite-Dimensional Case

In all this section, unless mentioned, we will suppose thatα and β are two interpolating Blaschke sequences with respective zeros amm≥1 and bnn≥1, then the kernel functions kaiαi≥1 and conjugate kernel functions k~aiαi≥1 form two Riesz bases for Kα (same for Kβ) (see [8, 9]). For all sequence of complex numbers fmm≥1 such that ∑m=1∞fm21−am2<∞, the unique solution f in Kα of the interpolation problem fam=fm is given by
(39)f=∑i=1+∞faiα′aik~aiα=∑i=1+∞f~ai¯α′ai¯kaiα.We will also keep the previous notations of the matrix representations of an operator with respect to the different bases (si,j,pi,j,ti,j, and ui,j).


### 5.1. Matrix Representation of ATHOs

Denote byalkk≥1=b¯lkk≥1 the subsequence of common elements between amm and b¯nn ordered such that alk=blk¯ and that 1=l1∈lkk≥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.
LetB be a bounded linear transformation from Kα to Kβ. Then, B∈Hα,β if and only if for any i,j(40)si,j=β′b1¯1−ajb1¯s1,j−β′b1¯1−a1b1¯s1,1+β′bi¯1−a1bi¯si,1β′bi¯1−ajbi¯.Proof.
Suppose thatB=Bφα,β and we will prove that it satisfies (40). As in the proof of Theorem 8, for any φ in L2T, there is a ϕ∈Kαβ# such that Bφα,β=Bϕ¯α,β, where Kαβ#=Kα⊕αKβ#. Then, there are χ∈Kα and ψ∈Kβ# such that Bφα,β=Bχ¯α,β+Bαψ¯α,β. Let β# be an interpolating Blaschke product with zeros bm¯m≥1. For the bases kamαm≥1 and k~bn¯β#n≥1, we have
(41)χ=∑m=1∞cm¯kamαandψ=∑n=1∞dn¯k~bn¯β#.
Replacing inBφα,β,
(42)Bφα,β=∑m=1∞cmBkamα¯α,β+∑n=1∞dnBαk~bn¯β#¯α,β.
By (6), we have
(43)Bkamα¯α,β=B11−am¯z¯α,β=Bamz−amα,β=amkam¯β⊗kamα,Bαk~bn¯β#¯α,β=Bαβ#¯z¯−bnα,β=k~bnβ⊗k~bn¯α.
So(44)Bφα,β=∑m=1∞cmamkam¯β⊗kamα+∑n=1∞k~bnβ⊗k~bn¯α.
Replacing in (20), we get
(45)si,j=1β′bi¯∑m=1∞cmam1−aj¯amk~biβam¯¯+∑n=1∞dn1−bi¯bnk~bn¯αaj¯.
Whenlkk is empty, we have
(46)si,j=1β′bi¯∑m=1∞cmam1−aj¯amβ#amam−bi¯+∑n=1∞dn1−bi¯bnα#bnbn−aj¯.
Whenlkk is not empty, we have
(47)k~biβam¯¯=i∈lkk:0:m∈lkk,m≠i,β′bi¯:m=i,β#amam−bi¯:m∈lkk,i∈lkk:0:m∈lkk,β#amam−bi¯:m∈lkk,(48)k~bn¯αaj¯=j∈lkk:0:n∈lkk,n≠j,α′aj¯:n=j,α#bnbn−aj¯:n∈lkk,j∈lkk:0:n∈lkk,α#bnbn−aj¯:n∈lkk.
Therefore, we have 4 cases, {j∈lkkandi∈lkk}, j∈lkkandi∈lkk, j∈lkkandi∈lkk, and j∈lkkandi∈lkk. In all these cases, we decompose, add, and subtract as in the finite-dimensional case. Using the fact that a1=b1¯, we obtain
(49)si,j=β′b1¯1−ajb1¯s1,j−β′b1¯1−a1b1¯s1,1+β′bi¯1−a1bi¯si,1β′bi¯1−ajbi¯.
Conversely, we proceed as in [9] for truncated Hankel operators but using the generalized characterization in Theorem 4.As for Theorem10, we also generalize the matrix representation of the ATHOs with respect to conjugate kernel bases.Theorem 17.
LetB be a bounded linear transformation from Kα to Kβ. We have B∈Hα,β if and only if for any i,j≥1, we have
(50)pi,j=β′b11−ajb1p1,j−β′b11−a1b1p1,1+β′bi1−a1bipi,1β′bi1−ajbi.Suppose thatα and β are inner functions such that Kα and Kβ have Clark bases vηjαj≥1 and vζiβi≥1, where ηjj and ζii are sequences of eigenvalues for some Uλ1α and some Uλ2β satisfying the equations (30). We have the relations α′ηj=kηjα and β′ζi=kζiβ. Reorder these sequences such that ηlk=ζlk¯, for any k≥1 and that 1=l1∈lkk. We have the following theorem.Theorem 18.
LetB be a bounded linear transformation from KαtoKβ. B∈Hα,β if and only if
(1)
Whenlkk≥1 is not empty, for i∈lkkandj∈lkkori∈lkkandj∈lkkandj≠i(51)ti,j=β′ζ1ηj¯−ζ1β′ζiηj¯−ζit1,j+α′ηiβ′ζ1η1¯−ζjα′ηjβ′ζiηj¯−ζit1,i.Fori∈lkkandj∈lkkori∈lkkandj∈lkk,
(52)ti,j=β′ζ1ηj¯−ζ1β′ζiηj¯−ζit1,j+α′η1η1¯−ζjα′ηjηj¯−ζiti,1.(2)
Whenlkk≥1 is empty, for any i,j≥1(53)ti,j=β′ζ1ηj¯−ζ1β′ζiηj¯−ζit1,j−α′η1β′ζ1η1¯−ζ1α′ηjβ′ζiηj¯−ζit1,1+α′η1η1¯−ζjα′ηjηj¯−ζiti,1.Proof.
The proof is similar to the one of Theorem14, 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∈Kα and ψ∈Kβ,
(54)B−Uλ2β∗BUλ1α∗=ψ⊗k0α+k~0β⊗χ.As for Theorem15, we deduce the matrix representation with respect to modified Clark bases.Theorem 19.
A bounded linear transformationB from Kα to Kβ belongs to Hα,β if and only if
(1)
Whenlkk≥1 is not empty, for i∈lkkandj∈lkkori∈lkkandj∈lkkandj≠i(55)ui,j=ω1βωiββ′ζ1ηj¯−ζ1β′ζiηj¯−ζiu1,j+ωjαω1αα′ηiβ′ζ1η1¯−ζjα′ηjβ′ζiηj¯−ζiu1,i.Fori∈lkkandj∈lkkori∈lkkandj∈lkk,
(56)ui,j=ω1βωiββ′ζ1ηj¯−ζ1β′ζiηj¯−ζiu1,j+ωjαω1αα′η1η1¯−ζjα′ηjηj¯−ζiui,1.(2)
Whenlkk≥1 is empty
(57)ui,j=ω1βωiββ′ζ1ηj¯−ζ1β′ζiηj¯−ζiu1,j−ωjαω1βω1αωiβα′η1β′ζ1η1¯−ζ1α′ηjβ′ζiηj¯−ζiu1,1+ωjαω1αα′η1η1¯−ζjα′ηjηj¯−ζiui,1.


### 5.2. Matrix Representation of ATTOs

To get the matrix representation of ATTOs on interpolating model spaces, we will work as in the proof of Theorem 2.2 in [8]. We first need to explore the action of Vξ,c on Tα,β, where Vξ,c acts from Kα into Kα∘τξ,c and is defined by Vξ,cf=τξ,c′f∘τξ,c, τξ,cz=ξc−z/1−c¯z. ([13]).Proposition 20.
Letα and β be two inner functions, φ∈L2 and Aφα,β∈Tα,β. Then for some c∈D and ξ∈T,
(58)Vξ,cAφα,βVξ,c−1=Aφ∘τξ,cα∘τξ,c,β∘τξ,c.Proof.
Forf∈Kα,φ∈L2,ξ∈T and a∈D, we have
(59)Vξ,cPβφfz=τξ,c′zPβφf∘τξ,cz=τξ,c′zφf,kτξ,czβ=τ′ξ,cz∫Tφχfχ1−βτξ,czβχ¯1−τξ,czχ¯dχ.
Letχ=τξ,cω. Since τ′ξ,cz=ξc2−1/1−c¯z2, we have
(60)Vξ,cPβφfz=τξ,c′z∫Tφτξ,cωfτξ,cω1−βτξ,czβτξ,cω¯1−τξ,czτξ,cω¯τξ,c′ωdω=τξ,c′z∫Tφτξ,cωfτξ,cω1−zω¯1−τξ,czτξ,cω¯kzβ∘τξ,cω¯τξ,c′ωdω.
Since(61)τ′ξ,czτξ,c′ω1−zω¯1−τ′ξ,czτ′ξ,cω¯=τ′ξ,cω,we have
(62)Vξ,cPβφfz=∫Tφτξ,cωfτξ,cωτ′ξ,cωkzβ∘τξ,cω¯dω=∫Tφ∘τξ,cωVξ,cfωkzβ∘τξ,cω¯ωdω=φ∘τξ,cVξ,cf,kzβ∘τξ,cω=Pβ∘τξ,cφ∘τξ,cVξ,cfz.
In conclusion,(63)Vξ,cAφα,β=Aφ∘τξ,cα∘τξ,c,β∘τξ,cVξ,c.
ApplyingVξ,c−1=Vξ,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 alkk≥1 the subsequence of common zeros between α and β ordered such that alk=blk, for any k≥1 and that 1=l1∈lkk when lkk is not empty.Theorem 21.
A bounded linear transformationA from Kα to Kβ belongs to Tα,β if and only if its matrix representation with respect to the kernel bases satisfies
(1)
Whenlkk≥1 is not empty, for i∈lkkandj∈lkkori∈lkkandj∈lkkandj≠i(64)si,j=β′b1¯aj¯−b1¯s1,j+β′b1¯a1¯−bi¯s1,iβ′bi¯aj¯−bi¯.Fori∈lkkandj∈lkkori∈lkkandj∈lkk,
(65)si,j=β′b1¯aj¯−b1¯s1,j+β′bi¯a1¯−bi¯si,1β′bi¯aj¯−bi¯.(2)
Whenlkk≥1 is empty
(66)si,j=β′b1¯aj¯−b1¯s1,j−β′b1¯a1¯−b1¯s1,1+β′bi¯a1¯−bi¯si,1β′bi¯aj¯−bi¯.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 Kα into Kβ whose matrix representation satisfies Theorem 21. To show that A∈Tα,β, or equivalently to show that A satisfies the characterization in Theorem 4, we will find a χ∈Kα and a ψ∈Kβ such that
(67)A−SβASα∗=ψ⊗k0α+k0β⊗χ.
This is equivalent to(68)Akajα,k~biβ−SβASα∗kajα,k~biβ=ψ⊗k0αkajα,k~biβ+k0β⊗χkajα,k~biβ,for any i,j≥1. Using the relations in Lemma 3, when bi≠0, for any i≥1, we have for every i,j≥1(69)1−aj¯bi¯Akajα,k~biβ+aj¯bi¯kajα,A∗k~0β=kajα0ψ,k~biβ+k~biβ0¯kajα,χ,or
(70)1−aj¯bi¯β′bi¯si,j+aj¯bi¯kajα,A∗k~0β=ψ~bi¯−β0¯bi¯χaj¯,for any i,j≥1.
Whenlkk is not empty, suppose that β0≠0. Since the matrix representation of A satisfies the formulas in Theorem 21, the above system is equivalent to
(71)1−aj¯b1¯β′b1¯s1,j+aj¯b1¯kajα,A∗k~0β=ψ~b1¯−β0¯b1¯χaj¯,j≥1;kalkα,A∗k~0β=ψ~blk¯−β0¯blk¯χalk¯,k≥1;1−a1¯bi¯β′bi¯si,1+a1¯bi¯ka1α,A∗k~0β=ψ~bi¯−β0¯bi¯χa1¯,i≥1,i∈lkk.
Set an arbitraryψ~b1, then the solution of the system is
(72)χaj¯=b1¯β0¯ψ~b1¯−1−aj¯b1¯β′b1¯s1,j−aj¯b1¯kajα,A∗k~0β,j≥1;ψ~bi¯=−kaiα,A∗k~0β−β0¯bi¯χai¯;i∈lkk.ψ~bi¯=1−a1¯bi¯β′bi¯si,1+a1¯bi¯ka1α,A∗k~0β+β0¯bi¯χa1¯.i≥1,i∈lkk.
To show that the solutions of the systemχ and ψ are in Kα and Kβ, respectively, it suffices to prove that χ and ψ are the unique solutions in Kα and Kβ of the interpolation problems corresponding to amm≥1 and bnn≥1 (39). In fact, since A∗k~0β,A∗k~b1βaj, Aka1α~∈Kβ, and ajj≥1,bii≥1 are Blaschke sequences, we have
(73)∑j=1+∞χaj21−aj2=∑j=1+∞b1¯β0¯ψ~b1¯−1−aj¯b1¯β′b1¯s1,j−aj¯b1¯kajα,A∗k~0β21−aj2≤C∑j=1+∞A∗k~b1βaj21−aj2+Cψb1~2∑j=1+∞1−aj2+C∑j=1+∞A∗k~0βaj21−aj2<∞.(74)∑i=1+∞ψ~bi21−bi2=∑i=1,i∈lkk+∞<kajα,A∗k~0β>+β0¯bi¯χai¯21−bi2+∑i=1,i∈lkk+∞1−a1¯bi¯β′bi¯si,1+a1¯bi¯<ka1α,A∗k~0β>+β0¯bi¯χa1¯21−bi2≤C∑i=1+∞A∗k~0βai21−ai2+C∑i=1+∞χai21−ai2+C∑i=1+∞Aka1α~bi1−bi2+Cχa12+A∗k0β~a12∑i=1+∞1−bi2<∞.
Ifβ0=0, βc≠0 for some c∈D. Let ξ=1, by Proposition 20, A∈Tα,β if and only if VcAVc∗∈Tα∘τc,β∘τc. Since β∘τc0=βc≠0, we need to show that the matrix representation of VcAVc∗, wi,j with respect to the kernel bases kτcamα∘τcm and kτcbnβ∘τcn satisfies the relations in Theorem 21. In fact, as in [8], we have
(75)Vc∗kτcajα∘τcz=i1−caj¯1−c2kajαz,Vc∗k~τcbiβ∘τcz=−i1−c¯bi1−c2k~biβz,(76)wi,j=1β∘τc′bi¯VcAVc∗kτcajα∘τc,k~τcbiβ∘τc=τc′bi¯β′bi¯i1−caj¯1−c2−i¯1−c¯bi¯1−c2Akajα,k~biβ=1−caj¯1−cbi¯si,j,for any i,j≥1. We also have
(77)β∘τc′τcbi=β′biτc′bi,τcz=c−z1−c¯zandτ′cz=c2−11−c¯z2, using these formulas we can prove that every wi,j satisfies first relation in Theorem 21,
(78)β∘τc′τcb1¯τcaj¯−τcb1¯w1,j+β∘τc′τcbi¯τca1¯−τcbi¯wi,1β∘τc′τcbi¯τcaj¯−τcbi¯=1−caj¯1−cbi¯si,j=wi,j.
Iflkk is empty, then the equation (70) becomes with the assumption β0≠0,
(79)1−aj¯b1¯β′b1¯s1,j+aj¯b1¯kajα,A∗k~0β=ψ~b1¯−β0¯b1¯χaj¯,j≥1;1−a1¯bi¯β′bi¯si,1+a1¯bi¯ka1α,A∗k~0β=ψ~bi¯−β0¯bi¯χa1¯,i≥2.
If we set an arbitraryψ~b1, then, the solutions are
(80)χaj¯=b1¯β0¯ψ~b1¯−1−aj¯b1¯β′b1¯s1,j−aj¯b1¯kajα,A∗k~0β,j≥1;ψ~bi¯=1−a1¯bi¯β′bi¯si,1+a1¯bi¯ka1α,A∗k~0β+β0¯bi¯χa1¯,i≥2.
It remains to check thatχ∈Kα and ψ∈Kβ by showing that they are solutions of the corresponding interpolation problems. The case β0=0 can be treated in the same way as in the previous case.To get the matrix representation of ATTOs with respect to conjugate kernel basesk~amαm and k~bnβn, we proceed as in Theorem 13 and use the fact that CβACα=Aα¯βφ¯α,β from the Theorem 5.Theorem 22.
A bounded linear transformation fromKα to Kβ belongs to Tα,β if and only if
(1)
Whenlkk≥1 is not empty, for i∈lkkandj∈lkkori∈lkkandj∈lkkandj≠i(81)pi,j=β′b1aj−b1p1,j+β′b1a1−bip1,iβ′biaj−bi.Fori∈lkkandj∈lkkori∈lkkandj∈lkk,
(82)pi,j=β′b1aj−b1p1,j+β′bia1−bipi,1β′biaj−bi.(2)
Whenlkk≥1 is empty(83)pi,j=β′b1aj−b1p1,j−β′b1a1−b1p1,1+β′bia1−bipi,1β′biaj−bi.
Suppose thatα and β are inner functions such that the spaces Kα and Kβ have Clark bases, vηjαj and vζiβi. Denote the subsequence of common elements of ηjj and ζii by ηlkk=ζlkk ordered such that ηlk=ζlk, for any k≥1 and that 1=l1∈lkk.Theorem 23.
A bounded linear transformation belongs toTα,β if and only if
(1)
Whenlkk≥1 is not empty, for i∈lkkandj∈lkkori∈lkkandj∈lkkandj≠i(84)ti,j=β′ζ11−ζ1ηj¯β′ζi1−ζiηj¯t1,j+α′ηiβ′ζ1ζ11−ζiη1¯α′ηjβ′ζiηi1−ζiηj¯t1,i.Fori∈lkkandj∈lkkori∈lkkandj∈lkk,
(85)ti,j=β′ζ11−ζ1ηj¯β′ζi1−ζiηj¯t1,j+α′η11−ζiη1¯α′ηj1−ζiηj¯ti,1.(2)
Whenlkk≥1 is empty(86)ti,j=β′ζ11−ζ1ηj¯β′ζi1−ζiηj¯t1,j−α′η1β′ζ11−ζ1η1¯α′ηjβ′ζi1−ζiηj¯t1,1+α′η11−ζiη1¯α′ηj1−ζiηj¯ti,1.Proof.
The proof is the same as in Theorem21, except using the equivalent characterization from Theorem 4 instead.We also deduce the matrix representation with respect to the modified Clark bases.Theorem 24.
A bounded linear transformation belongs toTα,β if and only if
(1)
Whenlkk≥1 is not empty, for i∈lkkandj∈lkkori∈lkkandj∈lkkandj≠i(87)ui,j=ω1βωiββ′ζ11−ζ1ηj¯β′ζi1−ζiηj¯u1,j+ωjαω1αα′ηiβ′ζ1ζ11−ζiη1¯α′ηjβ′ζiηi1−ζiηj¯u1,i.Fori∈lkkandj∈lkkori∈lkkandj∈lkk,
(88)ui,j=ω1βωiβ∣β′ζ1∣1−ζ1ηj¯∣β′ζi∣1−ζiηj¯u1,j+ωjαω1α∣α′η1∣1−ζiη1¯∣α′ηj∣1−ζiηj¯ui,1.(2)
Whenlkk≥1 is empty(89)ui,j=ω1βωiββ′ζ11−ζ1ηj¯β′ζi1−ζiηj¯u1,j−ωjαωiβω1βω1αα′η1β′ζ11−ζ1η1¯α′ηjβ′ζi1−ζiηj¯u1,1+ωjαω1αα′η11−ζiη1¯α′ηj1−ζiηj¯ui,1.Remark 25.
Note that sinceJ#kajα=kaj¯α# and J#k~ajα=k~aj¯α#, we have
(90)si,j=1β′bi¯Bkajα,k~biβ=1β′bi¯CβBJ#kaj¯α#,kbiβ¯=1β′bi¯J#BCαk~ajα,k~bi¯β#,and since CβBJ# and J#BCα 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.


## 5.1. Matrix Representation of ATHOs

Denote byalkk≥1=b¯lkk≥1 the subsequence of common elements between amm and b¯nn ordered such that alk=blk¯ and that 1=l1∈lkk≥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.
LetB be a bounded linear transformation from Kα to Kβ. Then, B∈Hα,β if and only if for any i,j(40)si,j=β′b1¯1−ajb1¯s1,j−β′b1¯1−a1b1¯s1,1+β′bi¯1−a1bi¯si,1β′bi¯1−ajbi¯.Proof.
Suppose thatB=Bφα,β and we will prove that it satisfies (40). As in the proof of Theorem 8, for any φ in L2T, there is a ϕ∈Kαβ# such that Bφα,β=Bϕ¯α,β, where Kαβ#=Kα⊕αKβ#. Then, there are χ∈Kα and ψ∈Kβ# such that Bφα,β=Bχ¯α,β+Bαψ¯α,β. Let β# be an interpolating Blaschke product with zeros bm¯m≥1. For the bases kamαm≥1 and k~bn¯β#n≥1, we have
(41)χ=∑m=1∞cm¯kamαandψ=∑n=1∞dn¯k~bn¯β#.
Replacing inBφα,β,
(42)Bφα,β=∑m=1∞cmBkamα¯α,β+∑n=1∞dnBαk~bn¯β#¯α,β.
By (6), we have
(43)Bkamα¯α,β=B11−am¯z¯α,β=Bamz−amα,β=amkam¯β⊗kamα,Bαk~bn¯β#¯α,β=Bαβ#¯z¯−bnα,β=k~bnβ⊗k~bn¯α.
So(44)Bφα,β=∑m=1∞cmamkam¯β⊗kamα+∑n=1∞k~bnβ⊗k~bn¯α.
Replacing in (20), we get
(45)si,j=1β′bi¯∑m=1∞cmam1−aj¯amk~biβam¯¯+∑n=1∞dn1−bi¯bnk~bn¯αaj¯.
Whenlkk is empty, we have
(46)si,j=1β′bi¯∑m=1∞cmam1−aj¯amβ#amam−bi¯+∑n=1∞dn1−bi¯bnα#bnbn−aj¯.
Whenlkk is not empty, we have
(47)k~biβam¯¯=i∈lkk:0:m∈lkk,m≠i,β′bi¯:m=i,β#amam−bi¯:m∈lkk,i∈lkk:0:m∈lkk,β#amam−bi¯:m∈lkk,(48)k~bn¯αaj¯=j∈lkk:0:n∈lkk,n≠j,α′aj¯:n=j,α#bnbn−aj¯:n∈lkk,j∈lkk:0:n∈lkk,α#bnbn−aj¯:n∈lkk.
Therefore, we have 4 cases, {j∈lkkandi∈lkk}, j∈lkkandi∈lkk, j∈lkkandi∈lkk, and j∈lkkandi∈lkk. In all these cases, we decompose, add, and subtract as in the finite-dimensional case. Using the fact that a1=b1¯, we obtain
(49)si,j=β′b1¯1−ajb1¯s1,j−β′b1¯1−a1b1¯s1,1+β′bi¯1−a1bi¯si,1β′bi¯1−ajbi¯.
Conversely, we proceed as in [9] for truncated Hankel operators but using the generalized characterization in Theorem 4.As for Theorem10, we also generalize the matrix representation of the ATHOs with respect to conjugate kernel bases.Theorem 17.
LetB be a bounded linear transformation from Kα to Kβ. We have B∈Hα,β if and only if for any i,j≥1, we have
(50)pi,j=β′b11−ajb1p1,j−β′b11−a1b1p1,1+β′bi1−a1bipi,1β′bi1−ajbi.Suppose thatα and β are inner functions such that Kα and Kβ have Clark bases vηjαj≥1 and vζiβi≥1, where ηjj and ζii are sequences of eigenvalues for some Uλ1α and some Uλ2β satisfying the equations (30). We have the relations α′ηj=kηjα and β′ζi=kζiβ. Reorder these sequences such that ηlk=ζlk¯, for any k≥1 and that 1=l1∈lkk. We have the following theorem.Theorem 18.
LetB be a bounded linear transformation from KαtoKβ. B∈Hα,β if and only if
(1)
Whenlkk≥1 is not empty, for i∈lkkandj∈lkkori∈lkkandj∈lkkandj≠i(51)ti,j=β′ζ1ηj¯−ζ1β′ζiηj¯−ζit1,j+α′ηiβ′ζ1η1¯−ζjα′ηjβ′ζiηj¯−ζit1,i.Fori∈lkkandj∈lkkori∈lkkandj∈lkk,
(52)ti,j=β′ζ1ηj¯−ζ1β′ζiηj¯−ζit1,j+α′η1η1¯−ζjα′ηjηj¯−ζiti,1.(2)
Whenlkk≥1 is empty, for any i,j≥1(53)ti,j=β′ζ1ηj¯−ζ1β′ζiηj¯−ζit1,j−α′η1β′ζ1η1¯−ζ1α′ηjβ′ζiηj¯−ζit1,1+α′η1η1¯−ζjα′ηjηj¯−ζiti,1.Proof.
The proof is similar to the one of Theorem14, 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∈Kα and ψ∈Kβ,
(54)B−Uλ2β∗BUλ1α∗=ψ⊗k0α+k~0β⊗χ.As for Theorem15, we deduce the matrix representation with respect to modified Clark bases.Theorem 19.
A bounded linear transformationB from Kα to Kβ belongs to Hα,β if and only if
(1)
Whenlkk≥1 is not empty, for i∈lkkandj∈lkkori∈lkkandj∈lkkandj≠i(55)ui,j=ω1βωiββ′ζ1ηj¯−ζ1β′ζiηj¯−ζiu1,j+ωjαω1αα′ηiβ′ζ1η1¯−ζjα′ηjβ′ζiηj¯−ζiu1,i.Fori∈lkkandj∈lkkori∈lkkandj∈lkk,
(56)ui,j=ω1βωiββ′ζ1ηj¯−ζ1β′ζiηj¯−ζiu1,j+ωjαω1αα′η1η1¯−ζjα′ηjηj¯−ζiui,1.(2)
Whenlkk≥1 is empty
(57)ui,j=ω1βωiββ′ζ1ηj¯−ζ1β′ζiηj¯−ζiu1,j−ωjαω1βω1αωiβα′η1β′ζ1η1¯−ζ1α′ηjβ′ζiηj¯−ζiu1,1+ωjαω1αα′η1η1¯−ζjα′ηjηj¯−ζiui,1.


## 5.2. Matrix Representation of ATTOs

To get the matrix representation of ATTOs on interpolating model spaces, we will work as in the proof of Theorem 2.2 in [8]. We first need to explore the action of Vξ,c on Tα,β, where Vξ,c acts from Kα into Kα∘τξ,c and is defined by Vξ,cf=τξ,c′f∘τξ,c, τξ,cz=ξc−z/1−c¯z. ([13]).Proposition 20.
Letα and β be two inner functions, φ∈L2 and Aφα,β∈Tα,β. Then for some c∈D and ξ∈T,
(58)Vξ,cAφα,βVξ,c−1=Aφ∘τξ,cα∘τξ,c,β∘τξ,c.Proof.
Forf∈Kα,φ∈L2,ξ∈T and a∈D, we have
(59)Vξ,cPβφfz=τξ,c′zPβφf∘τξ,cz=τξ,c′zφf,kτξ,czβ=τ′ξ,cz∫Tφχfχ1−βτξ,czβχ¯1−τξ,czχ¯dχ.
Letχ=τξ,cω. Since τ′ξ,cz=ξc2−1/1−c¯z2, we have
(60)Vξ,cPβφfz=τξ,c′z∫Tφτξ,cωfτξ,cω1−βτξ,czβτξ,cω¯1−τξ,czτξ,cω¯τξ,c′ωdω=τξ,c′z∫Tφτξ,cωfτξ,cω1−zω¯1−τξ,czτξ,cω¯kzβ∘τξ,cω¯τξ,c′ωdω.
Since(61)τ′ξ,czτξ,c′ω1−zω¯1−τ′ξ,czτ′ξ,cω¯=τ′ξ,cω,we have
(62)Vξ,cPβφfz=∫Tφτξ,cωfτξ,cωτ′ξ,cωkzβ∘τξ,cω¯dω=∫Tφ∘τξ,cωVξ,cfωkzβ∘τξ,cω¯ωdω=φ∘τξ,cVξ,cf,kzβ∘τξ,cω=Pβ∘τξ,cφ∘τξ,cVξ,cfz.
In conclusion,(63)Vξ,cAφα,β=Aφ∘τξ,cα∘τξ,c,β∘τξ,cVξ,c.
ApplyingVξ,c−1=Vξ,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 alkk≥1 the subsequence of common zeros between α and β ordered such that alk=blk, for any k≥1 and that 1=l1∈lkk when lkk is not empty.Theorem 21.
A bounded linear transformationA from Kα to Kβ belongs to Tα,β if and only if its matrix representation with respect to the kernel bases satisfies
(1)
Whenlkk≥1 is not empty, for i∈lkkandj∈lkkori∈lkkandj∈lkkandj≠i(64)si,j=β′b1¯aj¯−b1¯s1,j+β′b1¯a1¯−bi¯s1,iβ′bi¯aj¯−bi¯.Fori∈lkkandj∈lkkori∈lkkandj∈lkk,
(65)si,j=β′b1¯aj¯−b1¯s1,j+β′bi¯a1¯−bi¯si,1β′bi¯aj¯−bi¯.(2)
Whenlkk≥1 is empty
(66)si,j=β′b1¯aj¯−b1¯s1,j−β′b1¯a1¯−b1¯s1,1+β′bi¯a1¯−bi¯si,1β′bi¯aj¯−bi¯.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 Kα into Kβ whose matrix representation satisfies Theorem 21. To show that A∈Tα,β, or equivalently to show that A satisfies the characterization in Theorem 4, we will find a χ∈Kα and a ψ∈Kβ such that
(67)A−SβASα∗=ψ⊗k0α+k0β⊗χ.
This is equivalent to(68)Akajα,k~biβ−SβASα∗kajα,k~biβ=ψ⊗k0αkajα,k~biβ+k0β⊗χkajα,k~biβ,for any i,j≥1. Using the relations in Lemma 3, when bi≠0, for any i≥1, we have for every i,j≥1(69)1−aj¯bi¯Akajα,k~biβ+aj¯bi¯kajα,A∗k~0β=kajα0ψ,k~biβ+k~biβ0¯kajα,χ,or
(70)1−aj¯bi¯β′bi¯si,j+aj¯bi¯kajα,A∗k~0β=ψ~bi¯−β0¯bi¯χaj¯,for any i,j≥1.
Whenlkk is not empty, suppose that β0≠0. Since the matrix representation of A satisfies the formulas in Theorem 21, the above system is equivalent to
(71)1−aj¯b1¯β′b1¯s1,j+aj¯b1¯kajα,A∗k~0β=ψ~b1¯−β0¯b1¯χaj¯,j≥1;kalkα,A∗k~0β=ψ~blk¯−β0¯blk¯χalk¯,k≥1;1−a1¯bi¯β′bi¯si,1+a1¯bi¯ka1α,A∗k~0β=ψ~bi¯−β0¯bi¯χa1¯,i≥1,i∈lkk.
Set an arbitraryψ~b1, then the solution of the system is
(72)χaj¯=b1¯β0¯ψ~b1¯−1−aj¯b1¯β′b1¯s1,j−aj¯b1¯kajα,A∗k~0β,j≥1;ψ~bi¯=−kaiα,A∗k~0β−β0¯bi¯χai¯;i∈lkk.ψ~bi¯=1−a1¯bi¯β′bi¯si,1+a1¯bi¯ka1α,A∗k~0β+β0¯bi¯χa1¯.i≥1,i∈lkk.
To show that the solutions of the systemχ and ψ are in Kα and Kβ, respectively, it suffices to prove that χ and ψ are the unique solutions in Kα and Kβ of the interpolation problems corresponding to amm≥1 and bnn≥1 (39). In fact, since A∗k~0β,A∗k~b1βaj, Aka1α~∈Kβ, and ajj≥1,bii≥1 are Blaschke sequences, we have
(73)∑j=1+∞χaj21−aj2=∑j=1+∞b1¯β0¯ψ~b1¯−1−aj¯b1¯β′b1¯s1,j−aj¯b1¯kajα,A∗k~0β21−aj2≤C∑j=1+∞A∗k~b1βaj21−aj2+Cψb1~2∑j=1+∞1−aj2+C∑j=1+∞A∗k~0βaj21−aj2<∞.(74)∑i=1+∞ψ~bi21−bi2=∑i=1,i∈lkk+∞<kajα,A∗k~0β>+β0¯bi¯χai¯21−bi2+∑i=1,i∈lkk+∞1−a1¯bi¯β′bi¯si,1+a1¯bi¯<ka1α,A∗k~0β>+β0¯bi¯χa1¯21−bi2≤C∑i=1+∞A∗k~0βai21−ai2+C∑i=1+∞χai21−ai2+C∑i=1+∞Aka1α~bi1−bi2+Cχa12+A∗k0β~a12∑i=1+∞1−bi2<∞.
Ifβ0=0, βc≠0 for some c∈D. Let ξ=1, by Proposition 20, A∈Tα,β if and only if VcAVc∗∈Tα∘τc,β∘τc. Since β∘τc0=βc≠0, we need to show that the matrix representation of VcAVc∗, wi,j with respect to the kernel bases kτcamα∘τcm and kτcbnβ∘τcn satisfies the relations in Theorem 21. In fact, as in [8], we have
(75)Vc∗kτcajα∘τcz=i1−caj¯1−c2kajαz,Vc∗k~τcbiβ∘τcz=−i1−c¯bi1−c2k~biβz,(76)wi,j=1β∘τc′bi¯VcAVc∗kτcajα∘τc,k~τcbiβ∘τc=τc′bi¯β′bi¯i1−caj¯1−c2−i¯1−c¯bi¯1−c2Akajα,k~biβ=1−caj¯1−cbi¯si,j,for any i,j≥1. We also have
(77)β∘τc′τcbi=β′biτc′bi,τcz=c−z1−c¯zandτ′cz=c2−11−c¯z2, using these formulas we can prove that every wi,j satisfies first relation in Theorem 21,
(78)β∘τc′τcb1¯τcaj¯−τcb1¯w1,j+β∘τc′τcbi¯τca1¯−τcbi¯wi,1β∘τc′τcbi¯τcaj¯−τcbi¯=1−caj¯1−cbi¯si,j=wi,j.
Iflkk is empty, then the equation (70) becomes with the assumption β0≠0,
(79)1−aj¯b1¯β′b1¯s1,j+aj¯b1¯kajα,A∗k~0β=ψ~b1¯−β0¯b1¯χaj¯,j≥1;1−a1¯bi¯β′bi¯si,1+a1¯bi¯ka1α,A∗k~0β=ψ~bi¯−β0¯bi¯χa1¯,i≥2.
If we set an arbitraryψ~b1, then, the solutions are
(80)χaj¯=b1¯β0¯ψ~b1¯−1−aj¯b1¯β′b1¯s1,j−aj¯b1¯kajα,A∗k~0β,j≥1;ψ~bi¯=1−a1¯bi¯β′bi¯si,1+a1¯bi¯ka1α,A∗k~0β+β0¯bi¯χa1¯,i≥2.
It remains to check thatχ∈Kα and ψ∈Kβ by showing that they are solutions of the corresponding interpolation problems. The case β0=0 can be treated in the same way as in the previous case.To get the matrix representation of ATTOs with respect to conjugate kernel basesk~amαm and k~bnβn, we proceed as in Theorem 13 and use the fact that CβACα=Aα¯βφ¯α,β from the Theorem 5.Theorem 22.
A bounded linear transformation fromKα to Kβ belongs to Tα,β if and only if
(1)
Whenlkk≥1 is not empty, for i∈lkkandj∈lkkori∈lkkandj∈lkkandj≠i(81)pi,j=β′b1aj−b1p1,j+β′b1a1−bip1,iβ′biaj−bi.Fori∈lkkandj∈lkkori∈lkkandj∈lkk,
(82)pi,j=β′b1aj−b1p1,j+β′bia1−bipi,1β′biaj−bi.(2)
Whenlkk≥1 is empty(83)pi,j=β′b1aj−b1p1,j−β′b1a1−b1p1,1+β′bia1−bipi,1β′biaj−bi.
Suppose thatα and β are inner functions such that the spaces Kα and Kβ have Clark bases, vηjαj and vζiβi. Denote the subsequence of common elements of ηjj and ζii by ηlkk=ζlkk ordered such that ηlk=ζlk, for any k≥1 and that 1=l1∈lkk.Theorem 23.
A bounded linear transformation belongs toTα,β if and only if
(1)
Whenlkk≥1 is not empty, for i∈lkkandj∈lkkori∈lkkandj∈lkkandj≠i(84)ti,j=β′ζ11−ζ1ηj¯β′ζi1−ζiηj¯t1,j+α′ηiβ′ζ1ζ11−ζiη1¯α′ηjβ′ζiηi1−ζiηj¯t1,i.Fori∈lkkandj∈lkkori∈lkkandj∈lkk,
(85)ti,j=β′ζ11−ζ1ηj¯β′ζi1−ζiηj¯t1,j+α′η11−ζiη1¯α′ηj1−ζiηj¯ti,1.(2)
Whenlkk≥1 is empty(86)ti,j=β′ζ11−ζ1ηj¯β′ζi1−ζiηj¯t1,j−α′η1β′ζ11−ζ1η1¯α′ηjβ′ζi1−ζiηj¯t1,1+α′η11−ζiη1¯α′ηj1−ζiηj¯ti,1.Proof.
The proof is the same as in Theorem21, except using the equivalent characterization from Theorem 4 instead.We also deduce the matrix representation with respect to the modified Clark bases.Theorem 24.
A bounded linear transformation belongs toTα,β if and only if
(1)
Whenlkk≥1 is not empty, for i∈lkkandj∈lkkori∈lkkandj∈lkkandj≠i(87)ui,j=ω1βωiββ′ζ11−ζ1ηj¯β′ζi1−ζiηj¯u1,j+ωjαω1αα′ηiβ′ζ1ζ11−ζiη1¯α′ηjβ′ζiηi1−ζiηj¯u1,i.Fori∈lkkandj∈lkkori∈lkkandj∈lkk,
(88)ui,j=ω1βωiβ∣β′ζ1∣1−ζ1ηj¯∣β′ζi∣1−ζiηj¯u1,j+ωjαω1α∣α′η1∣1−ζiη1¯∣α′ηj∣1−ζiηj¯ui,1.(2)
Whenlkk≥1 is empty(89)ui,j=ω1βωiββ′ζ11−ζ1ηj¯β′ζi1−ζiηj¯u1,j−ωjαωiβω1βω1αα′η1β′ζ11−ζ1η1¯α′ηjβ′ζi1−ζiηj¯u1,1+ωjαω1αα′η11−ζiη1¯α′ηj1−ζiηj¯ui,1.Remark 25.
Note that sinceJ#kajα=kaj¯α# and J#k~ajα=k~aj¯α#, we have
(90)si,j=1β′bi¯Bkajα,k~biβ=1β′bi¯CβBJ#kaj¯α#,kbiβ¯=1β′bi¯J#BCαk~ajα,k~bi¯β#,and since CβBJ# and J#BCα 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.



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*Source: 4666376-2021-09-06.xml*