Matrix Representations of Asymmetric Truncated Toeplitz Operators

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Introduction
Let H 2 be the classical Hardy space. The space H 2 consists of all functions f (z) = ∞ k=0 a k z k analytic in the unit disk D = {z ∈ C : |z| < 1} and such that ∞ k=0 |a k | 2 < ∞. It can also be identified (via radial limits) with the closed linear span of analytic polynomials in L 2 = L 2 (T), T = ∂D. Let P be the orthogonal projection from L 2 onto H 2 .
For ϕ ∈ L 2 , the Toeplitz operator T ϕ is defined on the set of all bounded analytic functions H ∞ ⊂ H 2 by Since H ∞ is a dense subset of the Hardy space, the operator T ϕ is densely defined. Moreover, it can be extended to a bounded linear operator T ϕ : H 2 → H 2 if and only if ϕ ∈ L ∞ = L ∞ (T). Toeplitz operators have many applications and are wellstudied (for more details and properties see for example [2,19]). Two examples of these operators are the unilateral shift S = T z and the backward shift S * = T z .
The Toeplitz operator T ϕ can be seen as a compression to H 2 of the multiplication operator f → ϕ f defined on L 2 . Recently, compressions of multiplication operators to model spaces have been intensely studied. A model space is a closed subspace of H 2 of the form K α = H 2 α H 2 , where α is an inner function (α ∈ H ∞ and |α| = 1 a.e. on T = ∂D). Model spaces are the typical S * -invariant subspaces of H 2 . For each w ∈ D the point evaluation functional f → f (w) is bounded on K α and so there exists a kernel function k α w ∈ K α with the reproducing property f (w) = f , k α w for every f ∈ K α . We have k α w (z) = Truncated Toeplitz operators were introduced in 2007 in D. Sarason's paper [20]. Extensive study of this class of operators revealed many interesting properties and applications of truncated Toeplitz operators. For example, a truncated Toeplitz operator is not uniquely determined by its symbol; it is C α -symmetric [20]; the fact that it can be boundedly extended to K α does not imply that it has a symbol from L ∞ [1]. We refer the reader to the survey [10] for more results and references (see also [9]). A natural generalization of truncated Toeplitz operators, asymmetric truncated Toeplitz operators, were introduced more recently in [3,4] and [14].
Among the results on truncated Toeplitz operators are characterizations given by J.A. Cima, W.T. Ross and W.R. Wogen in [8]. The authors in [8] described truncated Toeplitz operators on finite-dimensional model spaces in terms of matrix representations with respect to some natural bases.
It is known that dimK α = m < ∞ if and only if α is a finite Blaschke product with m (not necessarily distinct) zeros a 1 , . . . , a m ∈ D [11,Chapter 5]. In that case, each f ∈ K α can be written as where q is a polynomial of degree at most m − 1 (and so f is analytic in an open set containing the closure of D and here K α = K ∞ α ). In particular, if α(z) = z m , then K α is the set of all polynomials of degree at most m − 1.
It is also known that if dimK α = m < ∞, then the set of kernel functions {k α w 1 , . . . , k α w m } corresponding to any distinct points w 1 , . . . , w m ∈ D is linearly independent (for proof see [20, proof of Thm. 7.1(b)] and thus a (non-orthogonal) basis for K α . Since C α is an antilinear isometry, the set of conjugate kernel functions { k α w 1 , . . . , k α w m } is also a basis for K α . In particular, if the zeros a 1 , . . . , a m are distinct, we have the kernel function basis K α m = {k α a 1 , . . . , k α a m } and the conjugate kernel function basis K α m = { k α a 1 , . . . , k α a m }. Note that in that case, for each j ∈ {1, . . . , m}, If dimK α = m < ∞, then α, as a finite Blaschke product, is analytic in an open set containing the closure of D. For any η ∈ T one can consider k α η defined by (1.1) with w replaced by η. It turns out that this k α η also belongs to K α and f (η) = f , k α η for every f ∈ K α . Moreover, [11,Chapter 7] for details). Now for a fixed λ ∈ T define Then, α λ ∈ T and equation α(η) = α λ has precisely m distinct solutions η 1 , . . . , η m , each from T ( [12, p. 6]). The corresponding kernel functions {k α η 1 , . . . , k α η m } are pairwise orthogonal: Hence, the set of normalized kernels is an orthonormal basis for K α . The functions v α η j can also be obtained as eigenvectors of the Clark operator U α λ -the one-dimensional unitary perturbation of the compressed shift S α = A α z , given by (see [7]). That is why the basis V α m = {v α η 1 , . . . , v α η m } is called the Clark basis (see also [11,Chapter 11]).
It is well known that a bounded linear operator T : H 2 → H 2 is a Toeplitz operator if and only if its matrix representation with respect to the standard basis is a Toeplitz matrix, that is, an infinite matrix with constant diagonals. It follows that truncated Toeplitz operators on K α , α(z) = z m , can be described as those linear operators on K α which are represented by finite Toeplitz matrices (with respect to monomial basis). In [8], the authors characterized the operators from T (α) (for α such that dimK α = m < ∞) using matrix representations of these operators with respect to the kernel basis K α m , with respect to the conjugate kernel basis K α m and with respect to the Clark basis V α m . In each of those cases the matrix representing a truncated Toeplitz operator turns out to be determined by entries from its first (or any other fixed) row and the main diagonal. For some infinite-dimensional cases, these results were generalized in [17].
In [16], the authors generalized the characterizations from [8] to the case of asymmetric truncated Toeplitz operators acting between finite-dimensional model spaces. For example, it was proved in [16] that if dimK α = m < ∞, dimK β = n < ∞ and the Blaschke products α, β have l common zeros, then a matrix representing an operator A ∈ T (α, β) (with respect to K α m and K β n ) is determined by its entries along the first row, first l entries along the main diagonal and last n − l entries along the first column (first row and first column if l = 0). In this paper, we continue the study of matrix representations of operators from T (α, β) (α, β -finite Blaschke products) and present some new characterizations. The novelty of our approach is that here we consider matrix representations which are computed with respect to two different types of bases (for example the kernel basis in K α and the conjugate kernel basis in K β ). The characterizations thus obtained turn out to be simpler than the ones from [16] as they do not depend on whether or not α and β have common zeros. Moreover, these results are new in that they were not known even for the case of truncated Toeplitz operators (α = β).
In Sect. 2, we assume that α and β are two finite Blaschke products, each with distinct zeros. We then characterize operators from T (α, β) using matrix representations computed with respect to the kernel basis in K α and the conjugate kernel basis in K β (and vice versa). In Sect. 3, we consider matrix representations computed with respect to the kernel (or conjugate kernel) basis in one space (assuming that the zeros of the corresponding finite Blaschke product are distinct) and Clark basis in the other (with no additional assumptions on the zeros of the corresponding finite Blaschke product).

Kernel Basis and Conjugate Kernel Basis
In what follows let α and β be two finite Blaschke products with zeros a 1 , . . . , a m and b 1 , . . . , b n , respectively. In this section we assume that the zeros a 1 , . . . , a m are distinct and that the zeros b 1 , . . . , b n are also distinct.
We first consider matrix representations with respect to Our reasoning is based on a duality relation between the kernels {k In other words, one can say that the set r j, p k β b j , the above equality follows from (2.1).
As in the proof of [16, Thm. 3.1], for A ∈ T (α, β) we will write Ak α a p as a linear combination of k

. . , b n and let A be any linear transformation from K
for each 1 ≤ p ≤ m. Therefore, for each 1 ≤ p ≤ m and 1 ≤ s ≤ n we have It is now easy to verify that (2. can be replaced by for all 1 ≤ p ≤ m and 1 ≤ s ≤ n.
can be obtained from M A using the following system of m + n − 1 equations: Using Theorem 2.1 we can now describe operators from T (α, β) in terms of their matrix representations with respect to K α If A is a linear transformation from K α into K β , then its matrix representation M A = (t s, p ) with respect to K α m and K β n is given by for all 1 ≤ p ≤ m and 1 ≤ s ≤ n (as before, this follows from (2.1)).

. . , b n and let A be any linear transformation from K α into K β . If M A = (t s, p ) is the matrix representation of A with respect to the bases
for all 1 ≤ p ≤ m and 1 ≤ s ≤ n.
Proof As above, let M A = (t s, p ) be the matrix representation of A with respect to K α m and K

Remark 2.4 (a) In this case, a matrix representing an operator from T (α, β) is also
completely determined by its entries along the first row and the first column. Again, the first row and column can be replaced by any other row and column: for any fixed p 0 ∈ {1, . . . , m} and s 0 ∈ {1, . . . , n}, condition (2.4) can be replaced by for all 1 ≤ p ≤ m and 1 ≤ s ≤ n.

Remark 2.5
In [18] the authors consider matrix representations of the so-called truncated Hankel operators. These operators may be defined in several ways. The authors of [18] follow the definition proposed by C. Gu [13]: the truncated Hankel operator B α ϕ , ϕ ∈ L 2 , is defined on a dense subset of K α by where J : It is shown in [18,Thm. 3.1] that, for a finite Blaschke product α with distinct zeros a 1 , . . . , a n , a linear transformation B on K α is a truncated Hankel operator if and only if its matrix representation M A = (r s, p ) with respect to the kernel basis K α n = {k α a 1 , . . . , k α a n } satisfies for all 1 ≤ s, p ≤ n.

Recall from [14] that B is a truncated Hankel operator if and only if
Note that if α is a finite Blaschke product with distinct zeros a 1 , . . . , a n , then β = α # is a finite Blaschke product with distinct zeros b 1 , . . . , b n , where b s = a s , s = 1, . . . , n. So, by Theorem 2.3, A ∈ T (α, β) if and only if its matrix representation M A = (t s, p ) with respect to K α n and K β n satisfies for all 1 ≤ p, s ≤ n. Since here β (b s ) = β (a s ) = α (a s ) and (see [6]), we get

Kernel Basis and Clark Basis, Conjugate Kernel Basis and Clark Basis
Let α and β be two finite Blaschke products with (not necessarily distinct) zeros a 1 , . . . , a m and b 1 , . . . , b n , respectively. For a fixed λ 1 ∈ T let η 1 , . . . , η m ∈ T be the distinct solutions of the equation . . , m, be the corresponding Clark basis. Similarly, for a fixed λ 2 ∈ T let ζ 1 , . . . , ζ n ∈ T be the solutions of

Vm and Kň , Vm and Kň
We first assume that the zeros b 1 , . . . , b n are distinct and we consider matrix representations with respect to V α m and K for all 1 ≤ p ≤ m and 1 ≤ s ≤ n.
Proof Assume that A ∈ T (α, β) and let M A = (r s, p ) be its matrix representation with respect to V α m and K  T (α, β). Hence, for some c 1 , . . . , c m+n−1 ∈ C. From this and (3.2) it follows that Indeed, for 1 ≤ p ≤ m and 1 ≤ s ≤ n we have The reminder of the proof is analogous to that of Theorem 2.1 (use the dimension argument).
To describe the operators from T (α, β) in terms of their matrix representations with respect to V α m and K T (α, β).
Here, the matrix representation M A = (t s, p ) of a linear transformation A : K α → K β with respect to the bases V α m and K β n is given by we get that for all 1 ≤ p ≤ m and 1 ≤ s ≤ n.
Proof Let M A = (t s, p ) be the matrix representation of A with respect to V α m and K

Remark 3.3 (a)
Here, a matrix representing an asymmetric truncated Toeplitz operator is also determined by its entries along the first row and the first column. Again, the first row and column can be replaced by any other row and column.
(b) To determine a symbol for A from M A = (r s, p ) note that by the proof of Theorem 3.1, To

Km and Vň , Km and Vň
We now assume that the zeros a 1 , . . . , a m are distinct and we first consider matrix representations with respect to K α m and V Observe that here, for 1 ≤ p ≤ m and 1 ≤ s ≤ n, where M A * = (r * p,s ) is the matrix representation of the adjoint A * : K β → K α with respect to V   [14]. To find a symbol for A ∈ T (α, β) it is therefore enough to find a symbol for A * ∈ T (β, α). Now if the matrix representation M A = (r s, p ) with respect to K α m and V β n satisfies (3.8), then a symbol for A * can be obtained from M A * = (r * p,s ) = 1 α (a p ) r s, p as in Remark 3.3. If the matrix representation M A = (t s, p ) with respect to K α m and V β n satisfies (3.9), then a symbol for A * can be obtained from M A * = (t * p,s ) = 1 α (a p ) t s, p .
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