Matrix Representations of Asymmetric Truncated Toeplitz Operators
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Matrix Representations of Asymmetric Truncated Toeplitz Operators Joanna Jurasik1 · Bartosz Łanucha1 Received: 16 May 2020 / Revised: 14 August 2020 / Accepted: 17 August 2020 © The Author(s) 2020
Abstract In this paper, we describe the matrix representations of asymmetric truncated Toeplitz operators acting between two finite-dimensional model spaces K 1 and K 2 . The novelty of our approach is that here we consider matrix representations computed with respect to bases of different type in K 1 and K 2 (for example, kernel basis in K 1 and conjugate kernel basis in K 2 ). We thus obtain new matrix characterizations which are simpler than the ones already known for asymmetric truncated Toeplitz operators. Keywords Model space · Truncated Toeplitz operator · Asymmetric truncated Toeplitz operator · Matrix representation Mathematics Subject Classification 47B32 · 47B35 · 30H10
1 Introduction Let H 2 be the classical Hardy space. The space H 2 consists of all functions k f (z) = ∞ k=0 ak z analytic in the unit disk D = {z ∈ C : |z| < 1} and such that ∞ 2 k=0 |ak | < ∞. 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 Tϕ is defined on the set of all bounded analytic functions H ∞ ⊂ H 2 by
Communicated by Mohammad Sal Moslehian.
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Bartosz Łanucha [email protected] Joanna Jurasik [email protected]
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Department of Mathematics, Maria Curie-Skłodowska University, Maria Curie-Skłodowska Square 1, 20-031 Lublin, Poland
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J. Jurasik, B. Łanucha
Tϕ f = P(ϕ f ),
f ∈ H ∞.
Since H ∞ is a dense subset of the Hardy space, the operator Tϕ is densely defined. Moreover, it can be extended to a bounded linear operator Tϕ : 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 = Tz and the backward shift S ∗ = Tz . The Toeplitz operator Tϕ 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 α ∈ K with the reproducing property f (w) = f , k α for exists a kernel function kw α w every f ∈ K α . We have α kw (z) =
1 − α(w)α(z) , z ∈ D. 1 − wz
(1.1)
Moreover, K α is preserved by the conjugation Cα : L 2 → L 2 (an antilinear, isometric involution) defined on L 2 by the formula Cα f (z) = α(z)z f (z), |z| = 1. α = C k α , w ∈ D, is given by Note that the conjugate kernel function kw α w
α(z) − α(w) α , z ∈ D. (z) = kw z−w A thorough a
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