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.
Highlights
Let H2 be the standard Hardy space of the unit disc D identified with the subspace of the boundary functions of its functions in L2ðT Þ.A function in H∞ðDÞ is inner if it is unimodular on the unit circle T
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 Kα has nontangential limit at η
Zω z−ω it, and we give the matrix representation of asymmetric truncated Hankel operators (ATHO) in finite dimensional model spaces associated to Blaschke products each with distinct zeros
Summary
Let H2 be the standard Hardy space of the unit disc D identified with the subspace of the boundary functions of its functions in L2ðT Þ. Since a finite Blaschke product is analytic on a neighborhood of the unit disc, it has an ADC everywhere on T and kαη ∈ Kα, for every η ∈ T. In this case, let m denote the multiplicity of a finite Blaschke product α. (1) Let ω in D or ω in T such that α and β has an ADC at ω kβω ⊗ ~kαω = Aαα,β ∈ Tðα, βÞ and ~kβω ⊗ kαω = Aαβ,β ∈ Tðα, βÞ: ð6Þ zω z−ω it, and we give the matrix representation of ATHOs in finite dimensional model spaces associated to Blaschke products each with distinct zeros.
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