On Dirichlet-type and n-isometric shifts in finite rank de~Branges--Rovnyak spaces

2023/10/30 by Shuaibing Luo, Luo, Shuaibing, Eskil Rydhe +1 · 1 citation
Mathematics · #Advanced Banach Space Theory #Advanced Harmonic Analysis Research #FOS: Mathematics #Functional Analysis (math.FA) #Holomorphic and Operator Theory

paper · pdf · doi:10.48550/arxiv.2310.19393

openalex publication_date 2023/10/30 · openalex created_date 2023/11/01 · openalex updated_date 2026/07/28

Abstract

This paper studies the function spaces D(μ) by Richter and Aleman, and D_μ by the second author. It is known that the forward shift M_z is bounded and expansive on D(μ), and therefore D(μ) coincides with a de~Branges--Rovnyak space H[B]. We show that such a B is rational if and only if μ is finitely atomic, and this happens exactly when the corresponding defect operator has finite rank. We also outline a method for calculating the reproducing kernel of D(μ) for finitely atomic μ. Similarly, we characterize the allowable tuples μ = ((|dz|)/(2π), μ₁, …, μ_n-1) such that M_z on D_μ is expansive with finite rank defect operator. This investigation provides many interesting examples of normalized allowable tuples μ.

Cited by

• Higher order weighted Dirichlet type spaces with poly-superharmonic weights and Dirichlet type operators of finite order

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