Riesz–Kolmogorov Type Compactness Criteria in Function Spaces with Applications

Mishko Mitkovski (), Cody B. Stockdale (), Nathan A. Wagner () and Brett D. Wick ()
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Mishko Mitkovski: School of Mathematical and Statistical Sciences, Clemson University
Cody B. Stockdale: School of Mathematical and Statistical Sciences, Clemson University
Nathan A. Wagner: Washington University in St. Louis, Department of Mathematics and Statistics
Brett D. Wick: Washington University in St. Louis, Department of Mathematics and Statistics

A chapter in Multivariable Operator Theory, 2023, pp 543-573 from Springer

Abstract: Abstract We present forms of the classical Riesz–Kolmogorov theorem for compactness that are applicable in a wide variety of settings. In particular, our theorems apply to classify the precompact subsets of the Lebesgue space $$L^2$$ L 2 , Paley–Wiener spaces, weighted Bargmann–Fock spaces, and a scale of weighted Besov–Sobolev spaces of holomorphic functions that includes weighted Bergman spaces of general domains as well as the Hardy space and the Dirichlet space. We apply the compactness criteria to characterize the compact Toeplitz operators on the Bergman space, deduce the compactness of Hankel operators on the Hardy space, and obtain general umbrella theorems.

Keywords: Compactness; Framed spaces; Spaces of holomorphic functions; Toeplitz operators; Hankel operators; Primary 46B50; Secondary 46E15; 46E30; 47B35 (search for similar items in EconPapers)
Date: 2023
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Persistent link: https://EconPapers.repec.org/RePEc:spr:sprchp:978-3-031-50535-5_22

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DOI: 10.1007/978-3-031-50535-5_22

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