On Generators of Hardy and Bergman Spaces

Valentin V. Andreev (), Miron B. Bekker () and Joseph A. Cima ()
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Valentin V. Andreev: Lamar University, Department of Mathematics
Miron B. Bekker: University of Pittsburgh at Johnstown, Department of Mathematics
Joseph A. Cima: University of North Carolina at Chapel Hill, Department of Mathematics

Chapter 73 in Operator Theory, 2026, pp 2311-2321 from Springer

Abstract: Abstract A function φ $$\varphi $$ that is analytic and bounded in the unit disk 𝔻 $${\mathbb D}$$ is called a generator for the Hardy space H 2 ( 𝔻 ) $$H^2({\mathbb D})$$ or the Bergman space A 2 ( 𝔻 ) $$A^2({\mathbb D})$$ if polynomials in φ $$\varphi $$ are dense in the corresponding space. We characterize generators in terms of φ − $$\varphi -$$ invariant subspaces, which are also z − $$z-$$ invariant, and study wandering properties of such subspaces. The density of bounded analytic functions in φ − $$\varphi -$$ invariant subspaces is also investigated.

Keywords: Hardy space; Bergman space; Generator; Invariant subspace (search for similar items in EconPapers)
Date: 2026
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DOI: 10.1007/978-3-032-16356-1_104

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