On Generators of the Hardy and the Bergman Spaces

A function which is analytic and bounded in the Unit disk is called a generator for the Hardy space or the Bergman space if polynomials in that function are dense in the corresponding space. We characterize generators in terms of sub-spaces which are invariant under multiplication by the generator a...

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Bibliographic Details
Main Authors Andreev, Valentin V, Bekker, Miron B, Cima, Joseph A
Format Journal Article
LanguageEnglish
Published 08.04.2023
Subjects
Mathematics - Complex Variables
Mathematics - Functional Analysis
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Summary:A function which is analytic and bounded in the Unit disk is called a generator for the Hardy space or the Bergman space if polynomials in that function are dense in the corresponding space. We characterize generators in terms of sub-spaces which are invariant under multiplication by the generator and also invariant under multiplication by z, and study wandering properties of such sub-spaces. Density of bounded analytic functions in the sub-spaces of the Hardy space which are invariant under multiplication by the generator is also investigated.
DOI:10.48550/arxiv.2304.04119