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Cyclicity of the shift operator through Bezout identities

Published online by Cambridge University Press:  28 January 2025

Emmanuel Fricain*
Affiliation:
Univ. Lille, CNRS, UMR 8524 - Laboratoire Paul Painlevé, F-59000 Lille, France e-mail: romain.lebreton@univ-lille.fr
Romain Lebreton
Affiliation:
Univ. Lille, CNRS, UMR 8524 - Laboratoire Paul Painlevé, F-59000 Lille, France e-mail: romain.lebreton@univ-lille.fr

Abstract

In this paper, we study the cyclicity of the shift operator $S$ acting on a Banach space $\mathcal {X}$ of analytic functions on the open unit disc $\mathbb {D}$. We develop a general framework where a method based on a corona theorem can be used to show that if $f,g\in \mathcal {X}$ satisfy $|g(z)|\leq |f(z)|$, for every $z\in \mathbb {D}$, and if g is cyclic, then f is cyclic. We also give sufficient conditions for cyclicity in this context. This enable us to recapture some recent results obtained in de Branges–Rovnayk spaces, in Besov–Dirichlet spaces and in weighted Dirichlet type spaces.

Type
Article
Copyright
© The Author(s), 2025. Published by Cambridge University Press on behalf of Canadian Mathematical Society

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Footnotes

The authors were supported by Labex CEMPI (ANR-11-LABX-0007-01).

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