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Cyclicity of the shift operator through Bezout identities

Part of: Spaces and algebras of analytic functions Special classes of linear operators

Published online by Cambridge University Press:  28 January 2025

Emmanuel Fricain  and
Romain Lebreton
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
*
e-mail: emmanuel.fricain@univ-lille.fr
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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.

Keywords

CyclicityBanach algebrashift operatorde Branges-Rovnyak spacesBesov–Dirichlet spaces

MSC classification

Primary: 30H10: Hardy spaces 30H80: Corona theorems 47B32: Operators in reproducing-kernel Hilbert spaces (including de Branges, de Branges-Rovnyak, and other structured spaces)
Type
Article
Information
Canadian Mathematical Bulletin , First View , pp. 1 - 25
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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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