Cyclicity of the shift operator through Bezout identities

In this paper, we study the cyclicity of the shift operator $S$ acting on a Banach space $\text{X}$ of analytic functions on the open unit disc $\text{D}$. We develop a general framework where a method based on a corona theorem can be used to show that if $f,g\in \text{X}$ satisfy $\mathrm{\mid }g(z)\mathrm{\mid }\le \mathrm{\mid }f(z)\mathrm{\mid }$, for every $z\in \text{D}$, 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.