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 of analytic functions on the open unit disc . We develop a general framework where a method based on a corona theorem can be used to show that if f,g∈ satisfy |g(z)|≤ |f(z)|, for every z∈, 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.