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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.
