Cyclicity of the shift operator through Bezout identities
Copyright policy )Cyclicity of the shift operator through Bezout identities
AbstractIn 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.
Complex Variables, shift operator, Hardy spaces, Linear operators in reproducing-kernel Hilbert spaces (including de Branges, de Branges-Rovnyak, and other structured spaces), Besov-Dirichlet spaces, [MATH.MATH-FA] Mathematics [math]/Functional Analysis [math.FA], Cyclicity, Banach algebra, shift operator, De Branges-Rovnyak spaces, Besov-Dirichlet spaces, cyclicity, Functional Analysis (math.FA), de Branges-Rovnyak spaces, Banach algebra, [MATH.MATH-CV] Mathematics [math]/Complex Variables [math.CV], FOS: Mathematics, Corona theorems, Complex Variables (math.CV), Functional Analysis
Complex Variables, shift operator, Hardy spaces, Linear operators in reproducing-kernel Hilbert spaces (including de Branges, de Branges-Rovnyak, and other structured spaces), Besov-Dirichlet spaces, [MATH.MATH-FA] Mathematics [math]/Functional Analysis [math.FA], Cyclicity, Banach algebra, shift operator, De Branges-Rovnyak spaces, Besov-Dirichlet spaces, cyclicity, Functional Analysis (math.FA), de Branges-Rovnyak spaces, Banach algebra, [MATH.MATH-CV] Mathematics [math]/Complex Variables [math.CV], FOS: Mathematics, Corona theorems, Complex Variables (math.CV), Functional Analysis
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