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ORBITAL LIMIT POINTS AND HYPERCYCLICITY OF OPERATORS ON ANALYTIC FUNCTION SPACES

On a separable, infinite dimensional Banach space X, a bounded linear operator T : X → X is said to be hypercyclic if there exists a vector x in X such that its orbit Orb(T, x)= {x, T x, T 2 x, . . .} is dense in X. It is known that for a weighted backward shift T to be hypercyclic it suffices to merely require the operator to have an orbit Orb(T, x) with a non-zero limit point. In this paper we show that while this result also holds true for the class of adjoints of multiplication operators on the Bergman spaces, it breaks down for the class of composition operators on the Hardy space.