On operators T such that f(T) is hypercyclic

A bounded linear operator A on a complex, separable, infinite-dimensional Banach space X is called hypercyclic if there is a vector $ x \in X $ such that $ \{x, Ax, A^{2}x, ...\} $ is dense in X. Let T be a bounded linear operator on X such that T is surjective and its generalized kernel $ ⋃_{n\geq1}N(T^n)$ is dense in X. In the present paper we show that for some admissible functions f without zeros in the spectrum of T and if X is a Hilbert space then f(T) is the limit of hypercyclic operators (Theorem 2).