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 ${\displaystyle x\in X}$ such that ${\displaystyle \{x,Ax,A^{2}x,...\}}$ is dense in X. Let T be a bounded linear operator on X such that T is surjective and its generalized kernel ${\displaystyle {\bigcup }_{n\ge 1}N\left(T^n\right)}$ 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).