Summary: 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 \(\{x, Ax, A^ 2 x,\dots\}\) is dense in \(X\). Let \(T\) be a bounded linear operator on \(X\) such that \(T\) is surjective and its generalized kernel \(\bigcup_{n\geq 1} N(T^ n)\) is dense in \(X\). In the present paper we show that for some admissible functions \(f\) without zeros in the spectrum of \(T\) the operator \(f(T)\) is hypercyclic (Theorem 1). If \(f\) has zeros in the spectrum of \(T\) and if \(X\) is a Hilbert space then \(f(T)\) is the limit of hypercyclic operators (Theorem 2).