  We provide a criterion for $\varepsilon$-hypercyclicity. Also, we extend the
ideas of Badea, Grivaux, M\"uller and Bayart to construct
$\varepsilon$-hypercyclic operators which are not hypercyclic in a wider class
of separable Banach spaces, including several classical Banach spaces. For
instance, our result can be applied to separable infinite dimensional $L^p$
spaces and $C(K)$ spaces.
