An epsilon-hypercyclicity criterion and its application on classical Banach spaces

We provide a criterion for $\epsilon$-hypercyclicity. Also, we extend the ideas of Badea, Grivaux, M"uller and Bayart to construct $\epsilon$-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.