  In this paper, we propose to define the concept of family of local atoms and
then we generalize this concept to the atomic system for operator in Banach
spaces by using semi-inner product. We also give a characterization of atomic
systems leading to obtain new frames. In addition, a reconstruction formula is
obtain. Next, some new results are established. The characterization of atomic
systems allows us to state some results for sampling theory in semi-inner
product reproducing kernel Banach spaces. Finally, having used frame operator
in Banach spaces, new perturbation results are established.
