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

Frames for operators in Banach spaces via semi-inner products