On a generalization of atomic decompositions

We generalize atomic decomposition for Banach spaces andcalled it T -atomic decomposition. A necessary condition for T -atomic decomposition is given. A characterization for a triangular atomic decomposition is also given. Finally, as an application of triangular atomic decompositions, we prove that if a Banach space has a triangular atomic decomposition, then it also has an approximative atomic decomposition, an atomic decomposition and a fusion Banach frame.