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For a fusion Banach frame for a Banach space , if is a fusion Banach frame for , then is called a fusion bi-Banach frame for . It is proved that if has an atomic decomposition, then also has a fusion bi-Banach frame. Also, a sufficient condition for the existence of a fusion bi-Banach frame is given. Finally, a characterization of fusion bi-Banach frames is given.
A stronger version of the notion of frame in Banach space called Strong Retro Banach frame (SRBF) is defined and studied. It has been proved that if is a Banach space such that has a SRBF, then has a Bi-Banach frame with some geometric property. Also, it has been proved that if a Banach space has an approximative Schauder frame, then has a SRBF. Finally, the existence of a non-linear SRBF in the conjugate of a separable Banach space has been proved.
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