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atomic decompositions; fusion Banach frames; fusion bi-Banach frames
For a fusion Banach frame $(\lbrace G_n, v_n\rbrace , S)$ for a Banach space $E$, if $(\lbrace v_n^*(E^*), v_n^*\rbrace ,T)$ is a fusion Banach frame for $E^*$, then $(\lbrace G_n, v_n\rbrace , S; \lbrace v_n^*(E^*), v_n^*\rbrace ,T)$ is called a fusion bi-Banach frame for $E$. It is proved that if $E$ has an atomic decomposition, then $E$ 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.
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