On $\boldsymbol{k}$-deficit Banach Frames. Armenian Journal of Mathematics - Kaushik, S. K.; Kumar, Raj (Armenian Journal of Mathematics, 2009)
$k$-deficit Banach frames have been defined and studied. It has been proved that if a Banach space has a Banach frame having $k$-deficit $(k≥ 0)$, then its second conjugate space has a retro Banach frame. Also, we prove results regarding existence of $k$-deficit Banach frames in subspaces and super spaces of a Banach space and deduce that $\ell^∞$ does not have a $k$-deficit Banach frame for any $k$. Finally, we prove the equivalence of two statements regarding Banach frames.