On $\boldsymbol{k}$-deficit Banach Frames. Armenian Journal of Mathematics

Abstract

$k$-deficit Banach frames have been defined and studied. It has been proved that if a Banach space has a Banach frame having \mbox{$k$-deficit} $(k\geq 0)$, then its second conjugate space has a retro Banach frame. Also, we prove results regarding existence of \mbox{$k$-deficit} Banach frames in subspaces and super spaces of a Banach space and deduce that $\ell^\infty$ does not have a \mbox{$k$-deficit} Banach frame for any $k$. Finally, we prove the equivalence of two statements regarding Banach frames.