On the existence of non-linear frames

Abstract

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 $\mathcal{X}$ is a Banach space such that $\mathcal{X^*}$ has a SRBF, then $\mathcal{X}$ has a Bi-Banach frame with some geometric property. Also, it has been proved that if a Banach space $\mathcal{X}$ has an approximative Schauder frame, then $\mathcal{X^*}$ has a SRBF. Finally, the existence of a non-linear SRBF in the conjugate of a separable Banach space has been proved.