# On the existence of non-linear frames

## Shah Jahan, Varinder Kumar, and S.K. Kaushik

Address:

S. Jahan, Department of Mathematics, University of Delhi, Delhi–110 007, India

V. Kumar, Department of Mathematics, Shaheed Bhagat Singh College, University of Delhi, Delhi–110 017, India

S.K. Kaushik, Department of Mathematics, Kirori Mal College, University of Delhi, Delhi–110 007, India

E-mail:

chowdharyshahjahan@gmail.com

varinder1729@gmail.com

shikk2003@yahoo.co.in

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.

AMSclassification: primary 42C15; secondary 46B15.

Keywords: Banach frames, retro Banach frames, approximative Schauder frames.

DOI: 10.5817/AM2017-2-101