Banach Pair Frames

In this article, we consider pair frames in Banach spaces and   introduce Banach pair frames. Some various concepts in the frame theory such as frames, Schauder frames, Banach frames and atomic decompositions are considered as   special kinds of (Banach) pair frames.  Some frame-like inequalities  for (Banach)  pair frames are presented. The elements that participant  in the construction of (Banach) pair frames are characterized. It is shown that a Banach space  \mathrm{X} has a Banach pair frame with respect to  a Banach scalar sequence space \ell, when  it  is precisely isomorphic to a complemented subspace of \ell. It is shown that  if we are allowed to  choose the scalar sequence space, pair frames and Banach pair frames with respect to the chosen scalar sequence space denote the same concept.

Keywords:

Banach frame , Atomic decomposition , (Banach)pair frame , (Banach)pair Bessel