Frames and Riesz bases for Banach spaces, and Banach spaces of vector-valued sequences (Q1943752)

The authors give some relationships between Bessel sequences (respectively, frames) and Riesz sequences (respectively, Riesz bases). Atomic decomposition and Banach frames in Banach spaces are also discussed. Finally, Banach spaces consisting of Bessel or Riesz sequences are introduced and it is proved that they are isometrically isomorphic to some Banach spaces of bounded linear operators, and that some subspaces of those Banach spaces are isometrically isomorphic to some Banach spaces of compact operators. The result given in Proposition \(4.1\), which provides necessary and sufficient conditions for the existence of a Banach frame in a Banach space, was already proved by \textit{P. K. Jain}, \textit{S. K. Kaushik} and \textit{L. K. Vashisht} [Indian J. Pure Appl. Math. 37, No. 5, 265-272 (2006; Zbl 1125.46013)]. Furthermore, \textit{P. G. Casazza} et al.[Contemp. Math. 247, 149--182 (1999; Zbl 0947.46010)] proved that every separable Banach space has a Banach frame. This result was further improved by Jain et al., where it is proved that for the existence of a Banach frame in a Banach space \(X\), separability of \(X\) is not required.