$\bm{p}$-frames, Hilbert-Schauder frames and $\bm{\sigma}$-frame operators

Abstract

In view of the fact that there are not appropriate frame operators of frames in Banach spaces, this paper considers a class of sequences satisfying certain conditions in Banach spaces which is called as the $\sigma$-frame, and the corresponding concept of frame operators is given. The $\sigma$-frames and $\sigma$-frame operators are natural generalizations of frames and frame operators in Hilbert spaces. This paper illustrates that $\sigma$-frame operators are positive, self-adjoint and they can be decomposed through $l_2$. The perturbation result under operators of $\sigma$-frame is obtained. This paper also shows that the kind of $\sigma$-frames contains two other kinds of frames in Banach space---$p$-frames ($1＜p\leq 2$) and $\sigma {\rm HS}$ frames which are a kind of frames according to the definition of the Hilbert-Schauder frames. The perturbation results under operators of $p$-frames ($1＜p\leq 2$) and $\sigma {\rm HS}$ frames are obtained.