K -frames for Banach Spaces

Abstract

In order to reconstruct elements from the range of a linear bounded operator $K$ on a separable Banach space, concepts of $K$ -frames and $K^{\ast }$ -atomic systems for Banach spaces are introduced, the relationship between the two is discussed, and the sufficient condition for a Banach space to have a $K$ -frame is given. In addition, the duality of $K$ -frames for Banach spaces is studied, the pair of $\left(K,K^{\ast }\right)$ -dual Bessel sequences and the pair of approximate $\left(K,K^{\ast }\right)$ -dual Bessel sequences for Banach spaces are introduced, and their important properties are discussed.