$K$-frames for Banach Spaces

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 ∗ -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 K , K ∗ -dual Bessel sequences and the pair of approximate K , K ∗ -dual Bessel sequences for Banach spaces are introduced, and their important properties are discussed.