Gavruta [L. Gavruta, Frames for operators, Appl. Comput. Harmon. Anal. 32 (2012) 139-144] introduced the notion of K-frame and atomic system for an operator K in Hilbert spaces. We extend these notions to Banach spaces and obtain various new results. A necessary and sufficient condition for the existence of an atomic system for an operator K in a Banach space is given. Also, a characterization for the family of local atoms of subspaces of Banach spaces has been given. Further, we give methods to construct an atomic system for an operator K from a given Bessel sequence and an E d -Bessel sequence. Finally, a result related to stability of atomic system for an operator K in a Banach space has been given.
Definition 1.1 (Ref. 10). Let E be a Banach space andGröcheing 14 introduced a more general concept for Banach spaces called Banach frame. Banach frames and atomic decompositions were further studied in Refs. 5, 6, 14 and 15. Christensen and Heil 6 proved perturbation results for Banach frames and atomic decompositions. Casazza et al. 4 studied E d -frame and E d -Bessel sequence in Banach spaces. They gave the following definition of an E d -frame.
Definition 1.2 (Ref. 4). A sequence {f(1.1)The constants A and B are called E d -frame bounds. If atleast (1) and the upper bound condition in (1.1) are satisfied, then {f n } is called an E d -Bessel sequence for E. The bounded linear operator U : E → E d given by2) is called an analysis operator associated with the E d -Bessel sequence {f n }. If {f n } is an E d -frame for E and there exists a bounded linear operator S : E d → E such that S({f n (x)}) = x, for all x ∈ E, then ({f n }, S) is called a Banach frame for E with respect to E d . In Ref. 19Stoeva gave some perturbation results for X d -frames and atomic decompositions. Gavruta 13 introduced the notion of K-frame and atomic system for an operator K in a Hilbert space. She gave the following definition.
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