Atomic systems for operators

摘要

Gavruta (L. Gavruta, Fames for operators, Appl. Cornput. 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.