Approximative Atomic Systems for operators in Banach spaces

Abstract

$K$-frames and atomic systems for an operator $K$ in Hilbert spaces were introduced by Gavruta \cite{12} and further studied by Xio, Zhu and Gavruta \cite{21}. In this paper, we have introduced the notion of an approximative atomic system for an operator $K$ in Banach spaces and obtained interesting results. A complete characterization of family of approximative local atoms of subspace of Banach space has been obtained. Also, a necessary and sufficient condition for the existence of an approximative atomic system for an operator $K$ is given. Finally, explicit methods are given for the construction of an approximative atomic systems for an operator $K$ from a given Bessel sequence and approximative $\mathcal{X}_d$-Bessel sequence.