Approximative Atomic Systems for operators in Banach spaces

TL;DR

This paper extends the concept of atomic systems from Hilbert spaces to Banach spaces, providing characterizations, existence conditions, and construction methods for approximative atomic systems related to operators.

Contribution

It introduces the notion of approximative atomic systems in Banach spaces and offers new characterizations, existence criteria, and explicit construction methods.

Findings

Characterization of approximative local atoms in Banach spaces

Necessary and sufficient conditions for approximative atomic systems

Explicit construction methods from Bessel sequences

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.