# Approximative $K$-Atomic Decompositions and frames in Banach Spaces

## TL;DR

This paper extends the concept of $K$-frames from Hilbert spaces to Banach spaces by introducing approximative $K$-atomic decompositions, providing characterizations, construction methods, and exploring their properties and applications.

## Contribution

It introduces the novel notion of approximative $K$-atomic decompositions in Banach spaces, expanding frame theory beyond Hilbert spaces.

## Key findings

- Characterizations of the existence of approximative $K$-atomic decompositions.
- Methods for constructing approximative $K$-atomic decompositions.
- Conditions linking $ ext{X}_d$-frames and $ ext{X}_d$-Bessel sequences to bounded operators.

## Abstract

[L. Gavruta, Frames for Operators, Appl. comput. Harmon. Anal. 32(2012), 139-144] introduced a special kind of frames, named $K$-frames, where $K$ is an operator, in Hilbert spaces, is significant in frame theory and has many applications. In this paper, first of all, we have introduced the notion of approximative $K$-atomic decomposition in Banach spaces. We gave two characterizations regarding the existence of approximative $K$-atomic decompositions in Banach spaces. Also some results on the existence of approximative $K$-atomic decompositions are obtained. We discuss several methods to construct approximative $K$-atomic decomposition for Banach Spaces. Further, approximative $\mathcal{X}_d$-frame and approximative $\mathcal{X}_d$-Bessel sequence are introduced and studied. Two necessary conditions are given under which an approximative $\mathcal{X}_d$-Bessel sequence and approximative $\mathcal{X}_d$-frame give rise to a bounded operator with respect to which there is an approximative $K$-atomic decomposition. Examples and counter examples are provided to support our concept. Finally, a possible application is given.