# Atomic subspaces for operators

## TL;DR

This paper introduces atomic subspaces relative to bounded linear operators, generalizing fusion frames to $K$-fusion frames, and explores their properties and characterizations.

## Contribution

It defines atomic subspaces for operators, extends fusion frames to $K$-fusion frames, and studies their properties and characterizations.

## Key findings

- Characterization of $K$-fusion frames
- Properties of $K$-fusion frames like direct sums and intersections
- Generalization of fusion frames through atomic subspaces

## Abstract

This paper introduces the concept of atomic subspaces with respect to a bounded linear operator. Atomic subspaces generalize fusion frames and this generalization leads to the notion of $K$-fusion frames. Characterizations of $K$-fusion frames are discussed. Various properties of $K$-fusion frames, for example, direct sums, intersection, are studied.