Controlled K-frames and their invariance under Compact Perturbation

TL;DR

This paper introduces controlled K-frames in Hilbert spaces, analyzing their stability under compact perturbations to enhance understanding of their robustness and potential applications.

Contribution

It extends the concept of K-frames by incorporating control operators and studies their invariance under compact perturbations, a novel theoretical development.

Findings

Controlled K-frames are stable under compact perturbations.

The study provides conditions for the invariance of controlled K-frames.

Results improve understanding of the robustness of controlled K-frames in Hilbert spaces.

Abstract

K-frames were recently introduced by L. G\v{a}vruta in Hilbert spaces to study atomic systems with respect to bounded linear operator. Also controlled frames have been recently introduced by Balazs, Antoine and Grybos in Hilbert spaces to improve the numerical efficiency of interactive algorithms for inverting the frame operator. In this manuscript, the concept of controlled K-frames will be studied and the stability of Controlled K-frames under compact perturbation will be discussed.