Fuzzy atomic system and fuzzy K-frame in fuzzy Hilbert space

TL;DR

This paper introduces fuzzy atomic systems and fuzzy K-frames in fuzzy Hilbert spaces, exploring their properties, stability, and construction methods, with implications for fuzzy functional analysis.

Contribution

It presents the first study of fuzzy atomic systems and fuzzy K-frames in fuzzy Hilbert spaces, including their characterizations, invertibility, stability, and methods to construct new fuzzy K-frames.

Findings

Fuzzy atomic systems exist for strongly fuzzy bounded linear operators.

Fuzzy K-frames have invertible frame operators under certain conditions.

Scalar and product operations preserve fuzzy K-frames.

Abstract

Atomic system in fuzzy Hilbert space is introduced and the existence of the fuzzy atomic systems for a strongly fuzzy bounded linear operator is studied. The notion of a K-frame in fuzzy Hilbert space is presented and some of their characterizations are given. We will see that fuzzy frame operator of a fuzzy K-frame in fuzzy Hilbert space is invertible under some sufficient condition and validates this by giving some examples. Fuzzy K-frame property in fuzzy Hilbert space preserve by a strongly fuzzy bounded linear operator is established. We will describe stability condition of fuzzy K-frame in fuzzy Hilbert space under some perturbations. We construct new types of fuzzy K-frame using fuzzy K-frame in fuzzy Hilbert space. Further, it is seen that scalar combinations and product of two fuzzy K-frames is also a fuzzy K-frame in fuzzy Hilbert space