Results on Continuous $K$-frames for Quaternionic (Super) Hilbert Spaces

TL;DR

This paper investigates continuous K-frames in quaternionic Hilbert spaces, extending the concept to super spaces and analyzing their relationships and dualities, thus broadening the theoretical framework of quaternionic frame theory.

Contribution

It introduces the concept of continuous K-frames in quaternionic Hilbert spaces and explores their properties, extensions to super spaces, and duality relations, which are novel contributions.

Findings

Established the structure of continuous K-frames in quaternionic Hilbert spaces.

Extended the theory to super quaternionic Hilbert spaces with multiple operators.

Analyzed duality and relationships between frames in component and combined spaces.

Abstract

This paper aims to explore the concept of continuous \( K \)-frames in quaternionic Hilbert spaces. First, we investigate \( K \)-frames in a single quaternionic Hilbert space \( \mathcal{H} \), where \( K \) is a right $\mathbb{H}$-linear bounded operator acting on \( \mathcal{H} \). Then, we extend the research to two quaternionic Hilbert spaces, \( \mathcal{H}_1 \) and \( \mathcal{H}_2 \), and study \( K_1 \oplus K_2 \)-frames for the super quaternionic Hilbert space \( \mathcal{H}_1 \oplus \mathcal{H}_2 \), where \( K_1 \) and \( K_2 \) are right $\mathbb{H}$-linear bounded operators on \( \mathcal{H}_1 \) and \( \mathcal{H}_2 \), respectively. We examine the relationship between the continuous \( K_1 \oplus K_2 \)-frames and the continuous \( K_1 \)-frames for \( \mathcal{H}_1 \) and the continuous \( K_2 \)-frames for \( \mathcal{H}_2 \). Additionally, we explore the duality between the continuous \( K_1 \oplus K_2 \)-frames and the continuous \( K_1 \)- and \( K_2 \)-frames individually.