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