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

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.