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.