On $K$-frames for Quaternionic Hilbert Spaces

The aim of this paper is to study $K$-frames for quaternionic Hilbert spaces. First, we present the quaternionic version of Douglas's theorem and then investigate $K$-frames for a quaternionic Hilbert space $\mathcal{H}$, where $K \in \mathbb{B}(\mathcal{H})$. Given two quaternionic Hilbert spaces $\mathcal{H}_1$ and $\mathcal{H}_2$, along with two right $\mathbb{H}$-linear bounded operators $K_1 \in \mathbb{B}(\mathcal{H}_1)$ and $K_2 \in \mathbb{B}(\mathcal{H}_2)$, we study the $K_1 \oplus K_2$-frames for the super space $\mathcal{H}_1 \oplus \mathcal{H}_2$ and their relationship with $K_1$-frames and $K_2$-frames for $\mathcal{H}_1$ and $\mathcal{H}_2$, respectively. We also explore the $K_1 \oplus K_2$-duality in relation to $K_1$-duality and $K_2$-duality.