On $K$-frames for Quaternionic Hilbert Spaces

Abstract

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.

arXiv admin note: text overlap with arXiv:2411.03790