On $K$-frames for Quaternionic Hilbert Spaces

TL;DR

This paper extends the theory of $K$-frames to quaternionic Hilbert spaces, establishing foundational theorems, exploring direct sums, and analyzing duality properties specific to the quaternionic setting.

Contribution

It introduces the quaternionic version of Douglas's theorem and investigates $K$-frames, their direct sums, and duality in quaternionic Hilbert spaces, which is novel in this context.

Findings

Established quaternionic Douglas's theorem.

Analyzed $K$-frames for direct sums of quaternionic Hilbert spaces.

Explored $K$-duality and its relation to $K$-frames.

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.