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On $K$-frames for Quaternionic Hilbert Spaces

Najib Khachiaa

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.

On $K$-frames for Quaternionic Hilbert Spaces

Abstract

The aim of this paper is to study -frames for quaternionic Hilbert spaces. First, we present the quaternionic version of Douglas's theorem and then investigate -frames for a quaternionic Hilbert space , where . Given two quaternionic Hilbert spaces and , along with two right -linear bounded operators and , we study the -frames for the super space and their relationship with -frames and -frames for and , respectively. We also explore the -duality in relation to -duality and -duality.

Paper Structure

This paper contains 4 sections, 30 theorems, 57 equations.

Table of Contents

  1. Introduction and preliminaries
  2. $K$-frames for quaternionic Hilbert spaces
  3. $K$-frames on quaternionic super Hilbert spaces
  4. $K_1$-duality, $K_2$-duality and $K_1\oplus K_2$-duality

Key Result

Theorem 1

11 If $\mathcal{H}$ is a right quaternionic Hilbert space, then for all $u,v\in \mathcal{H}$,

Theorems & Definitions (69)

  • Definition 1: the field of quaternions
  • Definition 2: Right quaternionic vector space
  • Definition 3: Right quaterninoic pre-Hilbert space
  • Definition 4: Right quaternionic Hilbert space
  • Example 1
  • Theorem 1: The Cauchy-Schwarz inequality
  • Definition 5: orthogonality
  • Proposition 1
  • Theorem 2
  • Definition 6
  • ...and 59 more