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Frames and Operators on Quaternionic Hilbert spaces

Najib Khachiaa

Abstract

The aim of this work is to study frame theory in quaternionic Hilbert spaces. We provide a characterization of frames in these spaces through the associated operators. Additionally, we examine frames of the form $\{Lu_i\}_{i \in I}$, where $L$ is a right $\mathbb{H}$-linear bounded operator and $\{u_i\}_{i \in I}$ is a frame.

Frames and Operators on Quaternionic Hilbert spaces

Abstract

The aim of this work is to study frame theory in quaternionic Hilbert spaces. We provide a characterization of frames in these spaces through the associated operators. Additionally, we examine frames of the form , where is a right -linear bounded operator and is a frame.

Paper Structure

This paper contains 4 sections, 33 theorems, 48 equations.

Table of Contents

  1. Introduction and preliminaries
  2. Auxiliary results
  3. Frame coefficients
  4. Frames and operators on quaternionic Hilbert spaces

Key Result

Theorem 1

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

Theorems & Definitions (73)

  • 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
  • Theorem 2
  • Definition 6
  • Theorem 3
  • ...and 63 more