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Biframes in Hilbert $C^{\ast}-$modules

Mohamed Rossafi, Abdelilah Karara, Roumaissae El jazzar

TL;DR

The paper develops a theory of biframes in Hilbert $C^{*}$-modules by introducing two sequences that bound the $\mathcal{A}$-valued inner products through $A$ and $B$, generalizing frames and controlled frames. It establishes an operator-theoretic framework via the biframe operator $S_{\Xi,\Upsilon}$, deriving key properties (positivity, invertibility, and reconstruction formulas) and exploring how biframes transform under admissible operators. The work also discusses Parseval and dual biframes, provides reconstruction identities, and demonstrates stability under operator mappings such as $S_{P\Xi,Q\Upsilon}=Q S_{\Xi,\Upsilon} P^{*}$, highlighting the interconnections with frames, dual frames, and controlled frame concepts in the $C^{*}$-module setting. These results extend frame theory to a two-sequence context within Hilbert $C^{*}$-modules and offer a rigorous operator framework with potential applications in operator theory and module analysis.

Abstract

In this paper, we will introduce the concept of biframes for Hilbert $ C^{\ast}- $modules produced by a pair of sequences, and we present various examples of biframes. Then, we examine the characteristics of biframes from the viewpoint of operator theory by establishing some properties of biframes in Hilbert $ C^{\ast}- $modules.

Biframes in Hilbert $C^{\ast}-$modules

TL;DR

The paper develops a theory of biframes in Hilbert -modules by introducing two sequences that bound the -valued inner products through and , generalizing frames and controlled frames. It establishes an operator-theoretic framework via the biframe operator , deriving key properties (positivity, invertibility, and reconstruction formulas) and exploring how biframes transform under admissible operators. The work also discusses Parseval and dual biframes, provides reconstruction identities, and demonstrates stability under operator mappings such as , highlighting the interconnections with frames, dual frames, and controlled frame concepts in the -module setting. These results extend frame theory to a two-sequence context within Hilbert -modules and offer a rigorous operator framework with potential applications in operator theory and module analysis.

Abstract

In this paper, we will introduce the concept of biframes for Hilbert modules produced by a pair of sequences, and we present various examples of biframes. Then, we examine the characteristics of biframes from the viewpoint of operator theory by establishing some properties of biframes in Hilbert modules.
Paper Structure (4 sections, 11 theorems, 60 equations)

This paper contains 4 sections, 11 theorems, 60 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. The Biframes in Hilbert $C^{\ast}-$modules
  4. The biframe operator

Key Result

Lemma 2.2

Pas Assume $\mathcal{H}$ is a Hilbert $\mathcal{A}$-module. Given an operator $\mathcal{T} \in End_{\mathcal{A}}^{\ast}(\mathcal{H})$, it follows that

Theorems & Definitions (29)

  • Definition 2.1
  • Lemma 2.2
  • Definition 2.3
  • Definition 2.4
  • Definition 2.5
  • Theorem 2.6
  • Theorem 2.7
  • Definition 3.1
  • Example 3.2
  • Definition 3.3
  • ...and 19 more