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Continuous biframes in Hilbert $C^{\ast}-$modules

Abdellatif Lfounoune, Abdelilah Karara, Mohamed Rossafi

TL;DR

This work extends continuous frame theory to Hilbert $C^{*}$-modules by introducing continuous biframes and detailing their basic operators, including the biframe operator, duals, and Bessel multipliers. It provides a comprehensive tensor-product framework: continuous biframes on product modules correspond to biframes in the factors, with operators factoring as tensor products and explicit bounds multiplying. The paper also analyzes stability under invertible transforms, showing that invertible maps preserve biframes and yield scaled bounds, both in single modules and their tensor products. A thorough treatment of continuous biframe multipliers and their duals yields reconstruction formulas and stability results, extended to the tensor-product setting, enabling robust signal representations in the context of Hilbert $C^{*}$-modules.

Abstract

In this paper, we will introduce the concept of a continuous biframe for Hilbert $ C^{\ast}- $modules. Then, we examine some characterizations of this biframe with the help of an invertible and adjointable operator is given. Moreover, we study continuous biframe Bessel multiplier and dual continuous biframe in Hilbert $ C^{\ast}- $modules. Also, we develop the concept of continuous biframes in the tensor product of two Hilbert $C^{\ast}$-modules over a unital $C^{\ast}$-algebra $\mathcal{A}$ and provide some properties of invertible transformed biframes and Bessel multipliers in the tensor product.

Continuous biframes in Hilbert $C^{\ast}-$modules

TL;DR

This work extends continuous frame theory to Hilbert -modules by introducing continuous biframes and detailing their basic operators, including the biframe operator, duals, and Bessel multipliers. It provides a comprehensive tensor-product framework: continuous biframes on product modules correspond to biframes in the factors, with operators factoring as tensor products and explicit bounds multiplying. The paper also analyzes stability under invertible transforms, showing that invertible maps preserve biframes and yield scaled bounds, both in single modules and their tensor products. A thorough treatment of continuous biframe multipliers and their duals yields reconstruction formulas and stability results, extended to the tensor-product setting, enabling robust signal representations in the context of Hilbert -modules.

Abstract

In this paper, we will introduce the concept of a continuous biframe for Hilbert modules. Then, we examine some characterizations of this biframe with the help of an invertible and adjointable operator is given. Moreover, we study continuous biframe Bessel multiplier and dual continuous biframe in Hilbert modules. Also, we develop the concept of continuous biframes in the tensor product of two Hilbert -modules over a unital -algebra and provide some properties of invertible transformed biframes and Bessel multipliers in the tensor product.
Paper Structure (6 sections, 17 theorems, 97 equations)

This paper contains 6 sections, 17 theorems, 97 equations.

Table of Contents

  1. Introduction and preliminaries
  2. Continuous biframe in Hilbert $C^{\ast}$-modules
  3. Continuous biframe Bessel multiplier and dual continuous biframe in Hilbert $C^{\ast}-$modules
  4. Continuous biframes in the tensor product
  5. Invertible transforms on biframes
  6. Bessel Multipliers in the Tensor Product

Key Result

Lemma 1.2

Pas Let $\mathcal{H}$ be a Hilbert $\mathcal{A}$-module. If $\mathcal{T}\in End_{\mathcal{A}}^{\ast}(\mathcal{H})$, then

Theorems & Definitions (45)

  • Definition 1.1
  • Lemma 1.2
  • Definition 1.3
  • Definition 2.1
  • Remark 2.2
  • Example 2.3
  • Definition 2.4
  • Theorem 2.5
  • proof
  • Proposition 2.6
  • ...and 35 more