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Bessel sequences in Hilbert $C^{\ast}$-modules

Abdelilah Karara, Khadija Mabrouk

Abstract

In this peaper we stady certain Bessel sequences $\left\{f_k\right\}_{k=1}^{\infty}$ in Hilbert C*- modules $\mathcal{H}$ for which operator $S$ defined by \ref{eq2} is of the form $\mathcal{T}+ξI$, for some real number $ξ$ and a adjointable linear operator $\mathcal{T}$. Additionally, we investigate frames known as compact-tight frames, which have frame operators that are compact perturbations of constant multiples of the identity. As a conclusion, we provide a theory regarding the weaving of specific compact-tight frames.

Bessel sequences in Hilbert $C^{\ast}$-modules

Abstract

In this peaper we stady certain Bessel sequences in Hilbert C*- modules for which operator defined by \ref{eq2} is of the form , for some real number and a adjointable linear operator . Additionally, we investigate frames known as compact-tight frames, which have frame operators that are compact perturbations of constant multiples of the identity. As a conclusion, we provide a theory regarding the weaving of specific compact-tight frames.
Paper Structure (3 sections, 9 theorems, 47 equations)

This paper contains 3 sections, 9 theorems, 47 equations.

Table of Contents

  1. Introduction
  2. From Bessel sequences to frames
  3. Compact and finite-rank-tight frames

Key Result

Lemma 1.2

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

Theorems & Definitions (22)

  • Definition 1.1
  • Lemma 1.2
  • Definition 1.3
  • Definition 1.4
  • Definition 1.5
  • Proposition 2.1
  • proof
  • Theorem 2.2
  • proof
  • Theorem 3.1
  • ...and 12 more