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Weaving continuous generalized frames for operators

Hafida Massit, Mohamed Rossafi, Choonkil Park

TL;DR

The atomic $g-system is introduced and it is shown that approximate $L$-duals are stable under small perturbation and that it is possible to remove some elements of a woven continuous $ L$-$g$-frame and still have a woven continuous $ L$-$g$-frame.

Abstract

Recently, Bemrose et al. \cite{BE} developed a theory of weaving frames, which was motivated by a problem regarding distributed signal processing. In this present article, we introduce the atomic $g$-system and we generalize some of the known results in continuous $L$-frames, weaving continuous and weaving continuous $ g$-frames, also we study weaving continuous $ L$-$g$-frames in Hilbert spaces. Moreover, we study the behaviour continuous $ L$-$g$-frames under some perturbations, and we show that approximate $L$-duals are stable under small perturbation and that it is possible to remove some elements of a woven continuous $ L$-$g$-frame and still have a woven continuous $ L$-$g$-frame.

Weaving continuous generalized frames for operators

TL;DR

The atomic L Lg Lg$-frame.

Abstract

Recently, Bemrose et al. \cite{BE} developed a theory of weaving frames, which was motivated by a problem regarding distributed signal processing. In this present article, we introduce the atomic -system and we generalize some of the known results in continuous -frames, weaving continuous and weaving continuous -frames, also we study weaving continuous --frames in Hilbert spaces. Moreover, we study the behaviour continuous --frames under some perturbations, and we show that approximate -duals are stable under small perturbation and that it is possible to remove some elements of a woven continuous --frame and still have a woven continuous --frame.
Paper Structure (3 sections, 12 theorems, 47 equations)

This paper contains 3 sections, 12 theorems, 47 equations.

Table of Contents

  1. Introduction and preliminaries
  2. Main results
  3. Conclusion

Key Result

Theorem 2.5

Let $\{\chi_{\varsigma}\in B(\mathcal{H},\mathcal{H}_{\varsigma}),\;{\varsigma\in\mathfrak{A}} \}$ and $\{\xi_{\varsigma}\in B(\mathcal{H},\mathcal{H}_{\varsigma}),\;{\varsigma\in\mathfrak{A}} \}$ be a continuous woven atomic $g-$system for $L$. Then $\{\chi_{\varsigma}\in B(\mathcal{H},\mathcal{H}_

Theorems & Definitions (29)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Theorem 2.5
  • proof
  • Lemma 2.6
  • proof
  • Proposition 2.7
  • proof
  • ...and 19 more