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Continuous $K$-$g$-frames in Hilbert $C^*$-modules

Jahangir Cheshmavar, Javad Baradaran, Asadollah Hossienpour

Abstract

This study aims at combining the concepts of $g$-frame and $K$-frame for a Hilbert $C^*$-module $U$, for an operator $K \in End^*_A(U)$, where $End^*_A(U)$ contains all adjointable $A$-linear maps on $U$. As a result, continuous $K$-$g$-frames for Hilbert $C^*$-modules are introduced and studied. Subsequently, some characterizations of continuous $K$-$g$-frames in Hilbert $C^*$-modules are proved. Next, continuous $K$-$g$-dual of a $c$-$K$-$g$-frame is introduced. Finally, some results, particularly, the existence of continuous $K$-$g$-dual, are derived.

Continuous $K$-$g$-frames in Hilbert $C^*$-modules

Abstract

This study aims at combining the concepts of -frame and -frame for a Hilbert -module , for an operator , where contains all adjointable -linear maps on . As a result, continuous --frames for Hilbert -modules are introduced and studied. Subsequently, some characterizations of continuous --frames in Hilbert -modules are proved. Next, continuous --dual of a ---frame is introduced. Finally, some results, particularly, the existence of continuous --dual, are derived.

Paper Structure

This paper contains 3 sections, 12 theorems, 75 equations.

Table of Contents

  1. Introduction and preliminaries
  2. c-K-g-frames in Hilbert $C^*$-Modules
  3. Dual $c$-$K$-$g$-Bessel systems

Key Result

Lemma 1.5

Fang.Yao.2009 Let $U$, $V$ and $W$ be Hilbert $\mathcal{A}-moduls$. Also, let $T^{'}\in End_{\mathcal{A}}^*(W, V)$ and $T \in End_{\mathcal{A}}^*(U, V)$ with $\overline{Ran}(T^{*})$ orthogonally complemented. The following statements are equivalent.

Theorems & Definitions (35)

  • Definition 1.1
  • Definition 1.2
  • Definition 1.3
  • Remark 1.4
  • Lemma 1.5
  • Definition 2.1
  • Remark 2.2
  • Example 2.3
  • Definition 2.4
  • Remark 2.5
  • ...and 25 more