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

TL;DR

This paper introduces and studies continuous $K$-$g$-frames in Hilbert $C^*$-modules, combining $g$-frames and $K$-frames, with characterizations and duality properties explored.

Contribution

It defines continuous $K$-$g$-frames in Hilbert $C^*$-modules and establishes their fundamental properties and duality, extending frame theory in this context.

Findings

Characterization of continuous $K$-$g$-frames

Introduction of continuous $K$-$g$-dual

Existence results for continuous $K$-$g$-dual

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.