$K$-$g$-frames in Hilbert module over locally-$C^*$-algebras

TL;DR

This paper introduces and studies $K$-$g$-frames in locally $C^*$-algebras, generalizing $g$-frames, and explores their properties, duals, and characterizations within this mathematical framework.

Contribution

It defines $K$-$g$-frames in locally $C^*$-algebras, establishes their relationship with $g$-frames, and introduces the $K$-dual $g$-frame, expanding the theory of frames in this setting.

Findings

$K$-$g$-frames are more general than $g$-frames.

The paper characterizes $K$-$g$-frames via related concepts.

Properties of the $K$-dual $g$-frame are established.

Abstract

This paper explores the concept of $K$-$g$-frames in locally $C^*$-algebras, which are shown to be more general than $g$-frames. The authors first introduce the notion of a $g$-orthonormal basis and utilize it to define the $g$-operator, a crucial element for studying the construction of $K$-$g$-frames in locally $C^*$-algebras. The paper establishes a relationship between $g$-frames and $K$-$g$-frames and introduces the $K$-dual $g$-frame along with its properties. Finally, the authors characterize $K$-$g$-frames through two other related concepts.