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

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.