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

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.