CONTROLLED $K$-FRAMES IN $2$-HILBERT SPACES

In this paper, we introduce a new generalization of controlled $K$-frames to the context of $2$-Hilbert spaces, thereby extending beyond classical Hilbert space theory. We develop foundational results by examining the operator-theoretic properties of controlled $K$-frames in this setting, establishing equivalent conditions that characterize them, and exploring their stability under suitable transformations.This builds directly on prior work introducing controlled $K$-frames in Hilbert $C^*$-modules, where the concept was first defined, equivalent conditions were established, relationships between $K$-frames and controlled $K$-frames were revealed, and invariance and perturbation properties were analyzed. Our work elevates these ideas by adapting them to the richer structure of 2-Hilbert spaces-a framework extending Hilbert spaces through inner products valued in $C^*$-algebras.