Abstract

 

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.

 

Keywords and Phrases

Frame, Controlled $K$-frame, Controlled $K$-frame operator, Controlled $K$- Bessel sequence.

A.M.S. subject classification

42C15, 46C50.

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