Characterizations and constructions of tight $K$-frames in quaternionic Hilbert spaces

This paper introduces the notion of tight $K$-frames in quaternionic Hilbert spaces and investigates their characterizations and constructions. By employing analysis and frame operators, we provide an operator-theoretic characterization and demonstrate that a tight $K$-frame with bound $C_1$ reduces to an ordinary tight frame with bound $C_2$ if and only if $C_{1}K{K}^{\ast }=C_2{I}_{\mathcal{H}}$. Several construction methods are presented, including operator actions and combinations of Bessel sequences, with refined results for Parseval $K$-frames. Moreover, we establish a bijection between tight $K$-frames and classical tight frames on the subspace $\mathrm{Ran}(K^{\ast })$, showing that each can be obtained from the other via the pseudo-inverse of $K$.