Admissibility and frame homotopy for quaternionic frames

We consider the following questions: when do there exist quaternionic frames with given frame spectrum and vector norms? When such exist, is it always possible to interpolate between any two while fixing their spectra In other words, first question admissibility for second a generalization of homotopy conjecture. give complete answers both questions. For question, existence criterion exactly same as in real complex cases. second, non-empty spaces specified norms are path-connected, just case. Our strategy proving these results based on interpreting equivalence classes adjoint orbits, which an approach that also well-suited study frames.