Admissibility and Frame Homotopy for Quaternionic Frames

TL;DR

This paper investigates the existence and homotopy properties of quaternionic frames with specified spectra and norms, establishing criteria and path-connectedness results similar to real and complex cases.

Contribution

It provides complete criteria for the existence of quaternionic frames with given spectra and norms, and proves their space is always path-connected, extending known results to quaternionic frames.

Findings

Existence criterion matches real and complex cases.

Quaternionic frame spaces are path-connected.

Results extend frame theory to quaternionic settings.

Abstract

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