Quaternionic Generalized Norm Retrieval in Quaternion Euclidean Spaces

Quaternion algebra \(\mathbb {H}\) is a noncommutative associative algebra, and recently quaternionic Fourier analysis has become the focus of an active research due to their potentials in signal analysis and color image processing. The problems related to quaternions are nontrivial and challenging due to noncommutativity of quaternion multiplication. This paper is devoted to establishing the framework of quaternionic generalized norm retrieval (QGNR) in quaternion Euclidean spaces \(\mathbb {H}^{M}\). We introduce the concept of QGNR in \(\mathbb {H}^{M}\) that is defined for general quaternionic self-adjoint matrix sequences. Recall that, even in \(\mathbb {C}^{M}\) (\(\mathbb {R}^{M}\))-setting, the existing literature on norm retrieval problems is only for orthogonal projection matrix sequences instead of general self-adjoint matrix sequences. We characterize QGNR-sequences in terms of their phaselift operators and induced real matrices, present an Edidin type theorem on QGNR for \(\mathbb {H}^{M}\), and investigate the topological property of QGNR-sequences. Finally, we turn to constructing more QGNR-sequences. We prove that a quaternionic self-adjoint matrix sequence \(\mathcal {F}=\{F_{n}\}_{n\in \mathbb {N}_{N}}\) is such that all \(\{TF_{n}T^{*}\}_{n\in \mathbb {N}_{N}}\) with quaternionic invertible matrices T allow QGNR for \(\mathbb {H}^{M}\) if and only if \(\mathcal {F}\) allows quaternionic generalized phase retrieval, and characterize quaternionic generalized norm retrieval multipliers that transform every QGNR-sequence into another QGNR-sequence.