A characterization of complex stable phase retrieval in Banach lattices

TL;DR

This work characterizes stable phase retrieval for subspaces of complex Banach lattices, extending the real-case framework by replacing disjointness with a complex-perpendicularity notion captured by $|\mathfrak{Re}(f\overline{g})|^{1/2}$. The authors define a precise complex SPR criterion and show that SPR fails exactly when uniformly separated $\varepsilon$-almost perpendicular pairs exist, with quantitative bounds on the separation constant. The central contribution is a Complex Stable Phase Retrieval theorem that establishes equivalences among several obstruction and witness conditions, using an orthogonal-reduction technique to relate near- and near-orthogonal pair behavior. This advances the understanding of phase retrieval in Banach lattices and informs the structural obstructions to stable recovery in complex settings, with potential implications for analysis in diffraction, optics, and related inverse problems.

Abstract

This note provides a characterization of the subspaces of a complex Banach lattice which do stable phase retrieval, in the spirit of the characterization of real stable phase retrieval established by D. Freeman, T. Oikhberg, B. Pineau and M. A. Taylor.

Theorems & Definitions (14)

• Proposition 2.1: Real phase retrieval
• proof
• Proposition 2.2: Complex phase retrieval
• proof
• Theorem 2.3: Real stable phase retrieval
• proof
• Theorem 2.4
• Proposition 2.5
• proof
• Definition 2.6