On phase and norm retrieval by subspaces

This paper studies phase and norm retrieval by subspaces. We first investigate norm retrieval by hyperplanes. We show that if $N$ hyperplanes $\{{\phi }_i^{\perp }{\}}_{i=1}^N\subset {\mathbb{R}}^N$ allow norm retrieval and the vectors $\{{\phi }_i{\}}_{i=1}^N$ are linearly independent, then these vectors must be an orthonormal basis for ${\mathbb{R}}^N$. We then present several new properties of subspaces that allow phase and norm retrieval. In particular, we provide a complete classification of two proper subspaces that perform norm retrieval. It is known that the collection of norm-retrievable frames $\{{\phi }_i{\}}_{i=1}^M$ in ${\mathbb{R}}^N$ is not dense in the set of all $M$-element frames when $M<2N-1$. We extend this result to subspaces. Several alternative proofs of fundamental results in phase and norm retrieval are also provided.