New properties and existence of exact phase-retrievable g-frames (Q6551229)

Let \(H\) and \(K\) be Hilbert spaces, and let \(\{K_i\}_{i \in I}\) be subspaces of \(K\). A sequence of bounded linear operators \(\{\Lambda_i\}_{i \in I}\) from \(H\) to \(K_i\) is called a generalized frame (or g-frame) for \(H\) with respect to \(\{K_i\}_{i \in I}\) if there exist constants \(A, B > 0\) such that\N\[\NA \|f\|^2 \leq \sum_{i \in I} \|\Lambda_i f\|^2 \leq B \|f\|^2\N\]\Nfor all \(f \in H\). A g-frame \(\{\Lambda_i\}_{i=1}^m\) for a Hilbert space \(H\) is called phase-retrievable if for any two vectors \(x, y \in H\), the magnitudes of the g-frame coefficients \(\|\Lambda_i x\|\) and \(\|\Lambda_i y\|\) determine the vectors up to a global phase factor. That is, if \(\|\Lambda_i x\| = \|\Lambda_i y\|\) for all \(i = 1, \dots, m\), then \(x\) and \(y\) must be identical up to a phase, i.e., \(x = e^{i\theta} y\) for some real number \(\theta\).\N\NThe phase-retrievable g-frame is called exact phase-retrievable if no subset of the g-frame can still recover the phase of the signal. In other words, the frame has the smallest number of frame elements required to achieve phase retrieval, so removing any frame element would make the frame no longer phase-retrievable. Formally, a g-frame \(\{\Lambda_i\}_{i=1}^m\) is said to have exact phase retrievability if, for any proper subset \(\Delta\) of \(\{1, \ldots, m\}\), there exist two vectors \(x, y \in H\) such that \(\|\Lambda_i x\| = \|\Lambda_i y\|\) for every \(i \in \Delta\), but \(\|\Lambda_i x\| \neq \|\Lambda_i y\|\) for some \(i \in \Delta^c\).\N\NThe authors study the properties and existence of exact phase-retrievable g-frames. The paper is mainly an extension of their previous work in [the authors, Linear Multilinear Algebra 70, No. 19, 4117--4132 (2022; Zbl 1512.42048)], to the setting of exact phase-retrievable frames. The authors establish several new results, including the preservation of exact phase-retrievability under certain operations like taking the canonical dual frame and the direct sum of frames, as well as proving the existence of exact phase-retrievable g-frames for real Hilbert spaces under specific conditions.\N\NThe main results are:\N\begin{itemize}\N\item If a g-frame is exact phase-retrievable, its canonical dual frame will also maintain this exact phase-retrievability. (Theorem 2.1).\N\item An exact phase-retrievable g-frame retains its phase-retrievability after a small perturbation. (Theorem 2.2).\N\item The direct sum of two g-frames, each having the exact phase-retrievability property, will also preserve this property. (Theorem 2.4).\N\item For a real Hilbert space, an exact phase-retrievable g-frame exists with cardinality \(N\) for \(2n - 1 \leq N \leq \frac{n(n+1)}{2}\), where \(n\) is the dimension of the Hilbert space. (Theorem 3.1 and 3.2.)\N\end{itemize}