New properties and existence of exact phase-retrievable g-frames

Due to the frame elements of the g-frames being operators, it has many differences from traditional frames. Hence some new characterizations of exact phase-retrievable g-frames from the perspective of operator theory are mainly discussed in this paper. Firstly, we find that for an exact phase-retrievable g-frame, its canonical dual frame will maintain the exact phase-retrievability. Then the stability of the exact phase-retrievability is discussed. More specifically, an exact phase-retrievable g-frame is still exact phase-retrievable after a small disturbance can be obtained in this paper. In addition, we show that the direct sum of two g-frames which have the exact PR-redundancy property also have the exact PR-redundancy property. With the help of these results, the existence of the exact phase-retrievable g-frames is discussed. We prove that for the real Hilbert space case, an exact phase-retrievable g-frame of length N exists for every \(2n-1\le N \le \frac{n(n+1)}{2}.\) $2n-1\le N\le \frac{n(n+1)}{2}.$