Some Results on Generalized Frames: Javad Baradaran, Zahra Ghorbani - Team, Support (Thai Journal of Mathematics, 2022)
The concept of a generalized frame or simply a g-frame in a Hilbert space H was introduced by Wenchang Sun in [4]. Given a g-frame {Λ_i}_i I in a Hilbert space H and a bounded operator T on H, we show that the sequence {Λ_iT}_i I is a g-frame for H if and only if T is invertiable on H. Moreover, we prove that add a g-frame to its canonical dual g-frame and the canonical Parseval g-frame are also g-frames. At the end, we provide sufficient conditions under which a subsequence of a g-frame in a Hilbert space H is itself a g-frame for H.