id	author	title	date	pages	extension	mime	words	sentence	flesch	summary	cache	txt
ma-220	Rossaf, Mohamed; Mabrouk, Khadija; Ghiati, M'hamed; Mouniane, Mohammed	Duals of Continuous Frames in Hilbert C∗-Modules	2024	15	.pdf	application/pdf	4965	310	78	Moreover, if B is an upper bound for {Fw}w∈Ω and S is frame operator of {Fw}w∈Ω, then B ∥∥S−1 ∥∥− 1 2 ‖TF‖−1 ‖Γ‖−1 is a lower bound for {gw}w∈Ω. Proof. Let F = {Fw}w∈Ω be a ∗-continuous frame for H with pre-frame operator TF and frame operator S. Then the set of all the operator duals of {Fw}w∈Ω is precisely the following {gw}w∈Ω = { ΓFw + ϕew − ∫ Ω 〈 S−1Fw , Fi 〉 ϕewdµ(ω) } w∈Ω , where {ew}w∈Ω is the standard orthonormal basis for L2(Ω,A), ϕ ∈ B∗ ( H, L2(Ω,A) ) , and Γ is an invertible adjointable operator on H. Proof.	cache/ma-220.pdf	txt/ma-220.txt