(F,G)-operator frames for ℒ(ℋ,𝒦)

Abstract

The concept of [Formula: see text]-frames was recently introduced by Găvruta7 in Hilbert spaces to study atomic systems with respect to a bounded linear operator. Let [Formula: see text] be a unital [Formula: see text]-algebra, [Formula: see text] be finitely or countably generated Hilbert [Formula: see text]-modules, and [Formula: see text] be adjointable operators from [Formula: see text] to [Formula: see text]. In this paper, we study a class of [Formula: see text]-bounded operators and [Formula: see text]-operator frames for [Formula: see text]. We also prove that the pseudo-inverse of [Formula: see text] exists if and only if [Formula: see text] has closed range. We extend some known results about the pseudo-inverses acting on Hilbert spaces in the context of Hilbert [Formula: see text]-modules. Further, we also present some perturbation results for [Formula: see text]-operator frames in [Formula: see text].

Some Generalizations of Relay Fusion Frames and F , G -Relay Fusion Frames in Hilbert

“…References [18,19] introduce the notions of K-frames and (F, G)-operator frames, respectively. We now introduce (F, G)-R-FFs on Hilbert C * -modules as a generalization of K-frames, (F, G)-operator frames, and RFFs in Hilbert C * -modules.…”