Generalized frames with C*-valued bounds and their operator duals

Abstract

Certain facts about frames and generalized frames are extended for the new g-frames, referred as *-g-frames, in a Hilbert C*-modules. As a matter of fact, some relations are establish between *-frames and *-g-frames in a Hilbert C*-module. Furthermore, the paper studies the operators associated to a given *-g-frame, the construction of new *-g-frames. Moreover, the operator duals for a *-g-frame are introduced and their properties are investigated. Finally, operator duals of a *-g-frame are characterized.

Some Generalizations of Frames in Hilbert Modules

“…Definition 6 (see [19]). We call a sequence Λ i ∈ End * A (H, V i ): i ∈ I a * -g-frame in Hilbert A-module H over a unital C * -algebra with respect to V i : i ∈ I if there exist strictly nonzero elements A, B in A, such that, for all x ∈ H,…”

*-g-frames in tensor products of Hilbert C*-modules

“…Recentely, A. Khosravi and B. Khosravi [14] introduced the g-frame theory in Hilbert C * -modules, and Alijani, and Dehghan [2] introduced the g-frame theory in Hilbert C * -modules. N. Bounader and S. Kabbaj [4] and A. Alijani [1] introduced the * -g-frames which are generalizations of g-frames in Hilbert C * -modules. In this article, we study the * -g-frames in tensor products of Hilbert C * -modules and * -g-frames in two Hilbert C * -modules with different C * -algebras.…”

Some Inequalities for g-Frames in Hilbert C*-Modules

“…In the past decade, much attention has been paid to the extension of frame and g-frame theory from Hilbert spaces to Hilbert C * -modules, and some significant results have been presented (see [15][16][17][18][19][20][21][22][23]). It should be pointed out that, due to the essential differences between Hilbert spaces and Hilbert C * -modules and the complex structure of the C * -algebra involved in a Hilbert C * -module, the problems on frames and g-frames for Hilbert C * -modules are expected to be more complicated than those for Hilbert spaces.…”