Some properties of g-frames in Hilbert C∗-modules

Abstract

In this paper we give some new results for g-frames in Hilbert C * -modules and then we introduce a bounded A-linear operator L, by means of this operator L we character the properties of the g-frames and g-Riesz basis in Hilbert C * -modules. Finally, we establish some important equalities and inequalities for frames and g-frames in Hilbert C * -modules.

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

“…In this paper, we establish several new inequalities for g-frames in Hilbert C * -modules, where a scalar λ in R, the real number set, and an adjointable operator with respect to two g-Bessel sequences are involved. Also, we show that some corresponding results in [29,31] can be considered a special case of our results.…”

“…Theorems 4.1 and 4.2 in[31] can be obtained if we take λ = 1, respectively, in Corollaries 1 and 2. Let Λ = {Λ j } j∈J be a g-frame for H with respect to {K j } j∈J .…”

“…Those inequalities have already been extended to several generalized versions of frames in Hilbert spaces [26][27][28]. Moreover, the authors in [29][30][31] showed that g-frames in Hilbert C * -modules have their inequalities based on the work in [24,25]; it is worth noting that the inequalities given in [30] are associated with a scalar in [0, 1] or [ 1 2 , 1]. In this paper, we establish several new inequalities for g-frames in Hilbert C * -modules, where a scalar λ in R, the real number set, and an adjointable operator with respect to two g-Bessel sequences are involved.…”

G-frames for operators in Hilbert $C^{\ast}$-modules

“…The following lemmas will be used to prove our main results. Lemma 2.7 (see [15]) Let Λ j ∈ End * A (H , V j ) for all j ∈ J , then {Λ j } j∈J is a g-frame for H with respect to {V j } j∈J if and only if there exist two constants C 2 , D 2 > 0 such that…”

New double inequalities for g-frames in Hilbert $$C^{*}$$ C ∗ -modules

“…Moreover, Poria Poria (2016) generalized those inequalities to the case of Hilbert–Schmidt frames, which possess a more general form. On the other hand, the authors of Xiao and Zeng (2010) have already extended the inequalities for Parseval frames and general frames to g-frames in Hilbert -modules:…”