ABSTRACT:We introduce the more general frame sequences and dual frames related to a linear bounded operator K in Hilbert spaces which we call K-frame sequences and dual K-frames, respectively. We give several equivalent characterizations for K-frame sequences. We also investigate the relationships among K-frame sequences, K-frames, and frame sequences, and give a new perturbation result for K-frames by using the associated dual K-frames. It turns out that in many ways K-frame sequences and dual K-frames behave completely differently from frame sequences and dual frames, respectively.
We present a generalization of g-frames related to an adjointable operator K on a Hilbert C * -module, which we call K -g-frames. We obtain several characterizations of K -g-frames and we also give conditions under which the removal of an element from a K -g-frame leaves again a K -g-frame. In addition, we define a concept of dual, and using it we study the relation between a K -g-frame and a g-Bessel sequence with respect to different sequences of Hilbert C * -modules.
In this paper, we present several new inequalities for weaving frames in Hilbert spaces from the point of view of operator theory, which are related to a linear bounded operator induced by three Bessel sequences and a scalar in the set of real numbers. It is indicated that our results are more general and cover the corresponding results recently obtained by Li and Leng. We also give a triangle inequality for weaving frames in Hilbert spaces, which is structurally different from previous ones.
We give new characterizations of Riesz-type frames, on equivalent conditions for a continuous frame to be a Riesz-type frame and on equivalency relations between Riesz-type frames and continuous frames. We characterize also the Riesz-type frames by using a bounded linear operatorL. Finally, we study the stability of alternate duals of continuous frames and we prove that if two continuous frames are close to each other, then we can find alternate duals of them which are close to each other.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.