On Some Inequalities-Equalities Concerning the continuous generalized Fusion Frame in Hilbert spaces

TL;DR

This paper extends the theory of continuous generalized fusion frames in Hilbert spaces by deriving new identities, inequalities, and properties for various types of these frames, including Parseval and tight frames, and explores dual frames and frame operators.

Contribution

It introduces new inequalities and identities for continuous generalized fusion frames, including dual frames and frame operators, advancing the mathematical understanding of these structures.

Findings

Derived new identities and inequalities for continuous generalized fusion frames.

Established properties of Parseval and λ-tight continuous generalized fusion frames.

Analyzed the frame operator and resolution of identity for pairs of Bessel continuous generalized fusion mappings.

Abstract

Continuous generalized fusion frame theory was recently introduced by Rahimi and al. Several equalities and inequalities have been obtained for frame, fusion generalized fusion frame, among others. In the present paper, we continue and extend these results to obtain some important identities and inequalities in the case of continuous generalized fusion frame, Parceval continuous generalized fusion frame, $ \lambda$-tight continuous generalized fusion frame. Moreover, we obtain some new inequalities for the alternate dual continuous generalized fusion frame. Finally, we obtain frame operator of a pair of Bessel continuous generalized fusion mapping and we derive some results about resolution of identity.